Dynamical and stochastic aspects of self-organized ... · The dynamical systems approach allows us...

Dynamical and stochastic aspects of self-organized critical

behavior in sandpiles

Martin Rypdal

A dissertation for the degree of Philosophiae Doctor

UNIVERSITY OF TROMSØ Faculty of Science

Department of Mathematics and Statistics

October 2008

Acknowledgements:

I thank my collaborators Boris Kruglikov and Kristo!er Rypdal who have con-tributed essentially to the work presented in this thesis. Boris has been my supervi-sor for many years, and I have truly appreciated his guidance and excellent teaching.I would also like to thank the members of the Sophus Lie Seminar for providingan inspiring scientific environment in our department. In particular Einar Mjølhus,Per Jakobsen, Valentin Lychagin, Tor Fla and Marius Overholt. Boris Kozelov,Odd Erik Garcia and Tanja Zivkovic of the Complex Systems Group in Tromsøhave also contributed to my work through collaboration and useful discussions. Ifinally want to mention Yakov Pesin who visited Tromsø in 2005. His lectures ondimension theory and multi-fractal formalism for dynamical systems were inspiringand useful.

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List of papers that constitute the thesis:

1. B. Kruglikov, M. Rypdal, Dynamics and entropy in the Zhang model of SOC ,Journal of Statistical Physics 122, 975 (2006)

2. B. Kruglikov, M. Rypdal, Entropy via multiplicity, Discrete and ContinuousDynamical Systems 16, 395 (2006)

3. B. Kruglikov, M. Rypdal, A piece-wise a!ne contracting map with positiveentropy , Discrete and Continuous Dynamical Systems 16, 393 (2006)

4. M. Rypdal, K. Rypdal, A stochastic model of the toppling activity in self-organized critical systems, Preprint: arXiv:0710.4010v2 (2008), (Accepted bythe New Journal of Physics).

5. M. Rypdal, K. Rypdal, Modeling of temporal fluctuations in sandpiles, Preprint:arXiv:08073416v1(2008), (Accepted by Physical Review E).

Other papers which are a part of this work, but not directly incorporated in themanuscript are:

6. K. Rypdal, B. Kozelov, S. Ratynskaia, M. Rypdal, B. Klumov, C. Knapek,Scale-free vortex cascade emerging from random forcing in a strongly coupledsystem, New Journal of Physics 10 (2008).

7. K. Rypdal, M. Rypdal, T. Zivkovic, B. Kozelov, Complexity in astro andgeospace systems: the turbulence versus SOC controversy, AIP ConferenceProceedings 932, 203 (2007).

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Contents

1 Introduction 11

2 Basic dynamical properties of the BTW and Zhang models 192.1 SOC and dynamical systems . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 The BTW model as a dynamical system . . . . . . . . . . . . 202.1.2 Symbolic coding of the BTW model . . . . . . . . . . . . . . 222.1.3 The natural invariant probability measure . . . . . . . . . . . 242.1.4 Characterization of chaos . . . . . . . . . . . . . . . . . . . . 272.1.5 Chaos in the BTW model . . . . . . . . . . . . . . . . . . . . 302.1.6 Decay of correlations . . . . . . . . . . . . . . . . . . . . . . . 322.1.7 Return maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1.8 Dhar’s theory for return maps in the BTW model . . . . . . 34

2.2 The Zhang model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.1 Symbolic coding of the Zhang model . . . . . . . . . . . . . . 402.2.2 Entropy in the Zhang model . . . . . . . . . . . . . . . . . . 402.2.3 Removability of singularities . . . . . . . . . . . . . . . . . . 422.2.4 Characterization of the attractor . . . . . . . . . . . . . . . . 43

3 Dynamics and entropy in the Zhang model of SOC 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Basic properties of the Zhang model . . . . . . . . . . . . . . . . . . 50

3.2.1 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.2 Random excitations . . . . . . . . . . . . . . . . . . . . . . . 523.2.3 Return maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Removability of singularities and coding . . . . . . . . . . . . . . . . 593.3.1 Construction of attractors . . . . . . . . . . . . . . . . . . . . 593.3.2 Symbolic Coding . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Metric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.1 Existence and characterization of SRB-measures . . . . . . . 623.4.2 Measures of maximal entropy . . . . . . . . . . . . . . . . . . 65

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3.4.3 Hyperbolic structure . . . . . . . . . . . . . . . . . . . . . . . 663.4.4 Entropy of physical vs. mathematical models . . . . . . . . . 67

3.5 Topological entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.1 Growth of the number of continuity domains . . . . . . . . . 683.5.2 Evaluation of topological entropy . . . . . . . . . . . . . . . . 71

3.6 Geometry of the attractor . . . . . . . . . . . . . . . . . . . . . . . . 763.6.1 Fractal structure . . . . . . . . . . . . . . . . . . . . . . . . . 773.6.2 Dimensional study of the attractor . . . . . . . . . . . . . . . 77

3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.8 Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.8.1 Statistics of observables . . . . . . . . . . . . . . . . . . . . . 913.8.2 Asymptotic of the statistical data . . . . . . . . . . . . . . . . 953.8.3 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . 993.8.4 Lyapunov spectrum . . . . . . . . . . . . . . . . . . . . . . . 101

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A Topological entropy of piecewise a"ne maps . . . . . . . . . . . . . . 105

A.1 The Buzzi theorem and its generalization . . . . . . . . . . . 105A.2 Entropy of conformal piecewise a"ne skew-products . . . . . 107A.3 Estimates on entropy by angular expansion rates . . . . . . . 109

B Generalization of Moran’s formula . . . . . . . . . . . . . . . . . . . 111

4 Entropy via multiplicity 1151 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 Definitions and main results . . . . . . . . . . . . . . . . . . . . . . . 116

2.1 Piecewise a"ne maps and topological entropy . . . . . . . . . 1162.2 Singularity entropy . . . . . . . . . . . . . . . . . . . . . . . . 1172.3 Expansion rates and multiplicity entropy . . . . . . . . . . . 1182.4 The spherization and angular expansions . . . . . . . . . . . 1192.5 Piecewise a"ne skew products . . . . . . . . . . . . . . . . . 121

3 Proof of the theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.1 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . 1223.2 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . 1243.3 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . 1253.4 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . 128

4 Proof of the corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1 Estimate for the angles expansion rates . . . . . . . . . . . . 1294.2 Proof of Corollary 4.6 . . . . . . . . . . . . . . . . . . . . . . 130

5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5 A piecewise a!ne contracting map with positive entropy 135

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6 Introduction to the stochastic model 1411 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422 From a dynamical to a stochastic point of view . . . . . . . . . . . . 1433 The branching model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.1 Self-similar processes . . . . . . . . . . . . . . . . . . . . . . . 1464.2 Construction of Brownian motion . . . . . . . . . . . . . . . . 1474.3 The di!usion equation . . . . . . . . . . . . . . . . . . . . . . 147

5 Stochastic di!erential equations . . . . . . . . . . . . . . . . . . . . . 1486 SDE formulation of the branching process . . . . . . . . . . . . . . . 1497 Extension to the BTW and Zhang models . . . . . . . . . . . . . . . 151

7.1 Fractional noise terms . . . . . . . . . . . . . . . . . . . . . . 152A Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.1 Brownian motion and white noise . . . . . . . . . . . . . . . . 156A.2 Fractional Brownian noise . . . . . . . . . . . . . . . . . . . . 158

7 A stochastic theory for the toppling activity in SOC 1611 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622 The stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633 The normalized toppling process is a fractional Brownian walk . . . 1674 Analysis of avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . 1675 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8 Modeling temporal fluctuations in sandpiles 1751 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1762 The stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793 Calculation of avalanche exponents . . . . . . . . . . . . . . . . . . . 1824 Finite-size scaling for type-II avalanches . . . . . . . . . . . . . . . . 1895 The PDFs of the toppling activity . . . . . . . . . . . . . . . . . . . 1906 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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Chapter 1

Introduction

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In 1987 Per Bak, Chao Tang and Kurt Wiesenfeld (BTW) constructed a computermodel of the cellular-automaton type, which has become widely known as the BTWsandpile model. Its appeal is based on the assumption that a large class of naturalsystems exhibit some common dynamical characteristics, which are shared by thedynamics and statistics of avalanches arising in a pile of sand as sand grains areslowly and randomly fed from the top of the pile. One property that these systemshave in common is that the attractors of the dynamics give rise to scale-invariantproperties of the statistical distribution of certain global variables. This propertyimplies that one can draw some parallels to the behavior of equilibrium systems atcritical points.

It is believed that the attractors of sandpile models generally have full basinsof attraction, and that statistical properties of the asymptotic dynamics to someextent are independent of the detailed algorithm describing the interaction betweenneighboring sand grains. Thus the “critical state” is attained without fine tuning ofparameters that we know is required to bring equilibrium systems to criticality. Thedynamics observed in the BTW model was named Self-Organized Criticality (SOC),and in the enthusiasm that followed the first BTW articles, SOC was proposed asthe governing dynamical mechanism in a large class of complex natural systemswhere scaling laws have been found empirically.

The view held by many scientists working in the field of natural complexityis that SOC represents a class of dynamical phenomena in the same way as forinstance turbulence or chaos. This means that SOC should be loosely characterizedby a set of mathematical properties, which enables us to recognize and identify theSOC properties when examining observation data from natural systems. However,contrary to the case of turbulence or chaos, the defining properties of SOC systemsare not yet understood in a satisfactory way. In fact, when a natural system isproposed as an example of SOC, there is often a significant lack of mathematicalevidence, and the arguments presented are often of a rather philosophical character.

It can also be problematic to distinguish SOC from other classes of dynamicalbehavior. In recently published articles [13, 14] evidence has been presented indicat-ing that a large class of di!erent complex systems exhibits global observables whichare distributed according to the so called Bramwell-Holdsworth-Pinton (BHP) dis-tribution. Among the quantities claimed to follow the BHP distributions are thetoppling activity in the BTW sandpile and fluctuations of the global dissipationrate for bounded turbulent flows, and this has been taken as evidence that SOCand turbulence share some fundamental dynamical mechanisms. It turns out thatthe toppling activity in the BTW sandpile is not BHP distributed, but nevertheless,the work of Bramwell et al. is a good example of how little we actually understand ofSOC dynamics. If we had better fundamental knowledge of the governing processesin sandpile models, then we would not base a comparison of SOC and turbulenceon the single-time statistics of one observable quantity.

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On the other hand, it is di"cult to compare two complex systems which are de-scribed within di!erent mathematical frameworks, and consequently a large numberof di!erent approaches to a description of SOC have been attempted. Scientists whowant to compare SOC with turbulence have a tendency to place sandpile modelsinto a framework where quantities have direct analogies in turbulence theory, forinstance by defining Reynolds numbers for sandpiles. Inversely, there exist so-called shell models of turbulence where the energy cascades can be interpreted asavalanches.

The scaling properties of the self-organized critical states have also tempted sci-entist to compare SOC phenomena with phase transitions in equilibrium systems.Although this approach may have provided some useful analogies, the mathemat-ical formalism of statistical physics is not very useful for attacking sandpiles. Forinstance, Ruelle’s thermodynamic formalism for lattice systems is only e!ectivewhen the equilibrium state is characterized by specified local interactions and onlysimple restrictions on the set of legal configuration. In sandpiles the set of re-current configurations has a very complex structure which cannot be captured bylocal interactions, and from a mathematical point of view, the mechanism lead-ing to self-organized criticality is fundamentally di!erent from phase transitions.Other approaches inspired by statistical mechanics, such as the dynamical drivenrenormalization group calculations, have been more successful, but providing littlegeneral dynamical insight.

The main problem is that we obtain poor descriptions of SOC when enforcingframeworks which are specifically designed for completely di!erent phenomena. Itis actually quite evident that successful descriptions of sandpile dynamics must relymore on the specific properties of these models. At the same time it is importantthat the approaches do not become too specific. An example of this is Dhar’s#-formalism [23], which depends crucially on the so-called abelian property, a non-generic property only found in particular sandpile models.

Two di!erent frameworks

In this thesis we propose to investigate SOC within two di!erent frameworks: dy-namical systems and stochastic di"erential equations. We claim that both of theseapproaches are su"ciently general to include a large class of sandpile systems andto allow general comparison with other complex phenomena, yet su"ciently specificto capture non-trivial elements of SOC.

The dynamical systems approach allows us to relate SOC to the more establishednotion of chaos. In particular, well defined dynamical invariants can be computed,and these can be compared with other dynamical systems. An example is thedynamical entropy of sandpile models. The BTW and Zhang models [3, 57] both

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have dynamical entropy equal to

log N

1 + !!"/!"" ,

where N is the number of sites on the model’s lattice, !!" is the average waitingtime between avalanches and !"" is the average avalanche duration. This entropyis always positive, which implies chaos. But for slowly driven sandpiles (no feedingof sand grains during avalanches) the avalanche duration is much greater than thewaiting time, and the entropy goes to zero as the number of sites on the latticegrows. For strongly driven sandpiles the entropy remains positive, illustrating afundamental di!erence between strongly and weakly driven sandpiles.

The other approach presented in this thesis involves modeling the so-called top-pling activity of sandpiles by a stochastic di!erential equation on the form

dX(t) = f(X) dt + #!

X(t) dBH(t) .

Here BH(t) represents a stochastic process known as a fractional Brownian motion.This model is used to compute several critical exponents in the BTW and Zhangmodels. Moreover, we show explicitly how the stochastic di!erential equationscan be used to illustrate essential di!erences between the toppling activity in aslowly driven sandpile, the toppling activity in a strongly driven sandpile, and thefluctuations of kinetic energy in a turbulent fluid.

The structure of the thesis

In the first part of the thesis we formulate the BTW and Zhang models as dy-namical systems following a construction of Blanchard, Cessac and Kruger [6, 7, 8].This involves modeling the random driving mechanism using a Bernoulli shift anddescribing the complete dynamics on an extended phase space $+

N #M , where $+N

is the set of all infinite words over an alphabet consisting of the sites in the lattice.The emphasis of our investigation will be calculation of dynamical invariants thatare useful for the characterization of chaos, such as dynamical entropy, Lyapunovexponents, attractor dimension and symbolic representations. Three research ar-ticles constitute chapters 3, 4, and 5. Chapter 3 focuses on the description of theZhang model [57] as a dynamical system. Chapters 4 and 5 deal with specific math-ematical results and ideas that where discovered through the work on the Zhangmodel. Chapter 2 is an introduction which contains definitions, elementary con-cepts and concrete examples essential for understanding the technical articles thatfollow.

The second part of the thesis presents the stochastic theory for the topplingactivity. It consists of two research articles (chapters 7 and 8) and an introduction(chapter 6). The introduction includes a derivation of the stochastic model for the

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so-called random neighbor model [20], a discussion of its generalization to the BTWand Zhang models and a brief survey of the required background on self-similarstochastic processes and stochastic di!erential equations.

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Part 1:

Sandpile models anddynamical systems

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Chapter 2

Basic dynamical properties ofthe BTW and Zhang models

Abstract: This chapter is an introduction to the first part of the thesis. Throughconcrete examples and calculations we show how the BTW and Zhang models can beformulated as dynamical systems. For the BTW model we preform symbolic codingin some simple situations. Characterization of sandpile models through dynamicalinvariants is discussed.

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2.1 SOC and dynamical systems

We begin by formulating the BTW sandpile model as a dynamical system. Laterin this chapter the formalism will be extended to the Zhang model.

2.1.1 The BTW model as a dynamical system

The BTW model is defined on a d-dimensional hyper-cubic lattice % $ Zd withN = Ld sites (or nodes). Every site i % % is associated with a non-negative integerzi % {0, 1, 2, . . . }, which we can think of as the number of grains in position i of thelattice. The collection z = {zi |i % %} is called the configuration of the sandpile,and the set of configurations {0, 1, 2, . . . }N is called the configuration space. Aconfiguration is called stable if zi < 2d for all i % % and non-stable otherwise. Theset of stable configurations is denoted M .

The time evolution of configurations is defined in two parts: If site i is overcriticalit will topple. This means that if zi & 2d, then site i looses one grain to each of itsnearest neighbors on the lattice. Sites on the boundary of the lattice have less than2d neighbors, so the total number of grains on the lattice is not conserved if thereare overcritical boundary sites.

Remark 2.1 The neighbors of i are defined as those sites j for which d!(i, j) = 1,where d! is the “Manhattan metric” on %.

All the overcritical sites in a configuration will topple simultaneously in a timestep, and the configuration evolves according to the rule z '( f(z), where f :{0, 1, 2, . . . }N ( {0, 1, 2, . . . }N is a map defined by

(fz)i = zi ) 2d &(zi ) 2d) +"

d!(i,j)=1

&(zj ) 2d) . (2.1)

Here & is the unit step function defined by

&(x) =

#0 x < 01 x & 1

.

The notations and definitions become clearer by considering a concrete example:

Example: The simplest non-trivial sandpile we can consider is d = 1 and N = 2.In this case each configuration is an ordered pair z = (z1, z2), where z1 and z2 arenon-negative integers. There are four stable configurations: (0, 0), (1, 0), (0, 1) and(1, 1). It is easy to see from equation 2.1 that f(z) = z if z is stable. Through the

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dynamics of the BTW model (which we have not defined completely yet), the onlynon-stable configurations that can be reached from a stable initial configuration are(2, 0), (0, 2), (2, 1) and (1, 2). Under f , these configurations are mapped as follows:

(2, 1) '( (0, 2) , (1, 2) '( (2, 0) , (0, 2) '( (1, 0) and (2, 0) '( (0, 1) .

The BTW model (as given by equation 2.1) is defined so that grains “fall o!”the lattice when there are overcritical sites on the boundary, and because of thiswe always reach a stable configuration after su"ciently many iterations of f . Toinitiate new avalanches we need to feed grains to the lattice, and this is usually doneby adding (dropping) a grain to a randomly chosen site whenever the configurationis stable. Since this part of the dynamics involves randomness, we must eventuallyintroduce an underlying probability space. But first we give a purely topologicaldescription.

The “trick” is to prefix all dropping positions in an infinite squence

t = (t1t2t3 · · · ), ti % % .

The first time the configuration is stable a grain is dropped onto site t1, the nexttime the configuration becomes stable a grain is dropped onto site t2, and so on.Equivalently, we can drop a grain in the position corresponding to the first entry oft whenever the configuration is stable, and then let t '( #(t), where

#(t1t2t3 · · · ) = (t2t3t4 · · · ) .

The dynamics of the BTW sandpile can then be described by a single map:

f(t, z) =

#(#(t), z + et1) if z is stable(t, f(z)) if z is non-stable

, (2.2)

where ei denotes the configuration with (ei)j = $ij .Since we are interested in the asymptotic properties of the dynamics, we can

restrict our attention to the set of configurations which can be reached more thanonce in an orbit.1 These configurations are called recurrent, and we denote set ofrecurrent configurations by R. Our dynamical system becomes

f : $+N #R( $+

N #R ,

where f is given by equation 2.2. Here $+N denotes the set of all infinite words over

the alphabet %.1This is a finite set.

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2.1.2 Symbolic coding of the BTW model

For a positive integer n we let $+n denote the set of all infinite sequences in the

symbols {1, 2, . . . , n}, and we let # : $+n ( $+

n denote the shift map defined in theprevious section. A topological Markov chain is the restriction of the map # to aparticular shift-invariant subset of $+

n . This subset is defined by a directed graphwhere the vertices are {1, 2, . . . , n}. It consists of those sequences (s1s2s3 · · · ) % $+

Nwith the following property:

*k & 1 : the graph has an arrow from sk to sk+1 .

The directed graph can be represented by a n # n matrix A = ||aij ||, whereaij = 1 if the graph has an arrow from i to j, and aij = 0 otherwise. The subset of$+

n defined by the graph (or matrix) is denoted by $+A, and the restriction of # to

$+A is denoted #+

A .

Remark 2.2 The reason why topological Markov are so interesting, is that theyserve as good models for a large class of chaotic dynamical systems. In fact, manyproperties of such systems can be proved by constructing a continuous change of vari-ables (toplological conjugacy) which transform the system to a topological Markovchain.

We now use an example to illustrate how the BTW model can be coded to atopological Markov chain:

Example: We continue to look at the case d = 1 and N = 2. There are sevenrecurrent configurations: (1, 0) , (0, 1) (0, 2) , (2, 0) , (2, 1) , (1, 2) and (1, 1). We de-note them by z1 . . . , z7 in the order that they are written. Our phase space is then$+

2 #R, where R = {z1, . . . , z7}.We partition the phase space into fourteen subsets

[i]# {zj}, i % {1, 2} , j % {1, . . . , 7} ,

where [i] := {t % $+2 | t1 = i} denotes the cylinder of the symbol i. Let

Z1 = [1]# {z1} , Z2 = [2]# {z1} , Z3 = [1]# {z2} , and so on.

The next step is to determine the images of the partition elements Zi. Consider theset Z1 = [1] # {z1}. We know that f(z1) = z4, and secondly (since z1 is a stableconfiguration) the shift map # is applied. Since #+

2 ([1]) = $+2 , we have

f(Z1) = $+2 # {z4} =

$[1]# {z4}

%+

$[2]# {z4}

%= Z7 + Z8 .

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Z2

Z1 Z4

Z6

Z5Z13Z14

Z7

Z8

Z12

Z11 Z10

Z9

Z3

Figure 2.1: The Markov graph defined by the dynamics of the BTW model for d = 1and N = 2.

In this case the partition element Z1 is mapped to two di!erent partition elements,Z7 and Z8. This is because z1 is a stable configuration, so that a dropping is pre-formed and # is applied. If we consider a partition element where the correspondingconfiguration is non-stable, then there is no shift, and the partition element is onlymapped to one other partition element. For instance,

f(Z5) = f([1]# {z3}) = [1]# {f(z3)} = [1]# {(1, 0)} = Z1 .

We construct a directed graph by drawing an arrow from the symbol i to the symbolj if f(Zi),Zj -= .. This means that there will be two arrows from partition elementscorresponding to stable sites and one arrow from partition elements corresponding tonon-stable sites. The complete graph is shown in figure 2.1 . The matrix A = ||aij ||is given by the general formula

aij =

#1 if f(Zi) , Zj -= .0 otherwise

.

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If we carry out all the computations we get:

A =

&

''''''''''''''''''''''(

0 0 0 0 0 0 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 1 1 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 1 1 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0

)

**********************+

. (2.3)

Remark 2.3 Topology on $+N and $+

A can be defined by metrics on the form

d(t, s) = a!k , k = min{l | tl -= sl} ,

where a > 0 is some positive real number. The number a is not essential for thetopology defined by these metrics, but other properties, like the Hausdor" dimensionsof subsets depend essentially on a.

With topologies in place, the equivalence between the maps f and #+A can be

described as a topological conjugacy. This means that there exists a homeomorphismg : $+

N #R ( $+A such that g / f = #+

A / g. Such a coding map always exists forthe BTW model, and in general it is given by the rule g(t, z) = s, where s =(s1s2s3 · · · ) % $+

A is the sequence defined by fn(t, z) % Zsn *n & 1.

2.1.3 The natural invariant probability measure

Usually the BTW model is defined such that whenever the configuration is stable, asite i % % is chosen randomly with respect to the uniform probability distribution on%, i.e. site i % % has probability pi = N!1 of being chosen. The choice of droppingposition at one time step is independent of the choice of dropping position at pre-vious times, so the uniform probability distribution p1 = · · · = pN = N!1 extendsto a uniform Bernoulli measure on $+

N . The probability that a dropping sequencebegins with i1i2 · · · ik is equal to pi1pi2 · · · pik = N!k. Written as a probabilitymeasure, this becomes

µBer([i1 · · · ik]) = pi1 · · · pik = N!k ,

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where [i1 · · · ik] = {t | t1 = i1, . . . , tk = ik} is the cylinder of the finite sequencei1, . . . , ik.

The uniform Bernoulli measure µBer induces a Markov measure µ on the topo-logical Markov chain $+

A. This measure can be described by a transition probabilitymatrix ' = ||%ij ||. This matrix has the same dimensions as the matrix A, and itsentries represent the probabilities of transitions from Zi to Zj .

If aij = 0, then %ij = 0, since the probability of an impossible transition mustbe zero. Also, if row i of A only contains one non-zero element, then %ij = 1.If row i of A has more than one non-zero element, then the partition element Zi

corresponds to a stable configuration. This implies that the shift map is applied,so if Zi = [j]# {z}, then

f(Zi) = f([j]# {z}) = $+N # {f(z)} =

,

j!"!

[j#]# {f(z)} .

Since f(Zi) intersects N di!erent partition elements, row i of A has N di!erent 1-s.Each of these correspond to a di!erent position of dropping, and since each positionhas the same probability, the corresponding entries in ' are equal to N!1. Thisgives the following formula:

%ij =

-./

.0

0 if aij = 01 if aij = 1 and

1j aij = 1

N!1 if aij = 1 and1

j aij = N

. (2.4)

Let Q = N # |R|, so that A and ' are Q#Q matrices. If we choose an initialpartition element Zi with respect to a probability distribution q = (q1, . . . , qQ)T ,then the probability of f(t, z) % Zj is

Prob[f(t, z) % Zj ] =Q"

i=1

Prob[(t, z) % Zi] · Prob[f(t, z) % Zj |(t, z) % Zi]

=Q"

i=1

qi %ij = ('T q)j = (Lq)j ,

where L = 'T . By repeating the argument we see that Prob[fn(t, z) % Zj ] =(Lnq)j , and thus a stable probability vector P is reached as the limit

P = limn$%

Lnq .

If this limit exists, then P must be a normalized egienvector of L.If the matrix A is mixing, which simply means that there exists an integer k

such that all the elements of Ak are positive, then the Perron-Frobenius theorem

25

ensures that L has a unique positive and normalized eigenvector P and that Lnq (P independently of q. In other words, the asymptotic probability distribution ofpartition elements is independent of the initial distribution.

Through the probability vector P we can now define the Markov measure µ:

µ([s1 . . . sn]) = Ps1 %s1s2 · · ·%sn"1sn .

The corresponding probability meausre g!1& µ on $+

N #R is such that its projectionto $+

N coincides with the uniform Bernoulli measure µBer.

Example: We continue illustrate the theory by looking at the simple exampled = 1, N = 2. In the last section we found the matrix A, and from equation 2.4 wesee that

' =12

&

''''''''''''''''''''''(

0 0 0 0 0 0 1 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 1 1 0 0 0 0 0 0 0 02 0 0 0 0 0 0 0 0 0 0 0 0 00 2 0 0 0 0 0 0 0 0 0 0 0 00 0 2 0 0 0 0 0 0 0 0 0 0 00 0 0 2 0 0 0 0 0 0 0 0 0 00 0 0 0 2 0 0 0 0 0 0 0 0 00 0 0 0 0 2 0 0 0 0 0 0 0 00 0 0 0 0 0 2 0 0 0 0 0 0 00 0 0 0 0 0 0 2 0 0 0 0 0 00 0 0 0 0 0 0 0 1 1 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0

)

**********************+

.

In this example the matrix A is not mixing. However, the matrix L = 'T hasa unique positive and normalized eigenvector P corresponding to the eigenvalue& = 1. A simple computation shows that

P =124

(2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2)T ,

which means that the partition elements Z9, Z10, Z11 and Z12 have probability 1/24,and the other partition elements have probability 1/12. In particular, the config-urations (2, 1) and (1, 2) have probability 1/12, and the five other configurationshave probability 1/6.

Notice that all stable configurations have the same probability, in this case 1/6.This is always true for the BTW model, a mathematical result first proved by Dhar[23]. We will return to this result in section 2.1.8.

26

2.1.4 Characterization of chaos

As we have seen, the symbolic coding g : $+N #R( $+

A gives us a complete topo-logical and metric (statistical) description of the dynamics of the BTW model. Inprinciple we can compute all desired information about the dynamics by determin-ing the matrix A and its properties. However this is not possible in practice sincethe size of A will grow very rapidly as we increase N . On the other hand, thereis some general information about the dynamics which can be deduced withoutknowing the detailed form of the matrix A.

As we have already mentioned, topological Markov chains serve as prototypesof chaotic dynamics, and since the BTW model is conjugated to #+

A it is natural tobelieve that it exhibits chaotic behavior. On the other hand, chaos is characterizedby a rapid loss of memory along orbits, contrary to the long range correlationswe expect to see along avalanches in sandpiles. In order to resolve this apparentparadox we will quantify the degree of chaos in the BTW model. The results showthat sandpile dynamics can be described as weakly chaotic, with predictability ontime scales comparable to the durations of system size avalanches.

Before we start our analysis of the chaotic properties of the BTW model we willbriefly discuss the dynamical invariants usually used to describe chaotic behavior.

Chaos is usually characterized by sensitivity to initial conditions. This meansthat in any small neighborhood in phase space we can find two initial conditionswhich orbits diverge essentially from each other after a finite number of iterations.This property is rather imprecise, so several mathematical invariants have beendeveloped to quantify this e!ect.

An example is the Lyapunov exponents. Let M be a manifold (not to be confusedwith the set of stable configurations in the BTW model) and G : M ( M a smoothmap. For any x % M and any tangent vector v % TxM one defines

'(x, v) = limn$%

1n

log ||dxfn(v)|| .

It is very easy to show that for each x % M there exists only dim(M) di!erentvalues of '. These numbers are called the Lyapunov exponents of the map G alongthe orbit starting at x. They are denoted 'k(x). Often there exists a naturalinvariant probability measure on M for which the numbers 'k(x) are constantalmost everywhere. In these situations we have a set of x-independent exponents.A positive Lyapunov exponent indicates that the map G (on average) has expandingproperties in at least one (varying) direction. This expansion causes sensitivity toinitial conditions and hence chaos.

Another characterization of chaos is given by dynamical entropies. There aretwo kinds of entropy, metric and topological. Just as the Lyapunov exponents,the metric entropy of a map G depends on an invariant probability measure µ. If

27

( = {Zi} is a partition of the phase space, then the quantity

Hµ(() = )"

i

µ(Zi) log µ(Zi)

is a measure of the (logarithmic) amount of uncertainty we are faced with in trying toguess which partition element a point (chosen randomly with respect to µ) belongsto. We then use the dynamical system to refine the partition by constructing

(n = ( 0G!1(() 0 · · · 0G!n+1(() ,

where G!1(() denotes the partition consisting of the sets G!1(Zi) and 0 denotesthe join of partition, i.e. {Zi}0{Xi} = {Zi ,Xj}. Knowing the position of a pointwith respect to this partition is the same as knowing the (-address of an all thepoints in an orbit segment of length n. Hence Hµ((n) must be greater than or equalto Hµ((). In particular, if the dynamics is chaotic and unpredictable, then Hµ((n)should grow with n. We define

hµ(G; () = limn$%

1n

Hµ((n) .

This quantity measures the exponential asymptotic growth rate of expHµ((n) asn ( 1. The metric entropy of G with respect to the measure µ is defined ashµ(G) = sup! hµ(G; (), where the supremum is taken over all “good” partitionsof the phase space. This supremum is attained for any partition ( for which thediameter of (n tends to zero as n (1. A dynamical system G : M ( M is chaoticwith respect to the invariant measure µ if hµ(G) > 0.

Remark 2.4 For smooth maps there is a close relationship between the metric en-tropy and the set of positive Lyapunov exponents. In fact, if G is a C1+" di"eomor-phism and µ a smooth invariant probability measure, then positive metric entropyis equivalent to the existence of a positive Lyapunov exponent. This follows fromthe Pesin entropy formula

hµ(G) =2 "

#i(x)>0

'i(x) dµ(x) ,

where the sums are taken over the positive Lyapunov exponents 'i(x). If a “good”invariant measure µ is not smooth, for instance if it is supported on a fractal attrac-tor, then it is typically absolutely continuous on the so called unstable manifolds.These measures are called SRB measures, and for them, the Ledrappier-Young for-mula states

hµ(G) =2 "

#i(x)>0

'i(x)(diµ(x)) di!1

µ (x)) dµ(x) ,

28

where the numbers diµ(x) are pointwise fractal dimensions in the point x of the the

measure µ on the i-th leaf of the unstable manifold. For non-invertible smooth mapsthe equality in Pesin’s formula does not always hold, but we have the Margulis-Ruelleinequality

hµ(G) 22 "

#i(x)>0

'i(x) dµ(x) .

Although metric entropy is one of the best existing tools for characterization ofchaos, it is important to keep in mind that it depends essentially on the choice ofinvariant measure. Any chaotic system has an infinite number of periodic orbits, andif we choose a measure supported on one of these orbits, then the metric entropyis zero. We can only assign physical meaning to the notion of metric entropy ifthe corresponding invariant measure reflects the “right” statistical properties ofthe dynamics. Such measures are characterized by the property that time-averagesequal ensemble averages, meaning that we for any continuous function ) have

1n

n!1"

k=0

)(Gkx) (2

) dµ , (2.5)

for µLeb-almost all initial conditions x. These ”good” measures are called naturalinvariant measures (or SRB measures if the dynamics is defined by a hyperbolicdi!eomorphism). It is important not to confuse the natural invariant measureswith ergodic measures. The famous Birkho! ergodic theorem only asserts thatequation 2.5 holds for µ-almost all initial conditions x.

Whenever a dynamical system admits a natural invariant measure µ, then thismeasure captures all the relevant statistical properties of the dynamics, and thenhµ(G) > 0 implies chaotic dynamics.

Another (and closely related) tool for characterization of chaos is topologicalentropy. Contrary to metric entropy, topological entropy does not depend on aninvariant measure, and it is defined as long as the phase space X is a metrizabletopological space. Given a metric d (compatible with the topology) we define iter-ated balls

B",n(x) = {y % X | d(fkx, fky) < * for k = 0, . . . , n) 1} .

The set B",n(x) consists of the points in phase space whose orbit segments of lengthn lie within an *-tube of the orbit segment x, f(x), . . . , fn!1(x) in Xn. Let N(*, n)be the number of such balls needed to cover the space space X. In other words,N(*, n) is the number of *-distinguishable orbit segments of length n. In chaoticsystems, this number grows exponentially with n, and topological entropy is definedas the exponential growth rate:

htop(G) = lim"$0

limn$%

1n

log N(*, n) .

29

The topological entropy depends on the topology only, i.e. the choice of (compatible)metric d is irrelevant. For continuous maps the topological entropy is related to themetric entropy via the variational principle:

htop(G) = supµ

hµ(G) ,

where the supremum is taken over all invariant Borel probability measures µ. Thevariational principle tells us that positive topological entropy implies that the metricentropy is positive for some invariant measure. For a large class of dynamicalsystems the supremum is attained, in which case we have a measure of maximalentropy. Typically the measure of maximal entropy is unique, and it often coincideswith the natural invariant measure.

Finally we mention that there exits other topological characterizations of chaos.First, the set of periodic points is typically dense in the attractor. Also, the numberof periodic points of period n grow exponentially with n, usually with a rate equalto the topological entropy. Other typical properties include topological transitivityin the attractor (there exists a dense orbit on the attractor) and topological mixing,which is a stronger property than transitivity.

2.1.5 Chaos in the BTW model

We know that the BTW model is topologically conjugated to a topological Markovchain #+

A : $+A ( $+

A. Topological entropy, growth in the number of periodic points,transitivity and topological mixing are all invariant under this conjugacy, and henceit su"ces to study the map #+

A in order to calculate these invariants. Fortunatelythis is an easy task given the transition matrix A. In particular, the topologicalentropy is equal to the logarithm of spectral radius of A. The reason for this is thatthe number of sequence segments of length n starting with the symbol i and endingwith the symbol j is equal to the matrix entry (An)ij . This number represents thenumber of (n, *)-balls (for * = a!1) needed to cover $+

A. This means that N(n, *)is equal to ||An||1, where || · ||1 denotes the matrix norm defined as the sum ofthe absolute values of all the entries. In finite dimensions we can always change toan equivalent matrix norm which is arbitrarily well approximated by the spectralradius +(A). Hence N(n, *) 3 +(A)n, and htop(f) = htop(#+

A) = log +(A).There is also a simple formula which relates the metric entropy of the Markov

measure µ to the matrix ':

hµ(#+A) = )

"

ij

Pi%ij log %ij .

Using equation 2.4 we see that"

i

Pi

$)

"

j

%ij log %ij

%=

"

i

Pi (i = !(" ,

30

where (i = log N if the partition element Zi corresponds to a stable site, and (i = 0otherwise.

It is also easy to see that !(" = log(N) · Prob[z % M ] where Prob[z % M ] isthe probability that a configuration is stable. This probability can be expressedvia avalanche observables. Under the dynamics, any stable configuration will be-come unstable after some time. When this happens an avalanche is initiated. Theavalanche terminates when the configuration is stable again. The duration of thisavalanche, i.e. number of subsequent time steps for which the configuration is unsta-ble, is denoted " . After an avalanche there is a quiet period before new avalanchestarts. The duration of the quiet period is called the waiting time and it is de-noted !. Since configurations are stable during quiet periods and unstable duringavalanches, it is clear that the probability that configuration is stable is given bythe ratio of the mean waiting time to the mean duration of a combined quiet periodand avalanche. That is:

Prob[z % M ] =!!"

!!"+ !"" =1

1 + !""/!!" .

Hence

hµ(f) =log N

1 + !""/!!" . (2.6)

Example: For d = 1 and N = 2, the transition matrix given by equation 2.3has spectral radius +(A) =

42. Hence the topological entropy of the system is

htop(f) = 12 log 2. We can also compute the metric entropy hµ(f) by using equation

2.6. We have seen that there are three stable configurations, each with probability1/6. This implies that that the probability of a stable configuration is 1/2, and thushµ(f) = 1

2 log 2.

In the example above we saw that the topological entropy was equal to the metricentropy hµ(f). This is always the case, and it means that µ always is a measureof maximal entropy. When the matrix A is mixing this result is very easy to proveby using a general result which states that the measures of maximal entropy ontopological Markov chains are Markov measures [32]. In fact, there is an algorithmfor constructing these measures, and by following this procedure one easily provesthe claim. Thus we have that

htop(f) = hµ(f) =log N

1 + !""/!!" .

31

Note that N(*, n) grows like 3 exp(n/(), where

( =1 + !""/!!"

log N.

The number ( represents an important time scale in the BTW model. On timescales comparable with ( (or longer), the BTW model behaves chaotically.

2.1.6 Decay of correlations

Another way of characterizing the loss of predictability in time is using auto-correlation functions. Suppose that ) : $+

N #R( R is a function which is constanton the partition elements Z = [j] # {z}. Then ) can be viewed as a function) : $+

A ( R, which is constant on cylinders of length one, i.e. )(s) = s1. We canthen think of ) as a vector with Q entries, ) = ()1, . . . ,)Q)T . The auto-correlationfunction with respect to the function ) is

C$(n) = !) · () / (#+A)n)" ) !)"2 ,

and when the averages are taken with respect to the natural invariant measure µ,we get

C$(n) ="

s1,...,sn

Ps1%s1s2 · · ·%sn"1sn)s1

3)sn )

"

j

Pj)j

4

="

s1

Ps1)s1

"

sn

3)sn )

"

j

Pj)j

4('n)s1sn

="

s1

Ps1)s1('n,)s1 = !) ·'n," ,

where , = )) !)".Define the Q#Q matrix J = [P, . . . , P ]T where all rows equal to the probability

vector P . Since !," =1

i Pi,i = 0 we have J, = 0. Thus we can write

C$(n) = !) · ('n ) J)," .

Let v = (1, . . . , 1)T % RQ. Since the sum over each row of 'n and each rowof J is equal to one, we have 'nv = Jv = v. If we assume that the matrixA is mixing, then the Perron-Frobenius theorem ensures that the matrix ' hasa unique maximal eigenvalue &1 = 1, which has multiplicity one and a positivecorresponding eigenvector which is proportional to v. The matrix J has rank one,so it is easy to express the eigenvalues of 'n)J in terms of the eigenvalues of '. If&1 = 1,&2, . . . ,&Q is the spectrum of ', then &1 = 0,&n

2 , . . . ,&nQ are the eigenvalues

32

of 'n)J . Indeed, if R is the coordinate change matrix that brings ' to its Jordanblock form &

'''(

1 0 · · · 00 B1 · · · 0...

.... . .

...0 0 · · · Bm

)

***+,

then

RJR!1 =

&

'''(

1 0 · · · 00 0 · · · 0...

.... . .

...0 0 · · · 0

)

***+,

and

'n ) J = R!1

&

'''(

0 0 · · · 00 Bn

1 · · · 0...

.... . .

...0 0 · · · Bn

m

)

***+R .

If the matrix ' is semi-simple the auto-correlation function can be written as

C$(n) =Q"

i=2

ci()) &ni ,

where ci()) are some constants depending on the function ). Since the Perron-Frobenius theorem ensures that |&i| < 1 for i -= 1, we have exponential decay ofthe auto-correlation function. In particular, the slowest mode of correlation decayis given by the second largest eigenvalue (in absolute value) &2. The same is trueeven if the matrix ' is not semi-simple. In this case we can bring ' to a Jordanblock form.

2.1.7 Return maps

The dynamical system associated with the BTW model can be significantly simpli-fied if we consider the return maps to the set of stable recurrent configurations. Thismeans that we compress the duration of avalanches to one time step by definingmaps

Fi(z) = fm(i,z)(z + ei) ,

where m(i, z) = min{n > 0 | fn(z + ei) % M}. This defines a dynamical systemF : $+

N #R( $+N #R by the formula

F (t, z) = (#+Nt, Ft1(z)) .

33

Remark 2.5 For some natural phenomena exhibiting SOC the avalanches occur ontime scales which are much shorter than the characteristic time scale of the drivingmechanism. For these systems the formulation of the return maps is not just amathematical trick, but a physically relevant reformulation of the model.

As with the map f , the dynamical system defined by F admits symbolic codingwith respect to a partition with elements on the form Zi = [j]#{z}. For the returnmap F there is random dropping in every time step, so the matrix A has N 1-s inall rows. The matrix ' has exactly N non-zero elements in each row, all equal to1/N . Hence the metric entropy is

hµ(F ) ="

ij

Pi%ij log %ij ="

i

Pi log N = log N .

Since A(1, . . . , 1)T = N(1, . . . , 1)T , the maximal eigenvalue of A is equal to N , andhence the topological entropy is htop(F ) = log N .

2.1.8 Dhar’s theory for return maps in the BTW model

In this section we briefly discuss some important results of D. Dhar [23]. He usedthe so called Abelian property of the BTW model to show that if we restrict to theset of stable recurrent configurations, then each map Fi is invertible. This meansthat for a given a partition element Z # = [i#] # {z#} and map Fi, there exists aunique partition element Z = [i] # {F!1

i (z)} whose pre-image under Fi intersectsZ #. Thus there are only N di!erent partition elements that can be mapped to Z #,and the column of A corresponding to the partition element Z # has exactly N 1-s.It follows that the the matrix L = 'T has N non-zero elements in each row, eachof them equal to 1/N . Then the P = (1/N, . . . , 1/N) satisfies LP = P . In otherwords: All stable recurrent configurations have the same probability.

If the matrix A is mixing, then the Perron-Frobenous theorem ensures that Pis the only invariant probability vector and that Lnq ( P (for positive q) withexponential speed. However, there is no general result that ensures mixing of thematrix A, and there are examples where Lnq does not converge to P for all q.On the other hand, in all of these examples there is coincidence between the timeaverage of almost all initial conditions and the ensemble averages taken with respectto the Markov measure given by P .

Another important result of Dhar is that the number of recurrent configurationsis equal to the absolute value of the determinant of a N #N matrix #. This matrixis defined by #ii = )2d, #ij = 1 if i and j are nearest neighbors in the lattice and#ij = 0 otherwise.

One of the applications of this result is construction of the matrix A usinga computer. The idea is simply to start with a configuration which we know is

34

Figure 2.2: The 1-s in the matrix A for the BTW model with d = 2 and N = 4.

recurrent (for instance the configuration with all entries equal to 2d ) 1). We cansimulate the dynamics of the model until we have collected |det #| di!erent stableconfigurations. We then know that we have all the stable recurrent configurations,and we can apply the maps Fi to each of them to construct A. The spectrumof A will provide us with important information about the correlations betweenavalanches.

Example: We consider the map F for the case d = 2 and N = 4. In this case the#-matrix is

# =

&

''(

)4 1 1 01 )4 0 11 0 )4 10 1 1 )4

)

**+ .

Thus the number of recurrent configurations is |det#| = 192. This gives Q =4 · 192 = 768 partition elements on the form Z = [i] # {z}, and the matrix A

35

!1.0 !0.5 0.0 0.5 1.0

!1.0

!0.5

0.0

0.5

1.0

Figure 2.3: The spectrum of the matrix ' for the BTW model with d = 2 andN = 4.

has dimensions 768 # 768. We construct A by computing Fi(z) for all z % R andi = 1, . . . , N . The non-zero elements of the matrix A are shown i figure 2.2. Fromthe matrix A it is easy to construct the matrix ', and the spectrum of ' is shownin figure 2.3. There are two eigenvalues with absolute value equal to one, 1 and)1. This implies that the matrix is not mixing. In fact the set of stable recurrentconfigurations can be divided into two disjoined sets (R = R1 +R2) which are ofequal cardinality. The dynamics of the model alternates between these two setswith period 2, giving rise to the eigenvalue )1.

Due to the alternation between the two sets R1 and R2 some observables )(s) =)(s1) (those who have components in the direction of the eigenvector correspondingto the eigenvalue )1) can give rise to auto-correlation functions that do not decayto zero. However, this correlation vanishes if we consider the map F 2 rather thanF . Hence we can view this as a trivial correlation. For observables )(s) = )(s1)that are not a!ected by this periodicity, the rate of decay in the autocorrelationfunction is no slower than 3 rn, where r = max{|&| |& % Sp(') \ {)1, 1}}. In ourexample we compute that r =

43/2.

36

0 500 1000 1500 2000 2500 3000

1

2

3

4

5

6

7

Time n

S!n"

Figure 2.4: Convergence of the time average (for total number of grains) to theensemble average 55/8 for the BTW model with d = 2 and N = 4.

Finding all the stable recurrent configurations also implies that we easily cancompute average quantities. For instance, the average value of the total number ofgrains in a stable recurrent configuration is 55/8. If we simulate the dynamics wecan compare time averages

Sn =1n

n!1"

k=0

)(F k(t, z))

with our ensemble average. The convergence of Sn to 55/8 is shown in figure 2.4.

2.2 The Zhang model

The Zhang model is similar to the BTW model, yet di!erent in two essential aspects.The first di!erence is that the occupation number of a site is allowed to be any realnumber, not just integers. This means that a configuration z = {zi}i"! is a N -tuple in RN rather than ZN . The occupation number of a site in the Zhang modelis usually called the site’s energy.

The other di!erence is in the relaxation rule. In the BTW model, a site alwayslooses a fixed number of grains as it topples, independently of the actual occupationnumber (as long as it exceeds the threshold). In the Zhang model the amount ofenergy a site looses is a fixed proportion of its energy. As site i topples its energy

37

is reduced to *zi and (1) *)zi is distributed in equal parts to its nearest neighbors.Here * % (0, 1) is a fixed parameter of the model. This defines a map f : Rn ( RN

given by

(fz)i = zi ) (1) *)zi&(zi ) Ec) +1) *

2d

"

d!(i,j)=1

zj&(zj ) Ec) .

The parameter Ec > 0 is called the critical threshold energy.The random dropping is preformed in the same way as in the BTW model, and

this can be described by extending the phase space to $+N # RN . As in the BTW

model we define return maps Fi : M ( M , where M = [0, Ec]N is the set of stableconfigurations. This gives us the dynamical system F : $+

N #M ( $+N #M , with

F (t, z) = (#+N (t), Ft1(z)) .

Example: In this example we will find formulas for the maps Fi in the case d = 1,N = 2 and Ec & (1 + *)/(1) *).

Let z = (z1, z2) % M be a stable configuration. Avalanches start with theaddition of 1 to a site. If this is added to the first site, then z '( z+e1 = (z1+1, z2).This configuration is stable if z1 2 Ec ) 1. Denote M11 = {z % M | z1 2 Ec ) 1}.

The configuration z + e1 is non-stable for z1 > Ec ) 1. Applying the map f toz + e1 we get

f(z + e1) =$*(z1 + 1), z2 +

1) *

2(z1 + 1)

%.

We have assumed that Ec & (1+*)/(1)*) & */(1)*), which implies that *(Ec+1) 2Ec, and hence

*(z1 + 1) 2 *(Ec + 1) 2 Ec .

This means that the first component of f(z+e1) always is less than Ec. The secondcomponent

z2 +1) *

2(z1 + 1)

can be both greater than Ec and less than Ec, depending on z. It is less than Ec

in the setM12 =

5z % M | z2 +

1) *

2(z1 + 1) 2 Ec

6,

and for z % M12 we have

F1(z) =$*(z1 + 1), z2 +

1) *

2(z1 + 1)

%.

38

Applying the map f to the configuration above yields$$1 + *

2

%2(z1 + 1) +

1) *

2z2, *z2 + *

1) *

2(z1 + 1)

%

Since Ec & (1 + *)/(1 ) *) we have (1 ) *)Ec & 1 + *, and hence the followingestimate for the first component:

$1 + *

2

%2(z1 + 1) +

1) *

2z2 2

$1 + *

2

%2(Ec + 1) +

1) *

2Ec

= Ec +1 + *

4

$(1 + *)) (1) *)Ec

%2 Ec .

The inequality Ec & (1 + *)/(1) *) is equivalent to

* 2 Ec ) 1Ec + 1

,

so the second component can be estimated by

*z2 + *1) *

2(z1 + 1) 2 Ec ) 1

Ec + 1Ec +

12

Ec ) 1Ec + 1

$1) Ec ) 1

Ec + 1

%(Ec + 1)

= Ec ) 1 2 Ec .

This shows that it is impossible to have non-stable configurations for more thantwo subsequent time steps in this example.

The map F1 has the formula

F1(z) =

-../

..0

T11(z + e1) for z % M11 =5

z % M | z1 + 1 2 Ec

6

T12(z + e1) for z % M12 =5

z % M \M11 | *z2 + 1!"2 (z1 + 1) 2 Ec

6

T13(z + e1) for z % M13 = M \ (M11 +M12)

,

where

T11 =7

1 00 1

8, T12 =

7* 0

1!"2 1

8and T13 =

9$1+"2

%21!"2

* 1!"2 *

:.

The map F2 is given by similar formulas, which are easy to derive using the sym-metry in the model.

In the example above we saw that the maps Fi : M ( M are piecewise a"ne. Thisis always true for the Zhang model. For each i % % there exists a finite partitionM = +kMik (consisting of polyhedra Mik $ M $ RN ) such that the restrictionsFi|Mik are a"ne maps on the form z '( Tik(z + ei) = Tikz + Tikei. Here Tik areN #N matrices.

39

2.2.1 Symbolic coding of the Zhang model

In the BTW model we used symbolic coding g : $+N #R( $+

A to calculate dynami-cal invariants such as topological and metric entropy. The topological Markov chain#+

A : $+A ( $+

A was also used to identify the natural invariant measure. The successof the coding technique for the BTW model suggests that one should attempt asimilar construction for the Zhang model.

To do this we need to fix a partition of $+N #M . One natural choice is to take

sets Z = [i] # Mik. We enumerate these sets Z1, Z2, . . . ZQ and for each initialcondition (t, z) we can construct a sequence s % $+

Q by the rule

Fn(t, z) % Zsn .

Let g : $+N #M ( $+

Q denote this map. The dynamics of the Zhang model has asymbolic representation $ := g($+

N #M) $ $+Q.

Unfortunately there are several problems with this construction. First, the mapg need not be invertible. For a fixed a sequence t % $+

N there can be two distinctconfigurations z, z# % M such that Ftn / · · · / Ft1(z) and Ftn / · · · / Ft1(z#) lie in thesame partition element of the partition M = +kMtn+1k for all n & 1. Then (t, z)and (t, z#) are mapped to the same sequence s % $+

Q under the coding map g. Thesecond problem is that the set $ typically has a more complicated structure than atopological Markov chain. It may not even be a closed set.

Remark 2.6 We can always construct a topological Markov chain $+A by defining

a Q#Q matrix A by the rule aij = 1 if F (Zi) , Zj -= . and aij = 0 otherwise. Byconstruction $ is a subset of $+

A, but we can not ensure equality. If aij = 1 andajk = 1, then there exists a sequence s % $+

A containing the segment (ijk), but thissegment may not exist in any sequence in the set $. In other words, we might have

$F (Zi) , Zj

% ; $F!1(Zk) , Zj

%= . .

This is impossible in the BTW model where each partition element only contains asingle configuration.

2.2.2 Entropy in the Zhang model

Using that the map #+N : $+

N ( $+N has topological entropy log N , it is very easy

to show that the topological entropy of the map F : $+N #M ( $+

N #M is at leastlog N . We must therefore ask if there exist situations where htop(F ) > log N . Inother words: Do the singularities in the Zhang model produce additional entropy?As it turns out, this question is much more di"cult to answer than one mightimagine, and the mathematics constructed in order to understand the problem has

40

provided new general results on the on entropy in dynamical systems with singu-larities (see chapters 4 and 5). Here we give a brief introduction to some aspects ofthis topic:

It is not di"cult to see that the modified map F : [0, 1]#M ( [0, 1]#M , with

F (x, z) = (Nx mod 1, F[x/N ](z)) ,

([·] denotes integer value) has the same topological entropy as the original map F .

Remark 2.7 The map #+N : $+

N ( $+N can be replaced with an expanding map

x ( Nx mod 1 on the interval [0, 1]. By expressing points x % [0, 1] in N -digitexpansions 0.i1i2i3 · · · we see that each point x corresponds to a sequence in $+

N ,and that the shift map #+

N corresponds to multiplication with N (mod 1). The shiftmap #+

N is not conjugated to the expanding map since there are real numbers in theinterval [0, 1] which have more than one representation in the N -digit system. Forinstance, the numbers 0.0111 · · · and 0.100 · · · are equal. However, this does note"ect the topological entropy.

Using this modification the Zhang model becomes a piecewise a"ne map on [0, 1]#RN $ RN+1 with domains of continuity on the form [(i)1)/N, i/M ]#Mik. In eachof these domains the map F is a"ne with a linear part given by a matrix with thefollowing block form

d(x,z)F =7

N 00 Tik

8.

Thus the linear parts of the map Fn are on the form

d(x,z)Fn =

7Nn 00 Tinkn . . . Ti1k1

8,

and we see that log N is a Lyapunov exponent of the map F . From the definition ofthe Zhang model it is easy to see that the spectral radius of the matrices Tik all areless than or equal to 1 (and any su"ciently long realized composition Tinkn . . . Ti1k1

has spectral radius less than 1). This implies that all the other Lyapunov exponentsare negative. If one then applies the Ruelle inequality and the variational principlewe get htop(F ) 2 log N (and hence htop(F ) = log N). However, these generalresults do not always hold for piecewise a"ne maps. In fact, there exist piecewisea"ne contracting maps with positive entropy. An example of such a map is givenin chapter 5, and even more examples are given in chapter 4.

There is a particular mechanism that can generate positive entropy in contract-ing a"ne maps with singularities. Even though the maps are contracting, anddistances between points do not increase under each a"ne mapping, the anglesbetween lines in phase space may grow with exponential speed. In the presence

41

of singularities, angular expansion may cause exponential orbit growth and posi-tive entropy. Actually, the entropy can be bounded by so-called angular Lyapunovexponents which measure the asymptotic angular expansion rates. (See chapter 4for details.) As a corollary, the entropy of a piecewise a"ne conformal contractingmap is zero. Unfortunately, the Zhang model is not defined by piecewise conformalmaps, and this approach fails to prove htop(F ) = log N .

In the end we were forced to use a di!erent technique, which only succeeds toprove that the entropy of the Zhang model is logN for generic * and Ec. This resultis presented in chapter 3.

Note that even though the BTW and Zhang model have the same topologicalentropy, the number of *-distinguishable orbit segments grows faster in the Zhangmodel. For Ec 5 1 and fixed * we have N(*, n) 3 Nn + w(n), where w(n) grows atleast linearly.

2.2.3 Removability of singularities

For some simple situations the question of entropy and coding is very easy to un-derstand. This is when the singularities (lines of discontinuity) do not a!ect theasymptotic dynamics.

We define the attractor

A = %s

$ ;

n'0

Fn($+N #M)

%.

Here %s : $+N #M ( M is the projection on to the configuration space M . Since

we are interested in the asymptotic dynamics we can restrict our attention to theset $+

N # A. It can be shown that it is non-empty for su"ciently large Ec. Notethat A can be written A0 ,A1 , . . . , where An is defined by

An = %s

$ n!1;

k=0

Fn($+N #M)

%.

Clearly these sets make up a nested sequence

· · · $ An $ An!1 $ · · · $ A1 $ A0 = M .

If the dynamics has a simpler description once we restrict to one of the sets An, thenthis restriction can be made with out loosing any information about the asymptoticbehavior of the model. This is especially useful if An is disjoined from the singu-larity set S = +ik-Mik for some n % N. In this case we say that singularities areremovable, and it implies that the restriction of F to $+

N # A is continuous. Wecan then use the variational principle combined with the Ruelle-Margulis inequalityto show that htop(F ) = log N . Another important consequence of removability of

42

singularities is that it ensures that F is semi-conjugated to a topological Markovchain.

Remark 2.8 If the set An is disjoined from the singularity set we can constructcoding by choosing a partition of $+

N #A with elements on the form Z = [i]#Kj,where K1,K2, . . . are the connected components of An. With respect to this partitionwe have

$ = g($+N #An) = $+

A ,

where A = ||akl|| is given by

akl =

#1 for F (Zk) , Zl -= .0 otherwise

.

The maps Fi are continuous on the connected sets Kj, and the continuous imageof a connected set can only intersect one of the (disjoined) partition elements. Usingthis it is easy to show that the number of 1-s in each row of A is equal to N . Thenthe maximal eigenvalue of A is N , and htop(#+

A) = log N .In the Zhang model there are examples when singularities are removable and

the columns of A have a varying number of 1-s. Hence Dhar’s result for Abeliansandpiles does not generalize to the Zhang model, not even when singularities areremovable.

2.2.4 Characterization of the attractor

We have already explained several essential di!erences between the dynamics in theBTW and Zhang models. These become even clearer if one studies the geometryof the attractors in configuration space. For the BTW model attractors are simplyequal to the sets of recurrent configurations, which are finite. In the Zhang modelthe attractors have a complex fractal structure (see figure 2.5). The setA is actuallyof full dimension for Ec 5 1, but numerical simulations show that the support ofthe SRB measure has non-integer dimension.

We can construct the SRB measure by a sequence .n of probability measures

.n =1n

n!1"

k=0

F k& (µBer # µLeb) .

where µLeb is the Lebesgue probability measure on M and µBer is the Bernoullimeasure on $+

N . We assume that {.n} has a unique limit point . (in the weaktopology) and let .s denote its projection to M . Numerically, the measure .s can be

43

5.0 5.5 6.0 6.5 7.0

5.0

5.5

6.0

6.5

7.0

Figure 2.5: An orbit on the attractor in the configuration space M for d = 1, N = 2,* = 2/3 and Ec = 7.

approximated by simulating a long orbit and counting the number of configurationsthat belong to each cell of a fine partition. This numerical approximation can beused to calculate several characteristics of the attractor.

An especially interesting characteristic is the multi-fractal spectrum. This isdefined as the function

/ '( dimH(G%) ,

where G% = {z % M |d&s(z) = /} and

dµs(z) = lim'$0

log .s(B'(z))log $

is the point-wise dimension of the measure .s in the point z. Using a standardtechnique based on calculations of moments

1i .s(Yi)q for partitions {Yi} of varying

diameter (see [44] for details) we can calculate this spectrum from our numericalapproximation of .s. The result is shown in figure 2.6. The non-trivial multi-fractalspectrum illustrates the rich dynamical structure of the Zhang model.

44

1 2 3 4 50.0

0.5

1.0

1.5

2.0

!

f!!"

Figure 2.6: The multi-fractal spectrum f(/) = dimH G% for the projection of thenatural invariant measure onto the configuration space M . The parameters in thiscomputation are d = 1, N = 2, * = 2/3 and Ec = 7.

45

Chapter 3

Dynamics and entropy in theZhang model of SOC

Joint with B. Kruglikov

Abstract: We give a detailed study of dynamical properties of the Zhang model,including evaluation of topological entropy and estimates for the Lyapunov expo-nents and the dimension of the attractor. In the thermodynamic limit the entropygoes to zero and the Lyapunov spectrum collapses.

Refrence for this paper: B. Kruglikov, M. Rypdal, Dynamics and entropy in the Zhang

model of SOC , Journal of Statistical Physics 122, 975 (2006)

47

3.1 Introduction

In 1987 the concept of Self-Organized Criticality (SOC) was introduced by Bak,Tang and Wiesenfeld [3]. The attempt was to give an explanation of the omnipres-ence of fractal structures and power-law statistics in nature, and the claim wasthat certain physical systems can self-organize into stationary states, reminiscent ofequilibrium system at the critical point, in the sense that one has scale invarianceand long range correlations in space and time.

SOC is proposed as an explanation for variety of phenomena in nature, such asearthquakes, forest fires, stock markets and biological evolution [30]. However, mostwork has been devoted to the study of idealized “sandpile-like” computer models,such as the sandpile model [3], the abelian sandpile [23] and the Zhang model [57],that one believes exhibit SOC in the thermodynamic limit. Despite this e!ort, asatisfactory understanding of the model is not yet achieved. Through numericalinvestigation one has observed that in the thermodynamic limit, observables havepower-law distributions. More precisely, the probability distribution function P (s),of an observable s will have the form P (s) 3 1/s(s in the thermodynamic limit.There is not a widely agreed upon method for computing the SOC-exponents "s

numerically, and due to the incomplete understanding of the dynamics of the modelsand lack of a formal treatment of the thermodynamic limit, it is di"cult to properlyexplain the observed behavior. Hence it is not clear what the SOC-exponents reallytell us about the dynamics of the SOC models.

In a series of papers by Cessac, Blanchard and Kruger [6, 7, 8] it was proposedthat further understanding of SOC models can be achieved by studying the modelsin the framework of dynamical system theory. They showed how a particular model,the Zhang model, could be formulated as a dynamical system of skew-product typewith singularities, where the randomness of the external driving is described by aBernoulli shift, and the threshold relaxation dynamics is given by piecewise a"nemaps.

In this paper we give a detailed study of the dynamical system defined in [6]. Weprove several basic properties, some of which are already stated in [6, 7, 8], beforediscussing fundamental dynamical properties. We observe that, depending on theparameters of the model, we can observe fundamentally di!erent types of behavior.For low values of the threshold energy (critical energy), the dynamics can be rela-tively simple, since the singularities only e!ect the dynamics in a finite number oftime-steps. In such situations we say that singularities are removable, and we showthat the system permits symbolic coding whenever singularities are removable. Wegive examples of how symbolic coding can give a complete topological descriptionof the dynamics as a topological Markov chain. Hence the dynamics is chaotic, butthe essential dynamical invariants are all inherited from the Bernoulli shift factor.Moreover we can identify the physical invariant measure, and hence understanding

48

of the statistical properties is reduced to the theory of Markov chains.As we increase the critical energy the role of the singularities becomes essential.

Techniques, which are based on codings, are no longer applicable and a very interest-ing dynamics is observed. The dimensional characteristics of the attractor are alsosensitive to the parameters of the system, as we show by generalizing the Moran’sformula for IFS’s. In addition, we observe the situation, when the dimension ofthe IFS-attractor increases to the maximum, while the support of the SRB-measureremains fractal.

To measure the complexity of the dynamics we study entropy and Lyapunovexponents. We show that the system is hyperbolic, with one positive exponentoriginating in the Bernoulli shift. However, due to the presence of singularitiesthe Ruelle inequality and the Pesin formula are not directly applicable. We showthat the metric entropy of any SRB-measure equals the topological entropy almostsurely, and we evaluate the latter generalizing the technique developed by Buzzi[16, 17]. The result is that the Pesin formula and the variational principle hold aposteriori.

To give a satisfactory physical interpretation of the dynamics we rescale time toprevent infinitely slow driving of the system. We prove that for this physical system,the Lyapunov spectrum collapses completely and that the entropy goes to zero inthe thermodynamic limit. This implies that the expanding (chaotic) properties arelost, so that we may expect power-laws statistics and long range correlation e!ects.

The statistical properties we obtain hold for any SRB-measure. The existence ofSRB-measures is in fact still an open problem. From the general theory of dynamicalsystems with singularities [33, 43, 53], we can give conditions that are su"cient forthe existence of SRB-measures, but it is not known if these conditions hold for themajority of parameters. We expect this to be true (it was also conjectured in [6])and derive statistical corollaries (some of which hold without the assumption ofmeasure existence).

Apart from the physical importance of the Zhang model, it is interesting froma mathematical point of view. It can be described as a piecewise a"ne hyperbolicmap of the form

F : $+N #M ( $+

N #M, ((t0t1t2 . . . ), x) '( ((t1t2t3 . . . ), ft0(x)),

where $+N is the set of right infinite sequences from finite alphabet and {fi} a col-

lection of piece-wise a"ne non-expanding maps of M to itself. Previously piecewisea"ne expanding maps and piecewise isometries have been studied, but the con-tracting property of the relaxation dynamics provides di"culties. Therefore severalmethods are developed in this paper, which hold far beyond the framework of theZhang model.

The structure of the paper is the following. In section 1 we describe the model,find bounds on avalanches size and time, which makes possible to use the Poincare

49

return to reformulate the systems in a skew-product form. Then we describe, whendegenerations occur (the original Zhang setting * = 0 is not the only possibility) andstudy the contraction property to conclude hyperbolicity of the model. In section2 we introduce the concept of removability of singularities, which appears in thecoding approach for the study of the model.

Section 3 is devoted to the study of measure entropy and Lyapunov spectrumand section 4 concerns the topological entropy. Since the standard theorems do notwork in the presence of singularities, we develop a new technique for evaluation ofthese basic quantities. Section 5 briefly describes the dimension issues of the model.

In appendices A and B we provide bounds for the entropy and the dimension,which in the presence of singularities, overlaps and degenerations (this was designedfor an application to the Zhang model) are new. The results are of interest in itsown and can be read independently.

3.2 Basic properties of the Zhang model

In the Zhang model each site on the lattice is associated with a non-negative realnumber, which we call the energy of the site. The collection of energies is calledan energy configuration, and can be represented as a point in N -dimensional space,where N is the number of sites in the lattice. If a configuration is unstable, theovercritical sites will lose some of their energy to their nearest neighbors, resultingin a new energy configuration. This transformation on RN is denoted by f . If aconfiguration is stable, a site is chosen at random and an energy quantum $ = 1 isadded to this site. In [6] it was shown how the relaxation and random excitationcan be formulated as a map of skew-product type on an extended phase-space. Thisextended phase-space has the configuration space as one factor, and the set of allpossible sequences of excitations as the other factor. In [6] it was also shown how onecan reformulate the dynamical system by considering the return maps to the set ofstable configurations. This gives a simplification, in the sense that each avalancheis associated with an a"ne transformation. The set of stable configurations ispartitioned into domains, where each domain corresponds to an avalanche.

3.2.1 Relaxation

Take d, L % N and let % $ Zd be the cube [1, L]d of cardinality N := Ld = |%|. Let) : % ( %# := {1, . . . , N} be a bijection. We define a metric d! on % by

d!(k, l) ="

1(n(d

|kn ) ln| ,

and let d!! := )&d!. In the following we omit primes when it is clear from thecontext that we are considering the metric space (%#, d!!). Elements of % will be

50

called sites. We say that sites i and j are nearest neighbors if d!(i, j) = 1. Theboundary -% is defined as those sites i % % that have less than 2d nearest neighbors.

Fix parameters Ec > 0 and * % [0, 1) and define f : RN'0 ( RN

'0 by

f(x)i = xi ) 0(xi ) Ec)(1) *)xi +1) *

2d

"

d!(i,j)=1

0(xj ) Ec)xj ,

where

0(a) =

#1 if a > 00 if a 2 0

.

Let 6x61 =1N

i=1 |xi| be the 1-norm on RN .

Proposition 3.1 For all x % RN'0 we have

1 + *

26x61 2 6f(x)61 2 6x61 ,

and 6f(x)61 = 6x61 if and only if xi 2 Ec for all i % -%. If there is i % -% suchthat xi > Ec then

6f(x)61 2 6x61 )1) *

2dEc .

Proof. Let {xik}mk=1 be the entries of the vector x that are greater than Ec. Let

nik be the number of nearest neighbors of xik . Then

6f(x)61 ="

i"!

f(x)i ="

i"!

xi ) (1) *)m"

k=1

xik +1) *

2d

m"

k=1

nikxik

="

i"!

xi ) (1) *)m"

k=1

(1) nik

2d)xik .

The statement follows from the fact that we always have d 2 nik 2 2d, and nik = 2dif and only if xik -% -%. !

We say that a site i % % of the configuration x is relaxed if xi 2 Ec, andexcited if xi > Ec. A configuration x is called stable if all sites are relaxed. Theset of stable configurations is M := [0, Ec]N . For each configuration x we definem(x) = min{n & 0 | fn(x) % M}.

Proposition 3.2 For all x % RN'0 we have:

m(x) 2 2dN

1) *

6x61Ec

$ 2d

1) *+ 1

%diam(!)/2.

51

We need the following lemma ([·] denotes integer part):

Lemma 3.3 For x % RN'0 and n % N let /i(n, x) be the cardinality of the set

{l 2 n | (f lx)i > Ec}. Let 1 = [2d/(1 ) *)] + 1. If d!(i, j) = 1, then /j(n, x) &[/i(n, x)/1].

Proof. There is a finite increasing sequence {mk}, such that (fmk(x))i > Ec. Weclaim that on each interval (mk,mk+) ] there is a number m such that (fm(x))j >Ec. In fact, in the opposite case

<f1+mk+!"1(x)

=j& 1

1) *

2dEc > Ec .

Since [0,/i(n, x)] contains 2 = [/i(n, x)/1] disjoined such intervals, we get /j(n, x) &2. Thus /j(n, x) & [/i(n, x)/1]. !

Proof of Proposition 3.2. By applying inductively Lemma 3.3 we get:

/j(n, x) &3/i(n, x)

1d!(i,j)

4

In fact, if j = j0, j1, . . . jk = i is a path with d!(js, js+1) = 1 and3/i(n, x)

1k

4= t ,

then /jk(n, x) & t1k,/jk"1(n, x) & t1k!1, . . . ,/j0(n, x) & t. By Proposition 3.1

/j(n, x) 2 2d

1) *

6x61Ec

for j % -%, so

/i(n, x) 2 /(x) :=2d

1) *

6x61Ec

1diam(!)/2 .

for all i % % and all n % N. Suppose fm(x) -% M for all m 2 T . If T > N/(x),then there must be a site i % % that is greater than Ec for more than /(x) di!erenttimes. This is impossible so m(x) 2 N/(x). !

3.2.2 Random excitations

Define $+N = %N to be the set of right-infinite %-sequences and let #+

N : $+N ( $+

N

be the left shift. We define a map f : $+N # RN

'0 ( $+N # RN

'0 by

f(t, x) =

#(#+

Nt, x + et0) if x % M

(t, f(x)) if x -% M,

52

where e1, . . . , eN is the standard basis in RN . We denote points in $+N # RN

'0 byx = (t, x), and we define %u and %s to be the projections to $+

N and RN'0 respectively.

Proposition 3.4 For all x % $+N # RN

'0 it holds:

min{m & 0 |*i % % 7m# 2 m : (%s / fm!(x))i > Ec} 2 n(Ec, *, %) ,

wheren(Ec, *, %) = N(NEc + 2)

$3 2d

1) *

4+ 1

%diam(!),

Proof. In N [Ec] + 1 time-steps, there must be an overcritical site. Since in therelaxation process there is always an overcritical site, then during arbitrary subse-quent N [Ec] + 2 time-steps an exited site can be found. Hence after N((N [Ec] + 2)time-steps either all sites have been overcritical or there is a site that has beenovercritical at least ( times. However it follows from the proof of Proposition 3.2that if one site is overcritical

( =$3 2d

1) *

4+ 1

%diam(!)

times, then all sites have been overcritical at least once. !

For x % RN'0 and i % % we define "(i, x) := min{n % N | fn(x + ei) % M}.

Proposition 3.2 assures us that this number is finite and

maxx"M

maxi"!

"(i, x) 2 "m(Ec, *, %) = N2$1 +

1NEc

%$ 2d

1) *+ 1

%diam(!)/2+1.

Thus we observe that neither n(Ec, *, %) nor "m(Ec, *, %) are uniformly bounded inEc, but there is the following alternative:

There exists a constant C0, not depending on the energy Ec, such that eithern(Ec, *, %) 2 C0 or "m(Ec, *, %) 2 C0.

In fact, we can set C0 = 3N2<

2d1!" + 1

=diam(!)+1. Thus we get that eitherrelaxation happen su"ciently fast or all the sites keep being excited su"cientlyoften (uniformly in Ec).

But there does not exist such a bound uniform in * or N .

3.2.3 Return maps

Let x = (t, x) % $+N # M . For n = 1, . . . , "(t0, x) define Cn(x) = {i % % | (%s /

fnx)i > Ec}, and A(x) =<C1(x), . . . , C((t0,x)(x)

=. We call A(x) the avalanche of

the point x. Let M := $+N #M and define an equivalence relation 3 on M by

x 3 y 8 A(x) = A(y) .

53

This gives a partition of M . From the definition it is clear that A(x) depends on t0and x only. Hence partition elements are of the form [i]#Mij , where

*i % % :,

j

Mij = M

and [i] = {t % $+N | t0 = i} is the cylinder of the symbol i. We see that for each

i % %, the domains Mi1,Mi2, . . . are separated by segments of at most N !(m =exp ("m(Ec, *, %) log N !) hyperplanes. Hence we have a finite number of domainsMi1, . . . ,Miqi for each i % %. By definition there is a unique avalanche for eachpartition element [i] # Mij . We denote this avalanche by Aij . Its duration is"ij := "(i, x), for x % Mij , and define its size to be sij =

1(ij

n=1 |Cn|.We define the piecewise continuous map F : M ( M by

(t, x) '( (#+Nt, Ft0x)

where Fi(x) := f((i,x)(x + ei). We define Fij := Fi|Mij .

Remark 3.1 From a mathematical point of view the formulation (M, F ) is a sim-plification compared to ($+

N # RN'0, f). However, the duration of avalanches are

suppressed so that all avalanches have the same duration. This is not satisfactoryfrom a physical point of view, and hence we call (M, F ) the mathematical model and($+

N#RN'0, f) the physical model. We will later make a rescaling of time in the phys-

ical model, so that the driving does not become infinitely slow in the thermodynamiclimit.

For each x % RN'0 we define a matrix Q(x) by

Qkl(x) =

#12d0(xl ) Ec) if d!(k, l) = 1 ,

0 otherwise ,

and a diagonal matrix J(x) by Jkl(x) = (1) (1) *)0(xl ) Ec))$kl. Set

S(x) = J(x) + (1) *)Q(x) (3.1)

and observe that f(x) = S(x)x. Let x(1) = x + ei and x(n) = f(x(n ) 1)) forn % {2, . . . , "(t, x)}. Then Fi(x) = Li(x + ei), where

Li(x) = S(x("(x, i))) . . . S(x(1)) .

If x, y % Mij , then "(i, x) = "(i, y) and the same components of x(n) and y(n) aregrater than Ec for each n = 1, . . . , "(t, x), so Li(x) = Li(y). We define the linearmap Lij := Li(x) for x % Mij . We get Fi|Mij (x) = Lij(x + ei).

54

Definition 3.1 A sequence {(in, jn) | 1 2 n 2 0} is said to be admissible if

*;

n=1

(Fin"1jn"1 / · · · / Fi1j1)!1(Minjn) -= . ,

Theorem 3.5 For all i, j 6Lij61 2 1. Moreover for every constant c % (0, 1) thereis a number T % N such that for every 0 > T and admissible sequence {(in, jn) | 1 2n 2 0} it holds:

6Li"j" . . . Li1j161 < c .

Proof. If A is an N #N matrix we let Ck(A) be its k-th column. Observe that forany matrices A and B we have the following formula:

6Ck(AB)61 ="

l

6Cl(A)61Blk . (3.2)

By the construction: 6Ck(S(x))61 2 1 for all k % %. Hence 6Lij61 2 1.To prove the second statement we note that for * > 0 the diagonal elements of

the matrices S(x) are non-zero and & *. Therefore

(S(x(m))S(x(m) 1)))kl & * ·max{Skl(x(m)), Skl(x(m) 1))}.

Moreover, Skl(x(m)) > 0 if x(m)l > 0 and d!(k, l) = 1. It follows that any admis-sible product Li"j" . . . Li1j1 of length 0 & n(Ec, *, %) is positive. By Proposition 3.1there must be at least one column such that the sum over this column is less than1, for some factor Litjt and hence for the whole product. Therefore the sum overeach column of any admissible product of length 2n(Ec, *, %) must be less than 1.Let c0 < 1 be the maximal norm of all admissible products of length 2n(Ec, *, %).For k > k0 := [log c/ log c0] + 1 we have ck

0 < c and hence T = 2k0n(Ec, *, %) is therequired number.

The above argument does not apply to the case * = 0, and a di!erent proofmust be given for this case (which actually works in general as well). Take x % Mand let x(t) % M be the projection of its orbit to M . Denote S(x(t)) by St(x), andlet

St(x) = St(x) · · ·S0(x) .

We make the following claims:

1. There exists n % N such that for all l,m % % and all x % M there is t 2 nsuch that (St(x))lm -= 0.

2. For all i & 0 there exists ni % N such that 6Cm(St(x))61 < 1 for all t & ni,x % M and all sites m % % with d!(m, -%) 2 i.

55

The second claim for i = 12 diam(%) implies the statement of the theorem.

To see the first claim we fix x and let U $ %2 be the subset of the pairs (l, m)with (St)lm = 0 for all su"ciently large t. By inductively applying (3.2) we see thatthe columns for St(x) are non-zero for all t & 0. So for all 2 % % there is / % %such that (/, 2) % %2 \ U . Given sites / and 2 we choose t such that St(x)%+ -= 0.Consider now column / of the matrix St+1(x) = St+1(x)St(x). If / is stable, i.e.x(t + 1)% 2 Ec, then (St+1(x))%% > 0 and (St+1(x))%+ -= 0, so we just repeat theargument. But the site / can not be stable for more than n(Ec, 0,%) iterations.Hence we can with no loss of generality choose t such that x(t + 1)% > Ec. Thenthe column / of St+1(x) has non-zero elements in all position that correspond toneighbors of /. Hence we obtain that (/#,2) % %2 \U for all /# with d!(/#,/) = 1.Any two points can be connected by a path of neighbors, so U = ., and the firstclaim follows. In fact, one can see that the bound n does not depend on a choice ofx and satisfies: n 2 diam(%) · n(Ec, 0,%).

To prove the second claim let us note that if 6Ck(St(x))61 < 1, then 6Ck(St+1(x))61 <1 because by (3.2): 6Ck(AB)6 2 maxl 6Cl(A)6 ·6 Ck(B)6.

We will use induction on i starting from i = 0. Take k % -% and t 2 n suchthat (St(x))kk -= 0. If x(t + 1)k > Ec, then 6Ck(St+1(x))61 < 1 and

6Ck(St+1(x))61 ="

l

6Cl(St+1(x))61(St(x))lk < 1 ,

and so we have the desired inequality. If x(t + 1)k 2 Ec, then (St+1(x))kk = 1 andhence (St+1(x))kk -= 0. Then we repeat the argument. Since no site can be stablefor more than n(Ec, 0,%) successive time-steps we obtain the claim for i = 0 withn0 = n(Ec, 0,%) + n.

Consider now the case i > 0. For a site m % % with d(m, -%) = i, we take l % %with d!(l,m) = 1 and d!(l, -%) = i) 1. By the first claim we find some t 2 n suchthat (St(x))lm -= ., and by the induction hypothesis for t# & ni!1 we have:

6Cl

<St+t!(x) · · ·St+1(x)

=61 < 1 .

Using (3.2) we obtain 6Cm

<St+t!(x)

=61 < 1. We can choose ni = ni!1 + n. !

Lemma 3.6 Let Ec & */(1) *). Then for any x % M , n % N and i, j % Cn(x) wehave d!(i, j) -= 1.

Proof. Take x % M and let En be the maximal energy of a site in Cn(x). ClearlyE1 2 Ec + 1 and

En+1 2 max>

max{*En, Ec}+ (1) *)En, Ec + 1?

.

From this we see by induction that

En 2 max>Ec

*, Ec + 1

?=

Ec

*,

56

so *En 2 Ec for all n % N, and this means that a site cannot be overcritical intwo successive time-steps (for * = 0 the above argument does not work, but thestatement holds obviously).

All avalanches start with a single site. Let C1(x) = {i}. Then d(i, j) = 1 for allj % C2(x). This implies that any two elements of C2(x) can be connected with apath of length 2, so no two sites of C2(x) are nearest neighbors. If there exists apath of even length between two points in %, then all paths connecting these pointsare of even length. Therefore we can repeat the argument proving by induction thatd!(i, j) % 2Z for all i, j % Cn(x). !

Proposition 3.7 The linear maps Lij are all invertible whenever * & 1/2 or * > 0and Ec & */(1) *). If we have Ec & */(1) *), then

detLij = *sij .

Proof. Take arbitrary x % RN'0. First we observe that since the sum over each

column of Q(x) is less than or equal to 1, we have 6Q(x)v61 2 6v61 for each v % RN .This implies that

6S(x)v61 = 6(J(x) + (1) *)Q(x))v61& 6J(x)v61 ) (1) *)6Q(x)v61& (2*) 1)6v61 .

If * > 1/2, then S(x)v -= 0 for all v -= 0, so we have invertibility.For * = 1/2 the claim follows since in the above chain of inequalities at least

one is strict if v -= 0. In fact, if J(x)v = *v, then vi = 0 for all relaxed sites i.We claim that the equality Q(x)v = v is impossible. To see this denote by Q theminor-matrix formed by the rows and columns of Q(x), corresponding to exitedsites, and denote by v be the respective reduced vector. Then Qv = v.

Let U be the set of overcritical sites k with vk = max vl (we suppose it is positive,multiplying by )1 in the opposite case). Choose a boundary site k % U , i.e. thenumber of neighbors l to k with vl = vk is less than 2d. Then:

vk ="

l

Qklvl < vk

"

l

Qkl 2 vk.

This contradiction yields the result.Finally consider the last statement about the case Ec & */(1)*). It is proved by

reducing the matrix S(x). If xi 2 Ec, then column Ci(S(x)) equals (0, . . . , 0, 1, 0, . . . 0)T ,where the 1 is in the ith position. We can start the decomposition of detS(x) withcolumn i, and hence we see that row i and column i can be removed from S(x) with-out changing the determinant. We remove all rows and columns that correspond torelaxed sites. If +(x) is the number of overcritical sites of x, we get a +(x) # +(x)

57

matrix Sred(x). If site k is overcritical then Jkk(x) = *. If Ec & */(1 ) *), then itfollows from Lemma 3.6 that all nearest neighbors of k are relaxed. Hence columnk of Q(x) has only zero entries. This shows that Sred(x) = diag(*, . . . , *). Then

detS(x) = det Sred(x) = *,(x) ,

so detLij = *sij . !

Remark 3.2 In the original model of Zhang one has * = 0 in which case detLij = 0if Lij -= 1. But it is not true that non-trivial kernels can occur for * = 0 only,contrary to what was stated in [6]. A simple counter-example is the case N = 2,Ec = 1/3 and * = 1/3. For x1 > 0 and 2x1 + 3x2 < 1 we have:

F1

$ @x1

x2

A%=

19

@2 32 3

A @x1 + 1

x2

A.

and so det L12 = 0.

Having non-degenerate maps in the model is more convenient from the pointof view of mathematical tools (though from a physical viewpoint it can make nobig di!erence between degenerate and close-to-degenerate systems). Fortunately,degenerations occur only for a negligible set of parameters.

Theorem 3.8 The maps Lij are invertible for almost all (*, Ec). In fact, they areinvertible for the parameters complimentary to the set ) $ [0, 1) # (0,1), whichconsists of a finite set of vertical intervals for fixed d and N .

Proof. Fix an avalanche Aij and let L"ij be the corresponding linear maps (we

stress dependence on *). These maps are the compositions of elementary matricesS"(x("(x, i))) . . . S"(x(1)), with the factors from (3.1)

S"(x(t)) = 1 + (*) 1)(dJd" )Q)(x(t))

being polynomial in * and independent of the choice of x = x(1) % Mij . Thecondition detL"

ij = 0 is equivalent to detS"(x(t)) = 0 for some t. There are onlyfinite number of possibilities for the matrix S"(x) (though a countable number fortheir compositions L"

ij , the length of which grow as Ec ( 0). Thus we obtaink = k(d, N) polynomial equations Pj(*) = 0. Since S"(x) ( 1 for all x as * ( 1we see that lim"$1 Pj(*) = det1 = 1. This shows that the polynomial is notidentically zero, so there can only be a finite number of roots. For each root *ja

there is the maximal value Eiac of Ec (finite if *ja -= 0), where the corresponding

matrix S"(x) can appear in the avalanche. Thus the set of degenerate systems is{(*, Ec) | * = *ja, 0 < Ec < Eja

c }. !By proposition 3.7 ) does not intersect the set {* & 1/2} +{ Ec & */(1) *)}.

58

3.3 Removability of singularities and coding

The map F may be considered as a piecewise a"ne map F : I#M ( I#M , whereI = [0, 1] and F (t, x) = (Nt mod 1, F[Nt](x)). The map t '( Nt mod 1 is notconjugated to #+

N since the points m/Nk % I do not have unique representations in$+

N . However the sets {m/Nk} #M $ I #M are singularities, and following thestandard approach for piecewise a"ne maps, should be removed.

In some physical systems, like the Belykh family, the singularities propagate,intersecting themselves transversally. The Zhang model is not a general positionsystem in this respect, because singularities {m/Nk}#M $ I #M = M map intothemselves, forming zero angle.

3.3.1 Construction of attractors

Define the (spatial) singularity set S(F ) = +ij-Mij . Then U = M \ S(F ) consistsof a collection of open connected sets Z = {Z}. Let U0 := U and

Un :=,

i"!

Fi(Un!1) , U .

We say that x % S(F ) is a non-essential singularity of order m if there exists * > 0and m > 0 such that

card{Z % Z |Un ,B"(x) , Z -= .} 2 1

for all n > m. Denote the set of non-essential singularities of order m by NES(F ;m)and let NES(F ) := +m'0NES(F ;m) be the set of all non-essential singularities.Define ES(F ) = S(F ) \ NES(F ) to be the collection of essential singularities.Observe that there is a natural extension of F to V0 = U + NES(F ). In thefollowing we let F denote the extended map. As above we define

Vn :=,

i"!

Fi(Vn!1) , V0 .

Let X = ,n'0Vn and D = $+N #X . Clearly F (D) = D. The set Y = X is called the

physical (or spatial) attractor of F , and A = $+N #Y = D is the extended attractor

of F .

Proposition 3.9 F |D is continuous.

Proof. The set D intersects non-essential singularities only. Hence we mustshow that if x is a non-essential singularity in D, then the extension of each Fi toNES(F ) is continuous at the point x. Choose m % N and 3 > 0 such that B-(x),Un

59

intersects only one partition element Z % Z for n > m and let y % B-/2(x) , D.Then B-/2(y) $ B-(x) and so B-/2(y) , Un intersects the same partition elementZ. So x and y are mapped by the same a"ne map Fi|Z for each i % %. The claimfollows. !

In general, the map F does not have a continuous extension to A, but only toA \ ($+

N # ES(F )). Actually, if x % ES(F ) , Y lies on the boundary of severalcontinuity partitions for Fi, then there are several extensions of F to (i, x). Thuswe can continuously extend F to A only when the essential singularities do notintersect the attractor (are removable).

3.3.2 Symbolic Coding

If the singularities can a!ect the dynamics only for a finite number of iterations,then the dynamics can be well approximated by a topological Markov chain.

Definition 3.2 We say that singularities are removable if there exists m % N suchthat S(F ) = NES(F,m).

The physically most relevant observables ) : M ( R are those that are deter-mined by avalanches. We say that ) is an avalanche observable if it is constant oncontinuity domains [i]#Mij .

Theorem 3.10 If singularities are removable, then the map F is well-defined andcontinuous on A and there is a topological Markov-chain ($+

A,#+A) and a continu-

ous semi-conjugacy g : A ( $+A such that for all x, y % A and for all avalanche

observables ) we have:

g(x) = g(y) 9 )(Fn(x)) = )(Fn(y)) *n & 0 .

The Markov-chain is determined by a matrix A which has a maximal eigenvalueequal to N .

Remark 3.3 It is clear that all properties related to distribution of avalanche size,duration, area, etc. are invariant under a semi-conjugacy such as this. Observe thatfor each avalanche observable ) on A, there is a unique observable )# : $+

A ( Rsuch that ) = )# / g. Suppose we have a measure µ on A, and let . = g&µ. If ) isan avalanche observable on A, then the statistical properties of ) with respect to µare equivalent to the statistical properties of )# with respect to .. In this (#+

A ,$+A)

is a good approximation to F |A. The coding gives estimates on entropy and growthof periodic points, but these estimates are asymptotically no better than what we getfrom the trivial semi-conjugacy M ( $+

N .

60

Proof. Singularities are removable so there exists an integer m % N such that%s / Fm(M) only intersects trivial singularities. Let X1, . . . , Xs be the closureof the connected components of %s / Fm(M). F is well defined and continuous onthese components. Let Y1, . . . , Ys be the intersections of the components X1, . . . , Xs

with Y. We construct the partition R = {[i] # Yk} and enumerate it so thatR = {R1, . . . , Rr}, where r = Ns.

Let A = 6aij6 be the r#r matrix defined by the rule: aij = 1 if F (Ri),Rj -= .,and aij = 0 otherwise. A sequence R.0R.1 . . . is legal if a.t"1.t = 1 for all t % N.Define g : A ( $+

A by g(x) = (!0!1 . . .!t . . . ), where F t(x) % R.t . To prove thatg is surjective it su"ces to show that for each legal sequence R.0R.1 . . . , there is apoint x % %s / Fm(M) such that F i(x) % R.t for all i % N. Note that each ! canbe written as a pair (t, k), where t % {1, . . . , N} and k % {1, . . . , s}. Hence we canwrite

%;

n=0

F!n(R.n) =%;

n=0

F!n([tn]# Ykn) = {t}#%;

n=0

F!1t0 / · · · / F!1

tn"1(Ykn) .

The continuous image of a connected set is connected, so for each i = 1, . . . , N andeach k = 1, . . . , s there is a unique l % {1, . . . , s} such that Fi(Xk) $ Xl. Thisimplies that we have a nested sequence

Y0 $ F!1t0 (Yk1) $ F!1

t0 / F!1t1 (Yk1) $ . . .

and hence the intersection is non-empty.It is clear that gR is continuous (see [49] for details). Since the partition R is a

refinement of the continuity partition the conjugacy will be injective up to the classesof points that follow the same continuity domains. Hence if )(Fnx) -= )(Fny) forsome avalanche observable ) and some n & 0, then g(x) -= g(y). !Remark 3.4 Suppose we modify the Zhang model by using a full shift ($N ,#N )as the excitation factor. It is then possible that the modified map F is injective on$N # Y. Since we have strict attraction in the spatial factor after a fixed numberof iterations it is clear that we can then obtain an injective coding, and hence atopological conjugacy. However, if we make this modification it is not clear that$N # Y equals the set

* =%;

n=!%Fn($N #M) .

In fact if the maps Fi|Y are all injective, then F |" is invertible, but F |#N)Y istypically non-invertible. The reason for this is that, due to contraction, a pointx % Y does not have preimages for all the maps Fi and Fi are invertible only onFi(Y) $ Y. So to obtain invertibility we must turn to the attractor *. From aphysical point of view the spatial attractor is of the great interest, so it is desirable

61

to have an attractor which is a Cartesian product of the Bernoulli shift and thespatial attractor Y.

We can always construct a coding of F |D (even in non-removable case) by choos-ing a partition R = {R1, . . . Rr}, and taking gR : D ( {1, . . . , r}N to be the mapsending a point x % D to the unique sequence ! % {1, . . . r}N such that F t(x) % R.t

for all t & 0. But there is no reason, however, to expect gR(D) to be a topologicalMarkov chain, cf. [6].

3.4 Metric properties

The natural volume on M is given by the product measure of the uniform Bernoullimeasure on $+

N and the Lebesgue measure on M . By iterating this measure (andaveraging) we can construct SRB-measures. However it can happen that the mea-sures constructed are supported on essential singularities, where it is not possibleto define the dynamics in such a way that the measure is invariant. Hence we mustgive some conditions to ensure the existence of SRB-measures. If there is an SRB-measure it is characterized by the fact that its projection to $+

N coincides with theuniform Bernoulli measure. From this it follows that any SRB-measure is a mea-sure of maximal entropy. In situations where the system allows symbolic coding theSRB-measure corresponds to the Perry measure on the topological Markov chain$+

A.

3.4.1 Existence and characterization of SRB-measures

Let m = mu # ms, where mu = µBer is the uniform Bernoulli measure on $+N ,

and ms = µLeb is the Lebesgue measure on M . We say that an invariant Borelprobability measure µ on M has the SRB-property if there exists a measurableinvariant set G $ M such that

1. m(G) > 0

2. mu(%u(G)) = 1

3. All points x % G are future generic with respect to µ, i.e.

1n

n!1"

t=0

)(F tx) (2

) dµ ,

for all x % G and all continuous functions ) : M ( R.

62

For Axiom A attractors one can ensure the existence of measures for which theset of generic points has full Lebesgue measure and this is equivalent to sayingthat the canonical family of conditional measures on the unstable manifolds areabsolutely continuous with respect to the Lebesgue measure. For non-invertiblemaps one can in general only expect the set of generic points to have positivemeasure and hence it is unreasonable to require that m(G) = 1. Condition 2 is(for physical reasons) important in the Zhang model. It means that the statisticalproperties do not depend on the choice of a generic sequence t of excitations. (Thisis always implicitly assumed in the numerical investigations of the Zhang modelthat can be found in the physical literature.) Moreover, condition 2 will be satisfiedfor the SRB-measures that can be constructed by iterating the measure m.

By a standard approach we can give conditions for existence of SRB-measuresthat hold if singularities are removable, but it is not known if these conditions holdin all non-removable situations.

Proposition 3.11 Let (*, Ec) does not belong to the negligible set ) of Theorem3.8. If there exists n & 0, C > 0 and q > 0 such that

*$ > 0,*t & 0 : m$F!t

<$+

N # U'(ES(F ;n)=%2 C$q ,

then there exists a set D $ M (constructed in §3.3.1), which may intersect singu-larities, and a natural extension of F to D such that F (D) = D. Moreover the setD carries an F -invariant Borel probability measure with the SRB-property.

Remark 3.5 Proposition 3.11 is a simple modification of the result of Schmelingand Troubetzkoy [53]. In their paper the conditions for existence are in generaltoo restrictive for the Zhang model. In fact, in Example A of §3.7 we show asituation where the SRB-measure constructed in [53] does not exist, but we clearlyhave existence of a physically relevant measure. The reason for this paradox is thatone in general remove all singular points on the construction of the attractor, evenif there is a natural extension of F to the points of singularity. (Proposition 3.11obviously applies to this example since ES(F ; 5) = ..)

Proof. In [53] it is shown that a piecewise smooth map f with singularity set S hasa measure, not supported on singularities, such that the set of generic points haspositive Lebesgue measure. They require that the following conditions are satisfied:

1. The restrictions of f to each of its continuity domains are di!eomorphismsonto their image.

2. The second di!erentials D2fx does not grow too fast close to singularities.(See [53] for a more precise formulation.)

63

3. f is hyperbolic. In this context this means that there are constants C > 0 and& % (0, 1) such that for all x -% S there is a splitting of the tangent space at xinto subspaces E+(x) and E!(x). There are cones C+(x) and C!(x) aroundE+(x) and E!(x) that are invariant under Dfx and Df!1

x respectively. Theangles between the C+(x) and C!(x) are bounded away from zero, and forall points x that do not intersect singularities in the first n iterations it holds:

6Dxfn(v)6 & C!1&!n6v6 for v % C+(x) ,

and6Dxfn(v)6 2 C&n6v6 for v % C!(x) .

4. There exists C > 0 and q > 0 such that m(f!t(U-(S))) 2 C3q for all 3 > 0and all t % N.

We apply this result to the map F |#+N)(Un\ES(F ;n)). The singularity set for this map

is contained in $+N #ES(F ), so by assumption condition 4 is satisfied. Condition 1

follows from Proposition 3.7, condition two is obviously satisfied since F is piecewisea"ne and condition 3 follows from Theorem 3.5 with E+ = R1 : 0, E! = 0 :RN and C± being the regular cones around them (actually Theorem 3.5 ensureshyperbolicity for some iterate FT , which implies the claim).

In [53] the measures are constructed by iterating m, averaging and taking aweak limit. It is clear that, in the Zhang model, any measure obtained in this waywill satisfy condition 2 in our definition of an SRB-measure. !

If an SRB-measure exists it can be characterized by a number of di!erent proper-ties. From a physical perspective it is reasonable to require that a relevant invariantmeasure should preserve the uniform Bernoulli structure on $+

N . This correspondsto the Lebesgue measure on [0, 1] in the alternative formulation of the map F , andhence to absolutely continuous measure conditional measures on the unstable space[0, 1].

Proposition 3.12 If µ is an SRB-measure on D, then µu := (%u)&µ is the uniformBernoulli measure on $+

N .

Proof. There is a set A = %u(G) of full mu-measure, such that all t % A are genericwith respect to µu. Take a continuous function ) : $+

N ( R. Then2

) dµu = limn$%

1n

n!1"

t=0

)((#+! )tt) =

2) dmu ,

where the left equality holds for t % A and the right one for t % B with B $ $+N

a subset of full mu-measure (from Birkho! ergodic theorem). Since A ,B -= ., weget:

B) dµu =

B) dmu for all continuous functions ). !

64

3.4.2 Measures of maximal entropy

Suppose that there exists an invariant Borel probability measure µ on M . Letµu := (%u)&µ and let {.t} to be the canonical family of conditional measures on thefibers %!1

u ({t}). By the Abramov-Rokhlin formula

hµ(F ) = hµu(#+N ) + hµ(F |#+

N ) ,

where

hµ(F |#+N ;Q) = lim

n$%

1n

2H&t

$ n!1C

k=0

(Ftk"1 / · · · / Ft0)!1(Q)

%dµu(t) ,

for a partition Q andhµ(F |#+

N ) = supQ

hµ(F |#+N ;Q) ,

The supremum is taken over all finite measurable partitions Q of M . This for-mula was originally proved for product measures by Abramov and Rokhlin [1] andextended to arbitrary skew products by Bogenschultz and Crauel [11].

Below we use the notation Ft for the dynamics over a pre-fixed sequence t =(t0t1 . . . ) % $+

N . By n-the iteration we mean the map Fnt = Ftn"1 / · · · / Ft0 .

Theorem 3.13 If µ is an invariant Borel probability measure on M , then hµ(F ) =hµu(#+

N ). In particular, for an SRB-measure hµ(F ) = log N .

Proof. We prove the proposition by estimating hµ(F |#+N ) from above. Let Q be a

partition of M and

Qt0...tn"1 :=n!1C

k=0

(Ftk"1 / · · · / Ft0)!1(Q) =

n!1C

k=0

F!kt (Q) .

Fix 3 > 0 and choose the partition Q such that hµ(F |#+N ;Q) + 3 & hµ(F |#+

N ). Themaps Fi are non-expanding so it follows from the Ruelle-Margulis inequality [32]that

1n

H&t

$ n!1C

k=0

F!kt (Q)

%)( h&t(Ft) = 0.

Moreover the convergence is µu-uniform and so the same holds for the integrals.Another way to see it is via the multiplicity notion of §3.5.2 (then Q should besubordinate to each continuity partition {Mij | j = 1, . . . , qi}):

hµ(F |#+N ;Q) 2 lim

n$%

1n

log max|t|(n

mult(Qt0...tn"1 , supp(.t)).

Therefore hµ(F |#+N ) 2 *. Let * ( 0. !

65

Remark 3.6 Theorem 3.13 is a partial case of Theorem 4 from [36].

We say that an invariant measure µ is maximal if hµ(F ) = sup& h&(F ), where thesupremum is taken over all invariant Borel probability measures on M . It followsfrom Theorem 3.13 that hµ(F ) 2 log N . So if htop(F ) > log N , the variationalprinciple fails (this can happen for piece-wise a"ne systems, see [36, 37]). But weshow in §3.5.2 that the abnormal growth of htop(F ) does not occur in the Zhangmodel, at least for generic values of parameters Ec, *.

Corollary 3.1 Any SRB-measure on D has entropy log N and is hence a maximalmeasure.

Corollary 3.2 Suppose singularities are removable and that µ is an SRB-measureon A. Let g : A ( $+

A be the semi-conjugacy constructed in the proof of Theorem3.10. If (#+

A ,$+A) is topologically transitive, then g&µ is the Perry measure on $+

A.

Proof. A transitive topological Markov chain has a unique measure of maximalentropy. This measure is called the Perry measure [32]. !

3.4.3 Hyperbolic structure

There are several ways to define Lyapunov exponents for the Zhang model. TheZhang model can be represented as a piecewise a"ne map, where Bernoulli shiftis represented as the expanding map t '( Nt mod 1 of the interval (see §3.5.2).Hence it is clear that there is one positive Lyapunov exponent '+

0 = log N . Wedefine the other exponents by introducing the co-cycle T : M ( GL(N, R), definedby T (x) = Lij , where x % [i]#Mij . For x % M and v % RN \ {0} we define

'(x, v) = limn$%

1n

log6T (Fn!1(x)) . . . T (F (x))T (x)v6

6v6 .

It is a general fact that the function '(x, ·) takes at most N di!erent values '!1 (x) &· · · & '!N (x).

Proposition 3.14 For all x % M the Lyapunov spectrum is:

0 > '!1 (x) & · · · & '!N (x)

and for (*, Ec) outside the negligible set ) from Theorem 3.8: '!N (x) > )1.

Proof. From Theorem 3.5 we know that there exists T % N and c % (0, 1) suchthat

6T (FT!1(x)) . . . T (F (x))T (x)6 2 c

66

for all x % M . It immediately follows that '(x, v) 2 T!1 log c < 0.For * & 1/2 and arbitrary Ec or for * > 0 and Ec & (1+*)/(1)*) all linear maps

are invertible, and so for all x % M and all v % RN \ {0} we have '(x, v) & log k,where k = minij minSp(Lij) > 0. !

If there exists a unique SRB-measure, then it follows from the Osceledec theoremthat there are numbers '!1 , . . . ,'!N such that 'i(x) = '!i for Lebesgue almost everyx % M . The numbers '+

0 ,'!1 , . . . ,'!N are the Lyapunov exponents of the Zhangmodel. From Proposition 3.14 it follows that the Zhang model is hyperbolic in thesense that the Lyapunov spectrum consists of:

'+0 = log N > 0 > '!1 & · · · & '!N > )1 .

We see from Corollary 3.1 that the Pesin formula hµ(F ) = '+ holds for any SRB-measure. If there is no SRB-measure then the Lyapunov spectrum should be definedas functions on M :

'i(x) =2

#+N

'i(t, x) dµBer ,

where µBer is the uniform Bernoulli measure on $+N .

3.4.4 Entropy of physical vs. mathematical models

We can reformulate the Zhang system as the map f : B ( B, where B = +i'0f i(M)is a compact f -invariant subset of $+

N#RN . We wish to compare this to the inducedtransformation F : M ( M (cf. Remark 3.1).

If µ is an F -invariant Borel probability measure on M , then there is an associatedf -invariant Borel probability measure µ on B (and vice versa). Abramov’s theorem([15]) relates the entropies of both systems:

hµ(f) = hµ(F ) · µ(M). (3.3)

One does not need to assume ergodicity and can allow degenerations [22], as happensfor the case of Zhang model. In ergodic situation by the recurrence theorem of Kac[15] for a µ-generic point x % M :

1µ(M)

= limn$%

1n

n!1"

k=0

"(F kx), (3.4)

where "(x) = "(i, x) is the avalanche time initiated by addition of ei to x % M (see§3.2.2). If µ = µSRB is a unique SRB-measure, the above point x can be chosen

67

Lebesgue generic. The resulting limit is the average avalanche time " (we discussit in more details in §3.8.1-3.8.2) and we obtain:

hµ(f) = hµ(F )/" resp. hµSRB(f) = hµu(#+N )/" .

In general non-ergodic situation to get equality (3.4) we should integrate theterms in right-hand side and then we again obtain the average avalanche size !"",but now it is the space-average. Substituting this into (3.3) we get:

hµ(f) = hµ(F )/!"". (3.5)

For SRB-measures this formula can be also obtained from Ledrappier-Youngtheorem [38], because we have only one positive Lyapunov exponent.

3.5 Topological entropy

To calculate the topological entropy of F we established in [36] a set of inequalities,using the technique developed by J. Buzzi [16], [17] for piecewise expanding mapsand piecewise isometries, see Appendix A. The contraction in the maps Fij providesdi"culties, so several results were generalized to fit the framework of the Zhangmodel. It is not however true (as was widely believed) that the contraction does notcontribute to topological (contrary to metric) entropy, the corresponding counter-example can be found in [37]. The Zhang model has a feature common to all suchexamples [36], namely angular expansion, but still for most values of the parametersthis abnormal increase of the entropy does not occur.

3.5.1 Growth of the number of continuity domains

Let P = {[i]#Mij} be the partition of continuity for F , and enumerate the elementsso that P = {P1, . . . Pr}. Let

[Pa0 . . . Pan"1 ] :=n!1;

m=0

F!m(Pam) ,

andPn = {[Pa0 . . . Pan"1 ] -= . | am = 1, . . . , r} .

We define the singularity entropy of F by

Hsing(F ) = limn$%

1n

log card(Pn) .

68

Remark 3.7 Define a map g : A ( $+r by letting g(x) be the unique sequence

a0a1 . . . such that Fn(x) % Pan . Then

Hsing(F ) = htop(#+r ; g(A)) .

For piecewise a!ne expanding maps and piecewise isometries it is clear that thisalso equals the topological entropy, but due to the contraction, this is not obviousin the Zhang model. In addition, if singularities are not removable, the map g hasdiscontinuities.

Definition 3.3 Call a point x % S(F ) an unstable singularity if for all i and k -= lwe have: limy$x Fik(y) -= limy$x Fil(y).

Theorem 3.15 If all singularities S(F ),Y are unstable, then htop(F ) = Hsing(F ).

We need the following technical lemma:

Lemma 3.16 If the singularities in Y are unstable, then there exists a constant1 > 0 such that for all $ > 0, x % Y ,Mik and y % Y ,Mil, k -= l, we have:

d(x, y) < $ 9 d(Fi(x), Fi(y)) > $ .

Proof. Suppose that for all $ > 0 there exists x % Mik , Y and y % Mil , Ysuch that d(x, y) < $ and d(Fik(x), Fil(y)) 2 $. There exist sequences {xm} $ Mik

and {ym} $ Mil such that d(xm, ym) ( 0 and d(Fik(xm), Fil(ym)) ( 0. Thesequence {xm} has a convergent subsequence xmn ( z. The point z lies in S(F )and ymn ( z. By the continuity of the maps Fik and Fil we have Fik(xmn) ( Fik(z)and Fil(ymn) ( Fil(z). Since the metric d is continuous on M #M we have

d(Fik(z), Fil(z)) = limn$%

d(Fik(xmn), Fil(ymn)) = 0 .

Hence Fik(z) = Fil(z). Contradiction. !

Proof of Theorem 3.15. Let

[Pa0 . . . Pan"1 ] = [i0 . . . in!1]#K ,

where K $ M is a convex polygon. Fix $ > 0 and set k = [log 1/$]. Let z1, . . . , zm(')

be a $-spanning set for K. Chose t % $+N and define Nk sequences

sr0...rk"1 = (i0 . . . in!1r0 . . . rk!1tn+ktn+k!1 . . . ) % $+N .

Since the maps Fij are contracting it is clear that the set {x = (sr0...rk"1 , zl)} isa (n, $)-spanning set for [Pa0 . . . Pan"1 ], and since the minimum number of balls

69

needed to cover a convex polygon K $ M is bounded by m($) 2 CN$!N we seethat the number of (n, $)-balls to cover [Pa0 . . . Pan"1 ] is bounded by m($)Nk ·#{[Pa0 . . . Pan"1 ]}. Therefore we get an estimate for the number of (n, $)-balls tocover M and so

htop(F ) 2 limn$%

log<C'$!N card(Pn)

=

n= Hsing(F ) .

To see the opposite inequality let A,B % Pn and take x1 = (t, x) % A andx2 = (s, y) % B. Suppose that A -= B and that t0 = s0, . . . , tn!1 = sn!1. Thenthere is m < n such that %s / Fm(x) % Mtmk and %s / Fm(y) % Mtml, k -= l. ByLemma 3.16 there is 1 > 0 such that for all ( < 1:

max{d(%s / Fm(x),%s / Fm(y)), d(%s / Fm+1(x),%s / Fm+1(y))} & ( .

Therefore for $ su"ciently small, no (n+1, $)-ball can contain points of both A andB and the minimal (n + 1, $)-spanning set has at least card(Pn) elements. Thenhtop(F ) & Hsing(F ). !

Theorem 3.17 For the Zhang models: htop(f) = Hsing(f), htop(F ) = Hsing(F ).

Proof. By Remark 3.7 all quantities are topological entropies. The correspondingsystems in the second equality are the Poincare return maps for the transformationsof the first equality. The map F : M ( M is already the return map for f : B ( Bby the very construction.

To achieve the same claim for the symbolic system we extend the partition Pof M to a partition P of B, which on $+

N # (RN'0 \ M) , B equals the product

of the standard partition $+N = +i[i] and the partition of the spacial part by the

hyperplanes {xi = Ec}. Denote by s 2 r + N(2N ) 1) the number of elements ofthe new partition P.

Let B $ B be the f -invariant closure ofA in $+N#RN

'0. Define a map g : B ( $+s

by letting g(x) be the unique sequence b0b1 . . . such that fn(x) % Pbn . We wish toprove that

htop(f) = Hsing(f) = htop(#+s |g(B)) . (3.6)

For this it is su"cient to check that the singularities S(f) , %s(B) are unstable forf (the second equality follows from the definition).

Consider a singular point x % RN'0. Let y, z tend to x by two di!erent domains

of the projected partition P in RN'0. Then we can subdivide {1, . . . , N} = A+B+C,

where yi = Ec + 0, zi = Ec ) 0 (with the obvious notations instead of limits) fori % A, yi = Ec ) 0, zi = Ec + 0 for i % B and yi, zi belong to the same side of Ec

70

for i % C. Then for S = S(y)) S(z) we have:

Sij =

-..../

....0

*) 1, if i = j % A,1) *, if i = j % B,1!"2d , if d!(i, j) = 1, j % A,

"!12d , if d!(i, j) = 1, j % B,

0 otherwise.

We should check that S · v -= 0 for any vector v % RN'0 with components vi = Ec for

i % A + B (this implies that x is unstable). Note that other components vi, i % Cdo not contribute to the product.

Let i be a site from A with not all neighbors from A (if the set A is emptyconsider B). Denote the number of A-neighbors of i by kA < 2d and the numberof B-neighbors by kB & 0. Then (S · v)i = Ec(2d) kA + kB) "!1

2d -= 0.Thus Theorem 3.15 implies (3.6). Moreover, the same reasons yield a more

general statement. Namely, since g is a semi-conjugacy, we get:

htop(f |K) = htop(#+s |g(K)) (3.7)

for any subset K $ B (recall that g may have discontinuities, but our argumentsare not injured by this fact). This subset needs not to be invariant, and in this casewe should use Bowen’s definition of entropy [12]. Thus Katok’s entropy formula[31] implies that hµ(f) = hg#µ(#+

s ) for all f -invariant measures µ, which are notsupported on singularities.

If the system (M, F ) possesses a measure µ of maximal entropy, then by theobtained result the second claim of the theorem follows from (3.5) and Kac’s theorem[47]. In general, we can apply the above arguments to the partition of M by thesubsets of equal return times and using the fact, that both returns of f and g havethe same combinatorics, we get: htop(F ) = htop(#+

r |g(A)) (see, for instance, theloop equation approach [46]). !

3.5.2 Evaluation of topological entropy

It was predicted on the base of variational principle in [6] that topological entropyof the Zhang model is htop(F ) = log N . However this principle does not applybecause the map is not well-defined (continuously) on the whole space (or thanksto non-compactness if we remove the singularities). In fact, there can be no invariantmeasures on the non-singular part at all.

While we support the claim that htop(F ) = log N , it will not be proved in fullgenerality. We start with the asymptotic statement.

Consider the bifurcation diagram on the (E, *) strip (0,+1) # [0, 1), where apoint is critical if in its neighborhood dynamics of the Zhang model can experience

71

e

Ec

Figure 3.1: Shows the avalanche type domains on (!, Ec)-bifurcation diagram for N = 2.

The top domain is Ec > 1+!1!! . The line from infinity to the origin is Ec = !

1!! . We see

infinitely many domains with di!erent avalanches, accumulating to two of the axes.

avalanches of di!erent types. Thus the strip is partitioned into di!erent avalanchetype domains. The partition depends on L, d. For N = 2 the diagram is shown onfigure 3.1.

Note that for all L, d there is the top avalanche type domain representing theshortest avalanche time. For N = 2 it is given by the relation E > 1+"

1!" . Also notethat some domains have *-projection strictly smaller than the interval [0, 1). Thenext statement concerns only the top domain and the avalanche type domains thatare adjacent to the line * = 1.

Theorem 3.18 For the Zhang model: 0 2 htop(F ) ) log N 2 0(E, *), wherelimEc$% 0(Ec, *) = 0 (* fixed) and lim"$1 0(Ec, *) = 0. In the latter case Ec

changes accordingly with * so that (Ec, *) belongs to the same avalanche type do-main (then Ec (1, though this case di"ers from the former).

We can remove the N -rational points from $+N and then represent it as the

72

subset of I = [0, 1] with points n/Nk being deleted (this process does not changethe topological entropy). In fact, we need to remove only points n/N for the oth-ers will be deleted by inverse iterations of the map of I # X with the formulaF (t, x) = (Nt, f[Nt](x)). Thus we represent our system as a piece-wise a"ne par-tially hyperbolic map of the subset of R1+N . This is important for an applicationof the results from Appendix A.

The proof uses the notions of multiplicity and multiplicity entropy, due to G.Keller and J. Buzzi. Given a finite partition Z = {Zk} of M we define

Zn = {[Z0 . . . Zn!1] -= .} ,

where[Z0 . . . Zn!1] = Z0 , F!1(Z1) , · · · , F 1!n(Zn!1) .

Define multiplicity of a partition by

mult(Z) = maxx"M

card{Z % Z |Z ; x} ,

If Z is the continuity partition of the map F we often denote the multiplicity ofthe Z by mult(F ). Then it is clear that mult(Fn) = mult(Zn). The multiplicityentropy of F is (the limit exists by subadditivity, cf. [32])

Hmult(F ) = supZ

limn$%

1n

log mult(Zn) ,

and we see that if Z is the continuity partition, then

Hmult(F ) = limn$%

1n

log mult(Fn) .

It is clear that Hmult(F ) = 0 if the singularities are removable, so htop(F ) =log N by the result in Appendix A. For big Ec 5 1 the singularities are generallynon-removable, but still we have the same e!ect asymptotically:

Proof of Theorem 3.18. We take 0(Ec, *) = Hmult(F ). Since the singularitiest = n/N % I of the map #+

N : t '( Nt do not intersect in inverse iterations, themultiplicity growth is only due to the spacial maps Fi : M ( M . Thus usingthe notation Ft for the dynamics over a prefixed sequence of excitations we obtainHmult(F ) = supt"#+

NHmult(Ft).

To show the first claim let us notice that when * = const, but Ec ( 1, thenthe avalanches map the critical part of the boundary {7i : xi = Ec} $ -M far from-M , namely to the distance 3 1(*, L, d)Ec, see Figures 3.16 for N = 2 and 3.17 forN = 3. To reach again the boundary and experience avalanche we need many shiftsFi by the basic vectors ei. Thus the singularity can meet only after big number

73

of iterations. Since the initial picture of singularities has bounded multiplicity (for(*, Ec) from the top avalanche type domain), the multiplicity decreases at least ask/Ec so that it vanishes in the limit.

To prove the second claim we use the inequality Hsing(F ) 21

+i(F ) fromAppendix A.3. If the avalanche type domain is fixed, the number of compositionsof matrices S(x) in one avalanche (see §3.2.3) is bounded. Every such a matrixtends to identity when * ( 1. Thus all the linear parts Lij of avalanche maps Fij

tend to identity and so angular expansions +i(F ) tend to zero. !Notice that if Ec is fixed, but * ( 1, then the number of avalanches has unlimited

grow and the previous argument do not work. However, due to estimates of §3.2.2 onthe maximal avalanche length "m we conclude that 0(Ec, *) ( 0 as either Ec (1or * ( 1, both quantities being related by the constraint Ec & N(1)*)! 1

2 diam(!)!/

for some # > 0 (we don’t require, but allow N ( 1 as well). This statement isstronger than in the theorem.

Now we are going to prove vanishing of 0(Ec, *) a.e. in the finite part.

Theorem 3.19 For generic (Ec, *) we have: htop(F ) = log N .

We are going to prove the statement not only for the Zhang model, but also fornearly Zhang models. By this we mean the following. The map F is a bundle overthe Bernoulli shifts #+

N with factors Fi being piece-wise a"ne partially contractingmaps. That is M = +jMij and for Fij = Fi|Mij we have Fij(x) = Lij(x) + bij ,bij = Lijei. We are going to make arbitrary small generic perturbations of thematrices Lij (still with spectrum within unit ball) and vectors bij (one should carethat Mij are mapped into M) and prove the statement for this modified system.

Due to round-o! errors there is no much di!erence between the original andthe perturbed systems in computer simulations. And from the point of view of theexperiment such perturbations (instrument instability) are indispensable – look at[52] for the discussion of physical relevance of variation of parameters as the noise.Proof of Theorem 3.19. We claim that for generic (Ec, *) the singularitiesdo not multiply. Actually, some intersections of singularities are deformable as wevary the parameters, just by the transversality reasons, but the other disappearwith small perturbations.

Namely, for a multi-index # = (/1, . . . ,/k), /s = (is, js) coding an orbit, denoteF/ = F%k / · · · / F%1 the corresponding map along the orbit. The singularities ofthis map are: Sing(F/) = +k

r=1F!1/[r]

Sing(F%r ), where #[r] = (/1, . . . ,/r).If z0 % Sing(F/), then its orbit (with multi-possibilities due to singularities:

mapping a singular point we extend the components of the map F/ in various ways)meets several singularity planes, i.e. for some cuts #s = #[rs] of #, s = 1, . . . ,m, wehave the following system:

F/1(z0) = z1, . . . , F/m(z0) = z1, lq1(z1) = 0, . . . , lqm(zm) = 0, (3.8)

74

where lqs(z) are equations for the singularity hyperplanes of the corresponding mapFi (there are also inequalities, which we don’t mention). When m 2 N there areoccasions, when (3.8) has a solution continuously depending on (Ec, *). Howeverfor m > N this is no longer the case. In fact, considering nearly-Zhang models wesee that for generic data the above system (3.8) is characterized by a collection ofnon-trivial polynomial in * equations (for each Ec).

More precisely, the set of (Lij , bij) giving trivial polynomials has positive codi-mension and hence zero Lebesgue measure. Uniting these sets over all choices ofmulti-indices (#1, . . . ,#m) we see that the complement has full measure and dimen-sion and so a generic perturbation yields the data (Lij , bij) from it. Since non-trivialpolynomials have only finite number of zeros, then for a generic nearly Zhang modeland every Ec there is a countable subset of {0 2 * < 1}, so that the correspondingsystems (3.8) have no solutions. This means that multiplicity of Fn

t does not growwith n and so Hmult(Ft) = 0.

Now let us look to the Zhang model. We restrict for simplicity of expositionto the case of two sites N = 2 (the other configurations involve more complicatedcombinatorics). In this case the linear parts of the a"ne maps are compositions ofL1 = 1 + (*) 1)A1 and L2 = 1 + (*) 1)A2 with

A1 =@

1 0)1/2 0

A, A2 = J!1A1J =

@0 )1/20 1

A, where J =

@0 1)1 0

A

(we exclude the obvious matrix 1, see Example A for details). Notice that detL1 =detL2 = *.

Suppose that (3.8) has a continuous solution in some domain of (Ec, *). Then itis algebraic in * and linear in Ec (the latter is because the singularity lines within

one avalanche type domain shift with velocity 1 in the direction of either e1 =7

10

8

or e2 =7

01

8. Di!erentiating this by Ec we obtain:

L,1(z#0) = z#1, . . . , L,m(z#0) = z#m, (3.9)

where L,i = L,i,ti/ · · · / L,i,1 for +i = (+i,1, . . . , +i,ti) is the linear part of F/i (+i

is di!erent from #i because dF/i is a composition of several maps Ls). The pointszk, 1 2 k 2 m, are constrained to the singularity lines and we can suppose theseare the lines {x1 = Ec} or {x2 = Ec} (all other singularities are mapped to themwithin one avalanche). Thus z#k = vk + ,kwk, where vk = e1 or e2 and wi = Jvi isthe other, ,k being an unknown scalar.

Let (k = L!1,k

(vk), 4k = L!1,k

(wk); these vectors depend meromorphically on *.System (3.9) is solvable i! the a"ne lines (k + ,k4k, 1 2 k 2 m, in R2 have acommon point. We can suppose m = 3.

75

Denote by *((, 5) = !J(, 5" the standard symplectic form on R2. The above 3lines intersect jointly i!

*((1, 41)*(42, 43) + *((2, 42)*(43, 41) + *((3, 43)*(41, 42) = 0.

Dividing byD3

k=1 *((k, 4k) and using the fact that *((k, 4k) = det(L!1,k

) = *!|,k|

we get the equivalent equation:

*(51, 52) + *(52, 53) + *(53, 51) = 0. (3.10)

Here 5k = *!|,k|L!1,k

wk = L,kwk, where L( = L(1 / · · · / L(t for " = ("1, . . . , "t) andLk = *!1L!1

k = 1 + (*) 1)Ak is the adjunct matrix for Lk, which gives

A1 =@

0 01/2 1

A, A2 =

@1 1/20 0

A.

Now equation (3.10) holds i! there exist 3 not simultaneously zero numbers21,22,23 such that 21 + 22 + 23 = 0 and 2151 + 2252 + 2353 = 0. Since 5k is apolynomial matrix of degree |+k| and the products of At are always proportional toe1 or e2 (depending on the left-most factor) this last equation is never satisfied ifthe multi-indices +k are di!erent. !

Remark 3.8 To support the usage of nearly-Zhang models note that the wholeparadigm of SOC should allow generic perturbations of the data, for if there isa fine tuning of parameters, the model is unappropriate for physical explanation (ofcourse, we should pass to the thermodynamic limit, but in practice this only meanssome large finite parameters).

Our computer experiments did not expose any exponential growth of multiplicityin the Zhang model (though we see growth in complications of singularities), so wesuggest that Hmult(F ) = 0 and hence htop(F ) = log N always. In addition, by theabove discussion we can disregard these exceptional values of (Ec, *) even if thereare any. This finishes discussion of topological entropy.

3.6 Geometry of the attractor

The construction of the spacial attractor Y can be interpreted as an iterated functionsystem (IFS), where the maps Fi are not a"ne as usually considered, but piecewisea"ne. Hence one might expect that attractors Y are fractal, but with various sizecharacteristics, like dimension and measure, depending on parameters Ec and *.

76

3.6.1 Fractal structure

Computer experiments show that in certain cases the spacial attractor Y has fractalstructure, see e.g. figures 3.2 and 3.3. We clearly see that Y consists of self-similarpieces. However the pieces overlap, making evaluation of the fractal dimensiondi"cult. So we can provide only estimates of the attractor’s size.

Nevertheless we observe from our experiments that Hausdor! dimension dimH(Y)and the Lebesgue measure µLeb(Y) of the attractor grow piece-wise monotonicallywith Ec and *. Thus the following e!ects occur in steps:

• The dimension and the measure of Y vanish.

• dimH(Y) is positive, while the measure is zero.

• Both dimH(Y) and µLeb(Y) are positive.

• The attractor Y contains an interior point.

We will demonstrate the dimensional part in the next section, while we disregardthe observation about the measure. The reason for this is that µLeb is not physicallymotivated and we should look for an SRB-measure.

The experiments show that such a measure exists and has support lying strictlyinside Y. (See figure 3.4 and figure 3.5. In figure 3.4 the attractor Y is shown forthe case N = 2, Ec = 20 and * = 2/3, and figure 3.5 shows the orbit of a randominitial condition. The latter corresponds to the support of the SRB-measure.) Thisis possible because the contraction rate of ft is smaller for the exceptional sequencest % $+

N , than for a generic one. Thus study of the IFS-attractor does not lead toconclusions about ergodicity or uniqueness of the SRB-measure. Still it provides aninformation about spacial distribution of the orbits in the Zhang dynamics.

3.6.2 Dimensional study of the attractor

The fractal properties of Y do not hold for all values of parameters (Ec, *). Anexample where Y has integer dimension is shown in figure 3.4.

It was noted in [6] that Hausdor! (fractal) dimension of the attractor is aboutto increase as Ec grows. The arguments were the following: For bigger Ec thecontraction rate decreases, so the theory of iterated function system (IFS) impliesincreasing of the Hausdor! dimension

DY(*, Ec) = dimH(Y)

as a function of Ec. While this seems to be true, the statement does not hold inprecise sense. For instance, for N = 2, Ec % [ 1+"

1!" ,2

1!" ] the attractor is the set of 3

77

Figure 3.2: Shows the set U10 for N = 2, Ec = 5 and ! = 1/2.

points, while it seems to have non-zero dimension for other parameters (computersimulations clearly show this).

The problem is with the framework of IFS, where usually only conformal mapsare considered and certain regularity of their mapping graph and overlaps is as-sumed. However we will show validity of the claim in the asymptotic sense:

Theorem 3.20 With fixed d, L and generic * we have: limEc$%DY(*, Ec) = N .Moreover, DY(*, Ec) = N for big values Ec 5 1/*.

On the other hand for all * % [0, 1) it holds: limEc$0 DY(*, Ec) = 0.

In the above statement ”generic” means both full Lebesgue measure and secondBaire category. In fact, the equality holds for all * outside a countable set. It seemsthough that the limit statement is valid for all *. Thus we see that DY(*, Ec) is notstrictly monotone in Ec for its large values as one might expect from the argumentscited before the theorem.

78

Figure 3.3: Shows the set U10 for N = 2, Ec = 3 and ! = 1/5.

Proof. Consider first the statement about big energies Ec 5 1/*. Let us start bydemonstrating the idea of the proof on the example N = 2, see Figure 3.16.

The image of the vertical continuity domain M13 of height 1 adjusted to theright-top corner is a trapezium with the slope depending on * (see (3.11) for thenumeration of domains). It is thin – of constant length of horizontal section equal *

near its bottom side, but long – with diameter approximately equal Ec1!"2

4 . Thus ifwe shift 3 C/* times this domain up and then all of the shifted images horizontallyto the right, so that its first coordinate satisfies the inequality Ec ) 1 < x1 2 Ec,then these shifts cover an open domain, including a unit square, in the verticalstrip K1 = {x1 % (Ec ) 1, Ec]} (note that we can leave a copy of the domainsince this corresponds to the shift – dropping of energy (x1, x2) '( (x1, x2 + 1) onM13 $ K1 before the avalanche). An easy calculation shows that such shifts coverthe whole upper part of K1, strictly including the continuity domain M13 $ K1

adjusted to (Ec, Ec) and covering a vertical part of M12. A similar scenario happento the second coordinate. Thus in iterating the dynamics we will always have twocontinuity domains M13 and M23 adjusted to (Ec, Ec) and the adjacent parts of

79

Figure 3.4: Shows the set Y in the top right corner of M for N = 2, Ec = 20 and ! = 2/3.

The dimension of Y is 2 in this example.

M12,M22 lying in the attractor.For general L and d we observe the same picture: With generic * a continuity

domain Mij adjusted to the upper-most corner is mapped under avalanche-map Fij

to the trapezoid-like polyhedron with irrational slopes. Its shifts cover then an opendomain in each of the strips Ki = {xi % (Ec) 1, Ec]} and so after more shifts – theupper part of this strip, whence the statement.

We illustrate this process on Figure 3.17. The 3 domains adjusted to the corner(Ec, Ec, Ec) are mapped into interior of the cube and they have di!erent irrationalslopes (we picture them of zero thickness that corresponds to large values of Ec 51), so that their shifts cover a big open domain near the faces adjusted to the abovecorner.

Consider now the second statement, Ec < 1. To estimate the fractal dimensionfrom above we use the generalization of the Moran’s formula from appendix B. Itimplies that if the IFS f1, . . . , fN satisfies 6fi6 2 $, then the Hausdor! dimensionof the attractor admits the following estimate: DY 2 log N6/ log 1

' , where 6 is themaximal multiplicity of the continuity partitions for fi.

80

13 14 15 16 17 18 19 20

13

14

15

16

17

18

19

20

Figure 3.5: Shows an orbit of a randomly chosen point (i.e. the support of the SRB-

measure) for N = 2, Ec = 13 and ! = 2/3.

Now we claim that as Ec ( 0 we have: $ = max 6fi6 2 (Ec)/ for some# > 0. To see this let us estimate the maximal duration of the avalanche "m =max(i,x)"!)M "(i, x). This quantity tends to 1 as Ec ( 0, but not as fast as"m(Ec, *, %) 3 C1/Ec (see §3.2.2). Namely we state that "m 3 C2 log 1/Ec. Ac-tually, if we drop energy 1 to an arbitrary site from a configuration in M , then ina finite Ec-independent time all the sites become overcritical. They remain over-critical, while the system does not loose a substantial amount of energy. Duringthis process the total energy is dissipating in geometric progression with an aver-age contraction rate 1 ) 1!"

N < 1. So the duration of this stage has asymptoticC3 log(1/Ec). The remaining time to finish the avalanche has a smaller asymptotic.

By §3.2.2 n(Ec, *, %) < C0 for small Ec. Thus the proof of Theorem 3.5 impliesthat for certain Ec-independent constant Cn = kC0 for each sequence of Cn steps inthe avalanche process the product of the corresponding S-matrices will have norm2 c(*) < 1, which is a uniform estimate in Ec < 1. The number of steps in oneavalanche grows as log 1/Ec. Therefore $ 2 C4c(*)C"1

n log 1/Ec 2 exp()# log 1/Ec),

81

where # = 12C!1

n log 1c(") as was claimed in the estimate.

Next we claim that 6 2 7(Ec) with 7 = o((1/Ec)0) *8 > 0. In fact, wedescribed above the avalanche process for small energies. The first stage is finiteand contributes only a bounded number of singularity hyperplanes. In its secondsstage all of the sites are excited, so there the corresponding number of singularityhyperplanes equals the duration. The last stage is shorter of time , = o(log 1/Ec),but the number of singularity hyperplanes grows faster, but still is bounded bye1 log N ! = o((1/Ec)0) (see §3.2.3) for any 8 > 0.

Finally DY(*, Ec) 2 log N6/ log 1' 2

C!+log 2/ log 1/Ec

( 0 as Ec ( 0. !

Corollary 3.3 For big Ec 5 1/* the system (A, F ) is not topologically transitive.

Proof. Suppose (t, x) is a point with the dense orbit in A. We know that for big Ec

the Lebesgue measure of the spatial part Y of the attractor A is a positive number! > 0. It follows from Theorem 3.5 that under iterations with fixed excitationsequence t the volume of the spatial part M decreases in geometric progressionwith the number of avalanches. Thus after a finite number of steps it becomes lessthan !. This iteration will be still a finite number of polyhedra, so that its closuredoes not coincide with Y. Since it contains all the points %s(Fn(t, x)), we obtaincontradiction. !

In the case N = 2 and * > 1/2, the value of Ec starting from which DY(*, Ec) = 2can be calculated precisely because even one shift of the sloped strip mentioned inthe above proof overlaps with itself and is su"cient for obtaining an open domain inthe attractor. This condition * > 1/2 together with Ec 5 1 from the theorem ideo-logically coincide with the su"cient conditions for invertibility of the di!erentials ofavalanche maps (Proposition 3.7). This makes an indication of a relation betweenthis invertibility and fractality of the attractor in the spirit of Ledrappier-Youngformula [38]. This latter is however unappropriate in our situation.

Note on the usage of the Ledrappier-Young formula. This formula, essen-tially used in [6] in the study of the Zhang model, cannot be used for the mapF : D ( D since this map is never invertible (in loc. cit. it was applied to F!1).In addition to invertibility the Ledrappier-Young theorem is based on the SRB-property. For the map F!1 this property is equivalent to absolute continuity ofthe stable foliation for F w.r.t. the measure µ. If the measure has fractal supportthis cannot happen. Therefore all formulas based on this property may turn to bewrong. We demonstrate this in Example A of §3.7.

On the other hand, as we have just shown in the theorem, in thermodynamiclimit Ec (1 the fractality is lost and so the absolute continuity property is restored(but only for the geometric attractor, the support of SRB-measure is smaller!).This however does not help with non-invertibility of the factor ($+

N ,#+N ). Even

82

if we change this factor to invertible two-sided sequences ($N ,#N ), the systemremains non-invertible since not all points of the attractor (which can be quitefat) admit negative iterations (Remark 3.4). In addition, a new negative Lyapunovexponent ) log N in the first factor appears and the formulas exploited in [6] becomecompletely inadequate.

3.7 Examples

In the examples below we consider the one-dimensional Zhang model with two sites,N = 2.

Example A: A computation shows that for Ec & (1+*)/(1)*) we have six domainsof continuity [i]#Mij , i = 1, 2. The domains M1j are given by

M11 = {x % [0, Ec]2 |x1 + 1 2 Ec}M12 = {x % [0, Ec]2 |x1 + 1 > Ec and (1) *)(x1 + 1)/2 + x2 2 Ec}M13 = {x % [0, Ec]2 |x1 + 1 > Ec and (1) *)(x1 + 1)/2 + x2 > Ec}

(3.11)

and the domains M2j are symmetric to these. The maps Fij are of the formFij(x) = Lij(x + ei), where

L11 =@1 00 1

AL12 =

@* 0

1!"2 1

AL13 =

E( 1+"

2 )2 1!"2

* 1!"2 *

F

and

L11 =@1 00 1

AL12 =

@1 1!"

20 *

AL13 =

E* * 1!"

21!"2 ( 1+"

2 )2

F

The maps F11 and F21 correspond to avalanches of size 0, the maps F12 andF22 correspond to avalanches of size 1, and the maps F13 and F23 correspond toavalanches of size 2.

It was discovered in [6] that the physical attractor Y has the following simplestructure:

Y =5<1 + *

1) *,

*

2) *

=,< *

2) *,1 + *

1) *

=,<1 + *

1) *,1 + *

1) *

=6.

for Ec % [ 1+"1!" ,

21!" ] We denote these points by a, b, c so that Y = {a, b, c}. The maps

F1|Y and F2|Y are permutations of Y:

F1|Y =$a b cb c a

%, F2|Y =

$a b cc a b

%.

83

a

b c

a

b c

Figure 3.6: The figure shows the physical attractor Y = {a, b, c} and the maps F1|Y , F2|Yfor Ec = 7/2 and ! = 1/2. The arrows on the picture to the left shows how the points of

Y are mapped under F1, and the picture on the right shows how the points are mapped

under F2.

Figure 3.6 shows the physical attractor Y and the maps F1|Y and F2|Y forEc = 7/2 and * = 1/2. We construct a partition R = {R1, . . . , R6} of $+

N # Y by

R1 = [1]# {a} R2 = [1]# {b} R3 = [1]# {c}R4 = [2]# {a} R5 = [2]# {b} R6 = [2]# {c} .

We let A = 6aij6 be the 6# 6 matrix where aij = 1 if F (Ri) ,Rj -= . and aij = 0otherwise. It is easy to verify that

A =

G

HHHHHHI

0 1 0 0 1 00 0 1 0 0 11 0 0 1 0 00 0 1 0 0 11 0 0 1 0 00 1 0 0 1 0

J

KKKKKKL

It is clear that gR : A( $+A is a topological conjugacy of the maps F |A and #+

A . Thematrix A is transitive and Sp(A) = {)1,)1, 0, 0, 0, 2}. Hence htop(F |A) = log 2.

If µ is the SRB-measure on A, with (%u)&µ being the uniform Bernoulli measureon $+

N , then (gR)&µ is the Perry measure on $+A. With respect to this measure it is

84

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

Figure 3.7: The region in the (!, Ec)-plane where the avalanches are the same as for

Ec = ! = 1/3. The point (1/3, 1/3) is shown in the interior of the region.

easy to see that the average avalanche size is s0 = 1. It then follows from that thesum of the negative Lyapunov-exponents is '1+'2 = log *. For instance we see thatfor Ec = 11/2 and * = 2/3 we have hµ(F |A) > |'1|+ |'2|. So even though F1|Y andF2|Y are invertible, the Ruelle inequality (and therefore the Pesin formula) cannotbe reversed.

We also remark that for Ec = (1+ *)/(1) *) we have Y $ S(F ), so the standardconstruction of SRB-measure will fail in this case. Still there is clearly a naturalinvariant measure.

The example with a trivial physical attractor can be generalized to all N ford = 1. Then Y consists of N + 1 points z0, z1, . . . , zN given by

zn =1 + *

1) *(1, 1, . . . , 1)) en

where e0 = 0 and e1, . . . , eN is the standard basis in RN .

Example B: For Ec = * = 1/3 the dynamics is very simple. The map F1 has twodomains of continuity: M11 = {(x, y) % M | y 2 )2x/3+1/3} and M12 = M \M11.The domains of continuity for F2 are given by symmetry. The maps are given by

85

F11(x) = L11(x + e1) and F12(x) = L12(x + e1), where

L11 =19

@2 32 3

Aand L12 =

227

@2 32 3

A.

We see that all four Fij are mappings to the diagonal line in M . In fact, F11(M11) =F21(M21) = [p1, p2], where p1 = (2/9, 2/9) and p2 = (1/3, 1/3). The interval iscontained in M12 ,M22 since the two lines of singularity intersect the diagonal inthe point (1/5, 1/5). The images of M12 and M22 also coincide and is an interval[p1, p3] $ [p1, p2], where p3 = (22/81, 22/81). This shows that singularities areremovable after one iteration. It is easy to see that for all t % $+

N the dynamics willcontract to the fixed point P = (4/17, 4/17). Hence the attractor of the system isA = $+

N # {P}. The dynamics on the attractor is of course conjugated to #+N .

The example can easily be extended to a neighborhood of (1/3, 1/3) in the(*, Ec)-plane. In the region

max5 1) 2* + 13*2

5 + 12*) 13*2,

*(7*2 ) 6* + 3)7*3 ) 10*2 + 7*) 4

62 Ec 2 min

5 *

1) *,1) 2* + 5*2

1 + 4*) 5*2

6

we have the same avalanches as for Ec = * = 1/3. This region is shown in figure3.7. For all points in the region the maps depend continuously on * and the lines ofsingularity depend continuously on * and Ec. The condition for an atomic spacialattractor is that the images of the domains do not intersect the singularities. ForEc = * = 1/3 the images are bounded away from the lines of singularity, and hencethere exists an open neighborhood of (1/3, 1/3) in the (*, Ec)-plane where the sameholds, i.e. the attractor is of the form $+

N # {P} (where P is a point in M) and thedynamics is conjugated to #+

N .

Example C: Let us consider Ec = 1/3 and * = 1/2. In this case there are 28domains of continuity, and 28 corresponding maps. A computer program is writtento compute the sets Un. The program uses exact calculations of the edges of thepolygons that make up Un, and hence it can be used to give rigorous “proof bycomputer” of removability of singularities. By using the program we obtain thatsingularities are removable. In fact U5 ,S = .. The set U5 consists of 13 connectedcomponents. Figure 3.8 shows the set U5 and the lines of singularity, and figure3.9 is a schematic illustration of how these connected components are situated withrespect to the lines of singularity. The intersection of the connected componentsof U5 with Y are denoted by Y1, . . . , Y13. Then we construct the partition R ={[i]# Yj}, and enumerate the elements so that R = {R1, . . . , R26}, where

R1 = [1]# Y1 R3 = [1]# Y2 . . . R25 = [1]# Y13

R2 = [2]# Y1 R4 = [2]# Y2 . . . R26 = [2]# Y13.

86

Figure 3.8: Shows the set U5 for N = 2, Ec = 1/3 and ! = 1/2. The points on the

attractor are magnified in order to make them visible in the figure, and hence it looks as

if they intersect singularities, but in fact they do not.

We construct the 26 # 26 matrix A = 6aij6 by letting aij = 1 if F (Ri) , Rj -= .,and aij = 0 otherwise. After making the computations, the matrix A becomes asshown in Figure 3.10. The black squares represent ones and white squares representzeros. Direct computation shows that the matrix A is transitive.

Since singularities are removable the map gR : A ( $+A is an avalanche con-

jugacy between FA and #+A . The SRB-measure projects to the Perry measure on

$+A, so it is possible to calculate properties such as average avalanche size. In this

example a computation gives s0 = 123/17. The spectral radius of the matrix A is2, and hence htop(#+

A) = log 2.

Example D: In the previous examples singularities are removable. This is howevernot the always the case. Figure 3.11 shows the set U20 for N = 2, Ec = 7 and* = 1/2. This is an example where singularities are non-removable. We will in thefollowing show that singularities are non-removable for Ec = (3 + *)/(1) *).

Since (3 + *)/(1 ) *) > (1 + *)/(1 ) *), the domains of continuity are given by

87

Y10

Y12

Y8Y6

Y3Y1

Y11

Y13

Y9

Y7

Y4

Y2

Y5

Figure 3.9: Shows how the 13 spatial partition elements are situated with respect to the

lines of singularity.

the formulas presented in Example A. Take the point p = (Ec, Ec ) 1) % M13. SeeFigure 3.12. Observe that p lies on the horizontal line x1 = Ec ) 1, and hencep % S(F ). Clearly p is in the interior of M13 so

F1(p) = F13

$3 + *

1) *,3 + *

1) *) 1

%=

$3 + *

1) *) 1,

3 + *

1) *) 2

%= p) (1, 2) .

Denote q1 := F1(p) and observe the it lies on the singularity line x2 = Ec)1. On theother hand q1 is in the interior of M21, so F2(q1) = p) (1, 1). Denote q2 := F2(q1).This point also lies in the interior of M21. Let q3 := F2(q2) = p) (1, 0). It is clearthat F11(q3) = p, and in this sense p is a periodic point. However, F1(q3) is notwell defined since q3 % -M12 , -M13.

In the following we let !z1, . . . , zn" denote the open convex polygon with edgesz1, . . . , zn. Define B'(p) = !p, p + (0, $), p + ()$, $), p + ()$, 0)". For small $ > 0 wehave B'(p) % M13, and hence

F1(B'(p)) = F13(B'(p)) = !q1, q1 + $a, q1 + $b, q1 + $c"

88

Figure 3.10: Shows the matrix A for the coding of Example C. Black squares are 1-s and

white squares are 0-s.

where

a =$1) *

2, *

%, b =

$1) 4*) *2

4,1 + *

2*%

, c =$)

<1 + *

2=2

,)1) *

2*%

.

The polygon F1(B'(p)) intersects the singularity line x1 = Ec) 1. See Figure 3.12.A simple computation shows that

F1(B'(p)) ,M11 = !q1, q1 + $a#, q1 + $b, q1 + $c"

wherea# =

$0,

< 2*

1 + *

=2%

.

It then follows from the above discussion that

F 1 / F 22 (F1(B'(p)) ,M11) = !p, p + $a#, p + $b, p + $c" .

It is then easy to verify that for all $ > 0 there is 1 > 0 such that B)(p) $%s / F 4($+

N # B'(p)). So for each n % N there is $n > 0 with B'n(p) $ Un. The

89

Figure 3.11: Shows the set U20 for Ec = 7 and ! = 1/2. In this example singularities are

non-removable.

image of B'n(p) under F1 intersects singularities, and its closure contains the pointq1, which thus is an essential singularity. Clearly the points p, q2 and q3 are alsoessential singularities.

3.8 Statistical properties

In order to evaluate the entropy and Lyapunov spectrum of the physical model inthe thermodynamic limit we need to derive several estimates for the asymptoticbehavior of observables like avalanche size, avalanche duration and ”waiting-time”between avalanches. The results are derived using only the uniform Bernoulli mea-sure on $+

N , and hence hold for any SRB-measure and for time-averages.In §3.8.3 we define the thermodynamic limit at the double limit Ec ( 1,

L = d4

N ( 1, contrary to [6] where the thermodynamic limit is defined as thelimit L ( 1 only. As Ec ( 1 we must make a scaling of time in the physicalmodel. Otherwise the influx of energy to the system will go to zero. With this new

90

F1F2

F2

F1

p

q1

q2

q3

B!!p"

Figure 3.12: Illustration of the fact that singularities are non-removable for Ec = 7 and

! = 1/2.

scaling we show that the entropy goes to zero and the Lyapunov spectrum collapsescompletely in the thermodynamic limit.

We do not provide strict mathematical proofs, but still think important to in-clude the discussion of our results from the physical point of view.

3.8.1 Statistics of observables

Let " be the coordinate measuring the duration of avalanche and let ! correspond tothe interval between avalanches (minimal value 1). We will also study the observables – the avalanche size (defined in §3.2.3). While in the first case we consider onlyactual avalanches, so that " > 0, in the second we make distinction between s0 –all avalanches including the trivial case of under-critical state (s = 0) and s+ – theactual avalanches, so that s+ > 0.

The reason for introducing two di!erent avalanche size observables is the follow-ing: s0 plays a crucial role in mathematical investigation of the model (see §3.8.4),while s+ is important from physical perspective. In §3.8.3 we will see that only

91

physical observables allow a thermodynamic limit.Denote by " , !, s0, s+ the corresponding mean time-average quantities, each of

which is a function on the space-factor M and is defined as follows:

#(x) =2

#+N

limk$%

1k

k!1"

i=0

#<f i(t, x)

=dµBer

with µBer being the Bernoulli measure on the one-sided shifts (by Birkho! ergodictheorem the time-average limit exists almost everywhere and is measurable). Thisfunction is invariant in the sense: #(x) = 1

N

1Ni=1 #(f(i, x)).

Whenever the system is ergodic with respect to an invariant measure µ thefunction # is constant µ-a.e. and equals the space (ensemble) average

!#"µ =2

#+N)M

# dµ.

If the system has a unique invariant SRB-measure the function # is constant µBer#µLeb-a.e. and equals the space (ensemble) average

!#" =2

#+N)M

# dµSRB.

We will need the maximal values of these observables in a sequel, which wedenote by "max, !max and smax = max s0 = max s+ respectively. We shall calculatetheir asymptotics in L and Ec.

Denote by 71 3 72 the asymptotic equivalence relation meaning that the ratio71/72 has subexponential grow/decay. We denote the equivalence by =, when thelimit of the ratio is 1.

Lemma 3.21 For Ec 5 1 the maximal avalanche time and size have the asymp-totics: "max 3 L)# , !max = LdEc, smax 3 Ld+)s , where 1( = 1s = 1 for d = 1 and1 < 1( , 1s < d for d > 1.

Proof. Consider at first the simple case d = 1. The maximum avalanche durationand size are achieved when all sites contribute to the avalanche, i.e. their energiesare su"ciently big and there is one site with energy greater than Ec ) 1 to initiatethe avalanche. Actually, if some site has small energy, it will serve as a boundaryand the avalanche wave reflects from it (to be explained below).

Assuming Ec > "1!" we know from Lemma 3.6 that in the avalanche process

each site, whenever overcritical, relaxes until in the next step it receives a su"cientportion of energy to become overcritical and relax etc. In other words, the sites

92

blink, being under- and over-critical in turn. But in this process they make over-critical their neighbors and the process propagates as a wave, with only di!erencethat its front excites new sites, while in the traversed region there remain blinkingovercritical sites.

This wave spreads along the interval B1L = [1, L] $ Z towards its boundary and

then it reflects from it, bearing now relaxation. In fact, as the front wave reachesthe boundary it losses a substantial part of the energy on the boundary sites, whichthus cannot be recovered and remain undercritical for the rest of this avalanche.They influence their neighbors to stop being critical and so forth. Thus we obtainthe reflected wave that, in contrast with the first one, turns overcritical sites torelaxed. When the wave hits itself, the avalanche process stops.

It is clear that the duration of this avalanche is "max = L. The number ofinvolved sites corresponds to the area of the triangle with a side B1

L = [1, L] andheight L/2 (recall that each site is overcritical only half time of its blinking period),whence smax = L2/2.

Consider now the case of dimension d > 1. Here the scenario of maximalavalanche is more complicated and consists of three stages (the proof is similarto the case d = 1, but quite lengthy and will be suppressed). Again the maximumavalanche duration and size are achieved when all sites contribute to the avalanche,though now if some isolated site has smaller energy it serves as a boundary onlyonce but then on the next several waves it receives the required portion of energyand follows the general scheme of motion.

At the first stage a site is excited and it initiates the rhombus-shape wave (acube in the Manhattan metric, see figure 3.13) that spreads to the boundary of thecube Bd

L = [1, L]d $ Zd (in time = L/2 if the center of the rhombus is placed nearthe center of the cube). The second stage begins as the wave reaches the boundaryface and reflects from it (See figure 3.14). The reflected wave is almost momentaryoverthrown by the coming overcritical wave, which again reflects from the boundary,come now deeper into the interior of the cube, but is overthrown too etc. If onelooks along the boundary face, the reflected wave travel along towards a vertexwith preserved form (like a soliton) and then disappears into this vertex (thereoccur strong interactions with other waves in this corner). But if one looks into theperpendicular direction, the collection of reflected waves oscillates (each reflectedwave enters deeper and deeper into the cube) contributing to the avalanche durationthe sum 1 + 2 + 3 + . . . , which stops with the end of the second stage (we do notspecify the sum precisely because after some oscillations the wave front becomesmore and more eroded by the interactions between overcritical and relaxing waves;this impairs the sum and decreases the exponent, but not too drastically).

The third stage begins as the main body of the overcritical sites becomes dis-connected and the avalanches behaves like worms crawling along the high energyfractal-like collection of states. We illustrate this in figure 3.15. From the descrip-

93

Figure 3.13: The picture is a ”snapshot” of the lattice " for d = 2, L = 10, Ec = 7 and

! = 1/2. A single site in the marginally stable configuration has been exited, and a great

avalanche is unfolding in a rhombus shape. This is the first stage of this avalanche. The

white squares are overcritical sites, the gray squares are sites with energy just bellow Ec

and the black squares have energy approximately equal !Ec.

tion of the maximal avalanche process it is clear that the asymptotic exponents1( , 1s do not depend on the energy Ec 5 1. Let us denote (the sequences areincreasing, so the limits exist):

1( = limL$%

log "max

log Land d + 1s = lim

L$%

log smax

log L.

Duration of the first stage of avalanche is 3 L. The above arguments show thatthe exponent 1#( of the second stage is > 1. The last stage can only increase it:1( & 1#( . To see that 1( < d we note that in average the number of critical sites onthe boundary is about 9L 3 L)$ with 0 < 13 < d. Thus we have a constant flowof energy out of the system with the average speed > 1!"

2d 9LEc, and the inequalityEtot 2 LdEc proves the claim.

To estimate the maximal avalanche size exponent 1s consider again the second

94

Figure 3.14: The picture shows a ”snapshot” of the second stage of the avalanche shown

in figure 3.13. We can see the well-shaped (soliton-like) waves of energy near the boundary

and observe how their form begins being eroded near the vertices.

stage of the above scenario. The number of involved sites corresponds to the volumeof the prism 'L over the cube Bd

L with height : = 12"max 3 L)!# , i.e.

Ld+)!s 3 Vold+1('L) =:

d + 1·Vold(Bd

L) 3 Ld+)!# .

Thus 1s & 1#s = 1#( > 1. Inequality 1s < d follows from the inequality from abovefor the duration exponent 1( .

The maximal value of the waiting time is obvious. !

3.8.2 Asymptotic of the statistical data

Now we can study statistics of the avalanche data asymptotically (as for the ther-modynamic limit).

We will need the following technical statement (informal only for Ec > 1):

95

Figure 3.15: The picture shows a ”snapshot” of the third stage of the avalanche shown

in figure 3.13. The energy configuration has a fractal structure where the avalanche can

sustain in regions of high energy.

Lemma 3.22 Almost every (w.r.t. a random excitation sequence) spatial trajectoryreturns to the cube B0 = (Ec ) 1, Ec]N .

Proof. It follows from Proposition 3.4 that K = {x % M |7i : xi % (Ec)1;Ec]} is areturn set, i.e. every trajectory Fn(t, x) meets $+

N#K. Partition the spatial part Yof the attractor according to the hyperplanes collection H = +i,m"N{xi = Ec)m}.

Each such a part can be shifted by an excitation sequence to the cube B0. Theprobability of all such sequences (where the avalanche starts from the set B0 ofmaximal energy in all sites) is positive. Let + > 0 denote the minimum of theseprobabilities over the finite set of all partition elements of Y by H. Then theprobability of not entering B0 in k successive avalanches is less than (1) +)k.

Therefore since the number of avalanches tend to infinity as we iterate the dy-namics, the measure of trajectories staying away from B0 is zero. !

Theorem 3.23 We have the following asymptotic estimates valid as Ec (1 (andN 5 1 fixed) or L = d

4N (1 (and Ec 5 1 fixed):

96

(i) " 3 L)# ;

(ii) ! 3 Ec;

(iii) s0 3 Ld+)s/Ec;

(iv) s+ 3 Ld+)s ,

where 1( , 1s are the same exponents as in Lemma 3.21 (thus 1. = limL$%

log .log L = 0).

Proof. The maximal avalanche size is achieved for a certain configuration of statesV $ M , which we can bound as follows: U1 $ V $ Ud, where

Uj = {x % M |*i :<1) j 1!"

2d

=Ec < xi 2 Ec and 7i0 : xi0 > Ec ) 1}.

Denote also Uj = {x % M |*i :<1) j 1!"

2d

=Ec < xi 2 Ec} > Uj .

To estimate the measure of the sites leading to the maximal avalanche weconsider preimages of $+

N # Uj under the map F . It is clear that one needsk" % [ 2

1!" ,2d

1!" ] di!erent backwards iterations F!is , s = 1, . . . , k", to cover thespatial attractor Y. Since the measure µ is F -invariant, we get for its %s-push-forward: µs(Uj) = +j/k", which is Ec-independent. Thus we get the same exponent(d + 1s) for s+ as for smax.

The same arguments yield the asymptotic of " .To obtain the asymptotic of ! in L we note that since the amount of lost energy

is < CL)$+'Ec (where 13 < d is the quantity from the proof of Lemma 3.21), theaverage remained energy in a site of configuration obtained from a maximal oneafter an avalanche is (LdEc)CL)$+'Ec)/Ld = Ec. Thus the waiting time does notgrow with L.

The asymptotic of ! in Ec is quite di!erent: If N is fixed but Ec grows, thenany state from -M becomes at distance 0N ·Ec after some relatively small numberof iterations.

Let us first demonstrate the idea in the simple case N = 2. For critical energyEc > */(1) *) the picture of avalanches is shown on Figure 3.16. We see that in afew steps of the dynamics the configuration becomes far from -M , i.e. it stronglycontracts in all directions. Actually, it is possible to imagine the situations whenthe point is mapped to the vertical strip and then is shifted horizontally for a longtime by excitations of the first site, but probability of this event exponentially goesto zero as Ec (1. Thus in a relatively short time the point from -M is mappedinto the square [0, 02Ec]2, where the constant 02 is Ec-independent. To achieve theboundary -M again it needs 3 (1) 02)2Ec random excitations.

The similar picture happens for N = 3, see Figure 3.17. In the general caseTheorem 3.5 insures that after some (few) number of steps we get strong contraction,so that the point becomes in the cube [0, 0NEc]N , i.e. far from the boundary -M .

97

Figure 3.16: The figure illustrates how the continuity domains are mapped under F1 and

F2 for N = 2 and Ec > !/(1 ! !).

Figure 3.17: The figure illustrates how the faces of the continuity domains are mapped

under F1, F2 and F3 for N = 3 and Ec > !/(1 ! !). The point of view is at (Ec, Ec, Ec).

98

Thus we need 3 (1 ) 0N )NEc excitations to make it overcritical and this impliesthe claim. Note that the asymptotic for ! is not for the double limit Ec ( 1,L (1, but for two partial limits only.

To obtain the estimate for s0 we need to estimate the conditional measureµs(Uj |Uj), which coincides with the probability that a randomly chosen configu-ration x % Uj and site i satisfy: xi > Ec ) 1. This probability is = b1/Ec. Thusµ($+

N # V ) 3 #"/Ec and the mean avalanche size is !s0" 3 Ld+)s/Ec.This is however the space-average (the arguments below work also for s+, ").

We would obtain the same for the time-average of Lebesgue a.e.-initial condition ifwe have an SRB-measure, or for µ-a.e. if we have an ergodic measure µ. But wecannot guarantee existence of an SRB-measure or an ergodic measure. However, fora non-ergodic measure, we can decompose A into ergodic components, where theBirkho! theorem works. By Lemma 3.22 each ergodic component (with non-trivialcontribution) intersects in its spatial part the top-energy cube B0 and so for each ofit the same asymptotic of space-averages holds with universal exponents but maybedi!erent coe"cients. Therefore we obtain the required asymptotic for s0.

Another way to get the last asymptotic is via the formula s0 = .·0+1·s+.+1 . !

For d = 1, Ec % [ 1+"1!" ,

21!" ] we already have " 3 N = L1 as the theorem states,

but s0 3 N = L1, N ( 1, which shows in the last respect the critical energyEc = 2/(1) *) is small.

In [6] the estimate 1( > 1 was predicted for all d, while this is a feature of thecases d > 1. In the latter cases the analytic calculation of exact values of exponents1( , 1s is a di"cult problem.

Remark 3.9 The di"erence between cases d = 1 and d > 1 demonstrated in thetheorem is known in the physical literature. The former case is usually consideredas the trivial SOC-model.

3.8.3 Thermodynamic limit

By thermodynamic limit of an observable ) we understand the double limit

[)]% = limL$%Ec$%

)

if it exists. It is assumed that for physical observables this limit exists (in [6] onlylimit L (1 was considered, though then the value of energy Ec could serve as anessential parameter, which is not desirable in the SOC-paradigm).

As an example of non-physical observable we expose s0 (in §3.8.1 we called itmathematically relevant): The double limit does not exists because the repeatedlimits are di!erent:

0 = limL$%

limEc$%

s0 -= limEc$%

limL$%

s0 = +1.

99

But s+ and " are good physical observables, for the thermodynamic limits exist:

[" ]% = 1, [s+]% = 1.

From §3.4.4 (use iteration arguments in the topological case) and §3.5.2 we obtain:

[htop(f)]% = [htop(F )/" ]% = 0 and [htop(F )]% = 1.

But with ! the situation is di!erent because the proof (rather than the vaguestatement of part (ii)) of Theorem 3.23 implies: limL$% limEc$% ! = 1, whilelimEc$% limL$% ! is finite. Since ! is definitely physically relevant observable oneneeds the following reparametrization: ! '( !/Ec, which corresponds to contractionof the waiting time via the following ansatz:

We let the energy quantum added at a unit time to the system equal $ = !(instead of 1 as before), but speed up time in the waiting intervals respectively:Ec = E0/!, !new = !!. Then the thermodynamic limit (the space part M can bequantized similarly via L = [l/!] with l a finite length) corresponds to the quasi-classical limit ! ( 0.

The duration of avalanche was suppressed in our definition of dynamics to lengthone. So if we want to find the entropy of the physical system, where each stepof avalanche has time-duration one, we should multiply it by the probability ofdropping energy into the system. For every trajectory this equals .

.+( . This ratiobehaves di!erently as L ( 1 or Ec ( 1, so we need the reparametrizationdescribed above.

In §3.5.2 we calculated the entropy of the ”return” Zhang model htop(F ) =log N . But after reparametrization it changes. Denoting by hZhang the entropy ofthe reparametrized system we get:

hZhangµ = hµu(#+

N ) · ! !new

!new + "", hZhang

top = d · log L ·< !new

!new + "

=max

.

This implies:[hZhang

µ ]% = [hZhangtop ]% = 0.

Therefore the expanding property is lost in the thermodynamic limit for the originalphysical system, as was already noticed in [6] for a bit di!erent situation.

Remark 3.10 Notice that in the reparametrized system !new < " , which is counter-intuitive for certain SOC-examples (sandpile, earthquakes etc, where one expects! 5 "). This indicates that the Zhang system should be modified by introducingthe local contraction of time depending on the avalanche size or speed. We will notconsider such gradient-type models here.

100

3.8.4 Lyapunov spectrum

In §3.4.3 we showed that the Zhang model is hyperbolic with one positive expo-nent and the remainder of the spectrum negative. This hyperbolicity is lost in thethermodynamic limit.

Proposition 3.24 For Ec & */(1) *) we have1N

i=1 '!i = s0 log *.

Proof. We cannot ensure the existence of an SRB-measure, so both the Lyapunovexponents and s0 should be seen as functions on M . From the general theory ofLyapunov exponents we know that

N"

i=1

'!i (x) = limn$%

1n

n!1"

t=1

log det T (F tx) .

From Proposition 3.7 we know that the formula det(Lij) = *sij holds for Ec &*/(1) *). Hence

N"

i=1

'!i (x) =N"

i=1

2

#+N

'!i (t, x) dµBer

=2

#+N

N"

i=1

'!i (t, x) dµBer

=2

#+N

limn$%

1n

n!1"

t=0

log(det T )(F t(t, x)) dµBer

=2

#+N

limn$%

1n

n!1"

t=0

s(t, x) dµBer log *

= s0(x) log * .

!

Corollary 3.4 |'!| = s0N log 1

" ( 0 as Ec (1, but |'!|(1 as L (1.

Thus we should study di!erently the following cases:

1. Ec ( 1, but L (and d) fixed. Since limEc$% s0 = 0, the negative partof the Lyapunov spectrum collapses: limEc$% '!i = 0 for every 1 2 i 2 N . Thehyperbolicity is lost, but the positive exponent '+

0 = log N survives. In particular,the entropy does not collapses.

101

2. L (1, but Ec 5 1 fixed. Here only a bounded piece '!1 , . . . ,'!k of the Lya-punov spectrum collapses, k = const. But the number of elements of this spectrumgrows and in average |'!| (1 . In particular, |'!|max ( 1 as L ( 1. Again,the positive exponent '0 and entropy are preserved and though we loose hyperbol-icity there are many non-degenerate Oscelledec modes. Moreover they prevail overcollapsing modes and so essentially the hyperbolicity is preserved as well.

3. Reparametrized model. This was introduced in §3.8.3 and require the renor-malization: multiplication of waiting time by the function .

.+( along the trajectory.In this case the Lyapunov spectrum collapses to zero in any limit Ec ( 1 andL (1 and the hyperbolicity is completely lost.

Thus the exponential grow of the statistics is suppressed and we can observepower law statistic as is the basic idea of SOC-phenomenon. The correspondingSOC-exponents are related to the asymptotic of the Lyapunov spectrum (as dis-cussed in [6, 7]), but are di"cult to calculate analytically.

3.9 Conclusion

It is of importance to the paradigm of SOC to understand the mechanisms behindthe behavior one observes numerically in the Zhang model and other sandpile mod-els, and the aim of this paper is to provide a first step to a rigorous mathematicalunderstanding of the dynamics of Zhang model.

Due to the singularities and non-invertibility of the model there existed veryfew applicable results, and hence we had to modify known results and develop somenew methods in order to describe the dynamical properties of the model.

The result of this work is that the singularities play a modest role in the sensethat do not change the main dynamical characterizations such as topological en-tropy, entropy of SRB-measures and the hyperbolic structure.

We show that any SRB-measure maximizes entropy, and based one this resultone could hope to develop a thermodynamic formalism for the Zhang model. How-ever, the presence of essential singularities prevents us from coding the system usingMarkov chains, except in some special cases where singularities only e!ect the dy-namics for a finite number of time-steps (singularities are removable).

In the physically interesting values of parameters (Ec relatively great comparedwith (1 ) *)!1) the attractor has a very complicated topological structure andin general cannot be described as a sub-shift of finite type. The e!ects of thesingularities can also be seen in the rich fractal structure of the spacial attractors.

Our analysis of the e!ect of the singularities allows us to take the thermodynamiclimit of the main dynamical quantities, showing that the entropy goes to zero andthat the Lyapunov-spectrum completely collapses to zero (for the re-scaled physical

102

model). From a physical point of view this is interesting. Typically chaotic dynamics(positive entropy and positive Lyapunov exponents) is an indication of exponentialspeeds of mixing (exponential decay of correlations) [5], which is not compatiblewith the power-law statistics of the SOC-hypothesis. The loss of hyperbolicity inthe thermodynamic limit hence supports the critical behavior observed numerically[30], [57], [25].

To conclude: We have shown that the Zhang model is chaotic hyperbolic dy-namical system, where all the entropy is produced by the random driving of thesystem, and due to the singularities the orbit structure is richer than for a topologi-cal Markov chain. The hyperbolicity implies that under weak conditions there existsa SRB-measure (self-organization), and in the thermodynamic limit the hyperbol-icity is lost and we may expect power-law statistics (criticality). In practice thesystems that are studied have finite size and finite critical energy. Hence they arechaotic, but with small entropy. The SOC-hypothesis is that these weakly chaoticsystems have SRB-measures, but that the rates of convergence to these measuresare exponential but slow compared to a unit step in an avalanche, causing whatappears to be power-law statistics.

103

Appendix

A Topological entropy of piecewise a"ne maps

A.1 The Buzzi theorem and its generalization

The statement as it is done in [16] does not apply to our situation. In [17] Buzzinoted that it extends to isometries and contractions. In fact, the assertion holdsalways, but since we cannot make a simple reference we write an adapted proof forthe convenience of the reader.

Theorem 3.25 Let X $ Rd be a bounded polytope and f : X ( X a piecewisea!ne map. Then

Hsing(f) 2 &+(f) + Hmult(f) ,

where&+(f) = lim

n$%supx"X

1n

maxk

log 6%kdxfn6 .

Proof. Let P = {Pi} be the continuity partition and fi := f |Pi be a"ne maps.Fix * > 0 and T = T (*) & d

" log(4

d + 1) such that for n & T we have:

mult(Pn) 2 exp((Hmult(f) + *)n), 6%dfn6 2 exp((&+(f) + *)n).

Take r = r(*) to be compatible with the partition PT (for fT ), i.e. any r-ballintersects maximally mult(PT ) partition elements.

We will prove that each non-empty cylinder C(a) = [Pa0 . . . PalT"1 ] of length|a| = lT can be partitioned into a collection Q(a) = {W} satisfying the followingproperties:

1.1

|a|=lT card(Q(a)) 2 C0 exp((&+(f) + Hmult(f) + 3*)lT )

2. diam(falT"1 / · · · / fa0(W )) 2 r .

105

Let us prove this claim by induction assuming it holds for some l & 0. The baseof induction is obvious and C0 is the minimal cardinality of an r/2-ball cover.

Take a partition element W % Q(a) that is used to cover the cylinder [Pa0 . . . PalT"1 ].By the induction hypothesis it has diameter less than r, so it can be continued tocover a non-empty cylinder of length (l + 1)T in at most mult(PT ) ways. So tocover the cylinders [Pa0 . . . PalT"1Pb0 . . . PbT"1 ] we make a division of W :

W =),

i=1

W #i , 1 2 mult(PT ) .

LetW ##

i = faT l"1 / · · · / fa0(W#i )

andW ###

i = fbT"1 / · · · / fb0(W##i ) .

By the assumption diam(W ##i ) < r for all i = 1, . . . , 1, but the sets W ###

i may havegreater diameter. We need to divide the sets W ###

i so that they have diameter lessthan r, and then pull this refinement back to the partition of sets W #

i .Let L be the di!erential of fbT"1 / · · · / fb0 on W ##

i . We can assume that L issymmetric and take {ek} to be a basis of eigenvectors corresponding to eigenvalues&1, . . . ,&d. Let {vk} be a basis in the vector subspace corresponding to W ###

i . Wecan choose this basis to be orthonormal and triangular with respect to {ek}. DivideW ###

i by the hyperplanes

,j(x) def= !vj , x" = pr4d, p % Z, j = 1, . . . , d .

This defines cells W of diameter less than r. Since ,j(W ###i ) = ,j(L(W ##

i )) hasdiam 2 |&i|r, the number of cells W needed to cover W ###

i is less than or equal to

(4

d + 1)d|&1|+ . . . |&d|+ 2 (4

d + 1)d6%dfn6 2 exp<(&+(f) + 2*)T

=,

where |&i|+ = max{|&i|, 1}.Therefore the total cardinality of the new partition is less than or equal to

mult(PT ) exp<(&+(f) + 2*)T

=exp

<(&+(f) + Hmult(f) + 3*)lT

=

2 exp<(&+(f) + Hmult(f) + 3*)(l + 1)T

=.

This proves the statement. !

The theorem holds as well for most degenerate piece-wise a"ne systems, butthere can be problems with Hmult(f). Namely the latter is not defined if the image

106

of continuity domain contains a boundary face of a continuity domain. But if weassume the image and the faces always meet transversally, no problems occur andthe above theorem applies literally.

In degenerate cases of the Zhang model, the above requirement holds for mostparameters. For instance, if N = 2 and * = 0, then all Ec /% Z satisfy the request.For integer Ec the theory fails, but one can look just to the whole image set, whichhas dimension < N : its Poincare return map is piece-wise a"ne and it satisfiesthe requirements. In the above example N = 2, * = 0 the dynamics is confined totwo one dimensional lines, whence the multiplicity entropy is zero (by dimensionalreasons) and htop(F ) = log 2 in this case.

A.2 Entropy of conformal piecewise a"ne skew-products

We say that an a"ne map is conformal modulo degenerations if the image on somesubspace transversal to the kernel is mapped conformally to its image. A piecewisea"ne map is said to be conformal modulo degenerations if all its a"ne componentsare conformal modulo degenerations. We will assume that degenerations satisfy thetransversality requirement of A.1.

In [36] we noticed that Hmult = 0 for piece-wise a"ne conformal maps. Thiseasily extends to allow degenerations. Now we consider a more general situation ofskew-product systems of Zhang’s type.

Theorem 3.26 Let fi : X ( X be piece-wise a!ne non-strictly contracting andconformal modulo degenerations, i = 0, . . . , N ) 1. Define F : $+

N #X ( $+N #X

by the formula F (t, x) = (#+N (t), ft0(x)), t = t0t1 · · · % $+

N , x % X $ Rd. Then wehave: htop(F ) = log N (so that a-posteriori the variational principle holds).

Remark 3.11 For N = 2 and * = 0 the a!ne components of have rank 1, andhence the above Theorem 3.26 shows that the topological entropy is log N . The sameholds for Ec = * = 1/3.

Proof. It follows from Theorem 3.25 that it su"ce to prove that Hmult(F ) = 0.To achieve the desired equality note that preimages of the time-like singular-

ity planes {t = n/N} never intersect under inverse iterations of F (t, x) = (Ntmod 1, f[Nt](x)). Thus Hmult(F ) = supt Hmult(ft). We will prove that Hmult(ft) =0 for all t % $+

N .A piecewise a"ne map can be considered as an ordered triple (X,P, f), where

X is a polytope in Rd, P = {Pi} is a partition of X made up of pairwise disjoinedpolytopes (with certain faces of boundary included, so that the whole boundary isdistributed between polytopes) and fi := f |Pi : Pi ( X are a"ne maps. We letX # = + Int(Pi) and Sing(f) = X \X #.

For a piecewise a"ne map (X,P, f) and a point x % X construct a piecewisea"ne map (Xx,Px, fx), called the di!erential of f at x, by letting

107

1. Xx = {y % Rd |7*0 > 0 s.t.** % (0, *0) : x + *y % X} $ Rd is the tangent coneto X.

2. Px is the partition of Xx consisting of non-empty sets

Px = {y % Rd |7*0 > 0 s.t.** % (0, *0) : x + *y % P} ,

where P % P.

3. fx : Px ( Xf(x) is the collection of maps

fx(y) = lim"$0+

f(x + *y)) lim'$0+ f(x + $y)*

, Px % Px .

Consider iterated di!erentials and denote fx1,...,xn := (. . . (fx1)x2 . . . )xn . Notethat (fn

t )x = (ftn"1)fnt (x) / . . . (ft1)f1

t (x) / (ft0)x.

Proposition 3.27 There exists a constant C % R+ such that for any subspaceW $ Rd and any (x1, . . . , xr) % X #W r!1 we have:

mult((fnt |W )x1,...,xr ) 2 C

$sup

V*Rd

supy1,...,yr+1

mult(ft|V )y1,...,yr+1

%n, *n & 0,

where the collection of points (y1, . . . , yr+1) runs over X # V r with the conditionrank(y2, . . . , yr+1) + codimV = rank(x2, . . . , xr) + codimW + 1.

The theorem follows from this, because for

µ(r) = limn$%

1n

log supy1"X,y2...,yr"V*Rd

rank(y2,...,yr)+codim V =r!1

mult(fnt |V )y1,...,yr

we have: Hsing(ft) = µ(1) 2 µ(2) 2 · · · 2 µ(d + 1) = 0.To prove the proposition note that when r = 0 we have (in this case we do not

need W,V ):

mult(fnt ) 2

$supx"X

mult(ft)x

%n=

$max

0(j<Nsupx"X

mult(fj)x

%n.

Let r & 1. Consider the continuity partition Ptx1,...,xr

for (ft)x1,...,xr , which is justthe continuity partition P(t0)

x1,...,xr of (ft0)x1,...,xr (one iteration), and let Pn,tx1,...,xr

for(fn

t )x1,...,xr be the iterated partition. Note that the latter is the collection of allnon-empty intersections

P t0 , (ft0)!1P t0 (P t1) , (ft0)

!1P t0 (ft1)

!1P t1 (P t2) , · · · , (ft0)

!1P t0 . . . (ftn"1)

!1P tn"1 (P tn),

108

where P ti are elements of P(ti)x1,...,xr and fP denotes the restriction of the (di!eren-

tial of the) map to the corresponding continuity domain. Every element of thesepartitions is invariant under the shift by vectors from span(x2, . . . , xr). Therefore itintersects the unit sphere S1(x2, . . . , xr)+ in the orthogonal complement. Considerthe induced partition on the sphere and refine it so that every element has diame-ter no greater than 3. Denote by n(Pn,t

x1,...,xr, 3) the minimal cardinality of such a

refinement. Let also m(P(ti)x1,...,xr , 3) be the maximal number of elements of P(ti)

x1,...,xr

that an 3-ball B(y, 3) , S+1 of S1(x2, . . . , xr)+ can meet.Denote by n(P,W, 3), m(P,W, 3) the corresponding quantities in the subspace

W . Then from the above formula for the iterated partition:

n(Pn+1,tx1,...,xr

,W, 3) 2 n(Pn,tx1,...,xr

,W, 3) ·m(P(tn+1)y1,...,yr

, V, 3),

where y1 = fnt (x1), y2 = fn

t#(x2), . . . , yr = fn

t#(xr) and V = fn

t#(W ) with fn

t# =

(fnt )x1,...,xr . Therefore

n(Pn+1,tx1,...,xr

,W, 3) 2 n(Pn,tx1,...,xr

,W, 3) · supV

supy1,...,yr

m(P(tn+1)y1,...,yr

, V, 3),

where the supremum is taken over all V $ Rd and (y1, . . . , yr) % X # V r!1 suchthat codimension of !y2 . . . , yr" in V equals codimension of !x2 . . . , xr" in W .

Since for a fixed 3 the number n(Pt0x1,...,xr

, 3) is finite and

m(P(ti)y1,...,yr

, V, 3) 2 supy"S1(y1,...,yr)$,V

|P(ti)y1,...,yr

,B(y, 3) , V |,

the claim follows from the following statement. Fix i % [0, N).

Lemma 3.28 There exists 3 > 0 (depending only on i) such that for all V $Rd and all (y1, . . . , yr) % X # V r!1, with rank(y2, . . . , yr) < dimV , and y %S1(y2, . . . , yr)+ , V there exists y# % (y2, . . . , yr)+ , V satisfying:

|P(i)y1,...,yr

,B(y, 3) , V | 2 mult((fi|V )y1,...,yr,y!).

This statement, modulo our notations and restrictions to V , is proved in [17]. Theproposition and hence the theorem follow. !

A.3 Estimates on entropy by angular expansion rates

It is possible to estimate the e!ect of angular expansion on the topological entropyof a piecewise a"ne map f : X ( X by its spherizations. Define the piecewisesmooth map d(s)

x f : STxX ( STf(x)X given at x % X # by the formula

d(s)x f(v) =

dxf(v)6dxf(v)6 .

109

For x % Sing(f) and v -% Tx Sing(f) (the tangent cone) we let d(s)x f(v) = lim

"$+0d(s)

x+"vf(v).

For other (x, v) % STX the map is not defined. The angular expansion of f is ex-actly the expansion in the fibers of its spherization.

If dxf is degenerate we restrict to the orthogonal component of its kernel, andconsider the map

dxf |Ker(dxf)$ : Ker(dxf)+ ( Im(dxf) .

Then the map Sx(f) = d(s)x f |Ker(dxf)$ : S Ker(dxf)+ ( S Im(dxf) between (rank(dxf))

1)-dimensional spheres is given by the formula

v '(dxf |Ker(dxf)$(v)6dxf |Ker(dxf)$(v)6 .

For i < d we define

+i(f) = limn$%

1n

sup(x,v)

max0(k(i

log 6%kdvSx(fn)6.

Let m& = minx dim Ker(dxf) and d& = d )m& = maxx rank(dxf), where d is thedimension of X. The numbers +i(f) can be non-zero only for i < d&.

We have: +0(f) = 0. The number +1(f) measures the maximal exponentialrate with which angles can increase under the map f . The numbers +i(f) for i < dmeasures the maximal rate of expansion of the restrictions to i-dimensional spheres.If f is conformal, then +i(f) = 0 for all i.

Theorem 3.29 ([36]). For piece-wise a!ne maps Hmult(f) 21d#!1

i=1 +i(f).

We define the maximal expansion rate

&max(f) = limn$%

supx

1n

log 6dxfn6 ,

and the minimal finite expansion rate

&min(f) = ) limn$%

supx

1n

log 6(dxfn|Ker(dxfn))!16 .

In [36] we show that+i(f) 2 i

$&max(f)) &min(f)

%.

This gives the following result (the same bound holds for htop(f)):

Theorem 3.30 For a piecewise a!ne map f it holds:

Hsing(f) 2 &+(f) +d&(d& ) 1)

2

$&max(f)) &min(f)

%.

110

B Generalization of Moran’s formula

Consider an IFS (Md, f1, . . . , fN ) on a Riemannian manifold M , where the mapsfi can possess singularities, but we assume that they are mild in a sense that thenumber of continuity domains is finite, any of them has piece-wise smooth boundaryand the map, restricted to any of the domains, smoothly extends to the adjacentsingularities, so that the number of continuity domains for fi is finite, each of themis a smooth manifold with boundary and fi has a smooth extension to the closureof every such domain (this is the case the case of the Zhang model).

Remark that the IFS can be interpreted as the dynamical system ($+N #M,F ),

f(t, x) = (#+Nt, ft0x). Attractor Y of the IFS can be defined via the attractor of

the extended system F , which has the form A = $+N # Y.

Let 6 ·6 be the norm on TM generated by the metric on M . Denote

s+i = max

x"M6dxfi6, s!i =

<maxx"M

6dxf!1i 6

=!1.

We assume that the maps are non-degenerate (this is just for simplicity of argu-ments) and strictly contracting, so that 0 < s!i 2 s+

i < 1.Let 5 = max

x"M#{i |x = fi(yi) for some yi % Y} be the maximal multiplicity of

overlaps and "i = maxx"Y

#{y % Y |x = fi(y)} be the maximal multiplicity of self-

overlaps (we assume it is finite) on the attractor. Denote also by 6i the multiplicityof the continuity partition for fi|Y , i.e. the maximal number of continuity domainsintersecting the attractor and meeting at one point of it.

Theorem 3.31 Let D = /, D = 2 be the solutions of the equations

N"

i=1

1!i|s!i |

% = 5,N"

i=1

6i|s+i |

+ = 1.

Then the Hausdor" dimension of the attractor satisfies:

D 2 dimH(Y) 2 D.

In addition to Hausdor! dimension we will need some other dimensional charac-teristics (see [44] for details). Denote by N(X, $) the minimal cardinality of coversof X by balls of radius $. Then the lower and upper box dimensions are defined bythe formula:

dimB(X) = lim'$+0

log N(X, $)/ log 1' , dimB(X) = lim

'$+0log N(X, $)/ log 1

' .

When these quantities are equal, their value is also called fractal dimension.

111

Consider a Borel probability measure µ %M(X) (an SRB-measure on the at-tractor can be taken in the SOC-context if it exists). The upper and lower pointwisedimensions are defined then as

dµ(x) = lim'$+0

log µ(B(x, $))/ log $, dµ(x) = lim'$+0

log µ(B(x, $))/ log $.

When they are equal and constant a.e. the measure µ is called exact-dimensional.This is precisely the case, when suppµ = X and we have equality in the generalchain of inequalities (together with (3.13) below):

ess. inf dµ(x) 2 dimH(X) 2 dimB(X) 2 dimB(X), (3.12)

where by essential infimum we mean its upper bound taken over all subsets U $ Xof measure 1 (and similar for ess. sup dµ(x)). The last two inequalities are knownand the first one follows from the inequality ess. inf dµ(x) 2 dimH(µ) ([44]), wheredimH(µ) = lim'$+0 inf{dimH(Z) |µ(Z) > 1) $}.

Other dimensional characteristics of the measure are defined similarly and sat-isfy:

dimB(µ) 2 dimB(µ) 2 ess. sup dµ(x). (3.13)

Note that dimB(µ) 2 dimB(X), while the quantities ess. sup dµ(x) and dimB(X)are in general incomparable.

The known formulas for the Hausdor! and other dimensions are generalizationsof Moran’s result ([42, 28]) and are based on the Bowen’s equation (using the ideaof coding); in this case one usually obtains exact-dimensionality [44]. In the SOC-context coding becomes problematic in the presence of singularities (unless theproperties of the SRB-measure are clarified) and thus we cannot easily establishexact-dimensionality or formula for the dimension.

We prove instead the inequality of the theorem for all the various dimensionsfrom (3.12) and (3.13), which we denote just by dim(Y):

D 2 dim(Y) 2 D.

Proof. Let us consider at first the upper box dimension dimB(Y). The functionN satisfies the inequalities:

1!i

N(Y, $/s!i ) 2 N(fi(Y), $) 2 6iN(Y, $/s+i ).

The inequality from above is obtained as follows. Let S = {xj} be a $-spanningset, i.e. a collection of points from X with U'(S) = X. Then fi(S) = {fi(xj)}may fail to be a $ · s+

i -spanning set thanks to singularities. Whenever $ < 1, every$-ball intersects maximally 6i domains of continuity for fi meeting Y. Then weneed to add maximally 6i points for each ball U'(xj) intersecting singularities. Theinequality from below is proved similarly.

112

Now we have:Y = f1(Y) + · · · + fN (Y)

and the same for U'-neighborhoods. This implies:

N(Y, $) 2N"

i=1

N(fi(Y), $) 2N"

i=1

6iN(Y, $/s+i ). (3.14)

Denote #($) = N(Y, $)$dimB(Y). This functions grows sub-polynomially:

lim'$+0

log #($)log 1/$

= 0. (3.15)

Lemma 3.32 Let &i > 1 be some numbers and pi > 0 be some probabilities,1Ni=1 pi = 1. Then (3.15) implies:

lim'$+0

1Ni=1 pi#(&i$)

#($)2 1.

Proof. Suppose the lower limit is > 9 > 1. Then for every su"ciently small $there exists i % [1, N ] such that #(&i$) & 9#($).

Denote & = max1(i(N &i. Let C = max'"[1/4,1] #($). Then:

#($) 2 19

#(&i1$) 2192

#(&i1&i2$) 2 · · · 2 19s(')

#(&i1 . . .&is(%)$),

where s($) is the first number such that &i1 . . .&is(%)$ % [1/&, 1]. This number canbe estimated as follows: s($) & ) log $/ log &) 1, whence:

log #($)log 1/$

2 log C ) s($) log 9

log 1/$2 log(C9)

log 1/$) log 9

log &.

Therefore lim'$+0log /(')log 1/' 2 )

log 3log 4

< 0 and we get a contradiction. This proves thelemma. !

Now to obtain the inequality from above for dimB(Y) divide (3.14) by N(Y, $).Denoting ; =

1Ni=1 6i|s+

i |dimB(Y), &i = 1/s+i and pi = 6i|s+

i |dimB(Y)/; we get:

1;2

N"

i=1

pi#(&i$)#($)

.

Thus Lemma 3.32 implies that 1/; 2 1 or

1 2N"

i=1

6i|s+i |

dimY

113

and the first claim dimB(Y) 2 D follows from the contraction |s+i | < 1. The same

arguments show another statement that dµ(x) 2 D.The inequality from below follows from

N(Y, $) & 15

N"

i=1

N(fi(Y), $) & 15

N"

i=1

1"i

N(Y, $/s!i ),

which implies1N

i=11

!i|s!i |dimB(Y) 2 5 and the same for the lower pointwise dimen-

sion: dµ(x) & D a.e.In this case we should define

#($) = N(Y, $)$dimB(Y) or #($) =<ess. inf ) log µ(B(x, $))

=$dimB(µ)

respectively and use

Lemma 3.33 Let &i > 1 be some numbers and pi > 0 be some probabilities,1Ni=1 pi = 1. Then:

lim'$+0

log #($)log 1/$

= 0 =9 lim'$+0

1Ni=1 pi#(&i$)

#($)& 1.

This is proved similarly to Lemma 3.32. The inequalities for the Hausdor! dimen-sion follows now from (3.12). !

114

Chapter 4

Entropy via multiplicity


Abstract: The topological entropy of piecewise a"ne maps is studied. It is shownthat singularities may contribute to the entropy only if there is angular expansionand we bound the entropy via the expansion rates of the map. As a corollary wededuce that non-expanding conformal piecewise a"ne maps have zero topologicalentropy. We estimate the entropy of piecewise a"ne skew-products. Examples ofabnormal entropy growth are provided.

Refrence for this paper: B. Kruglikov, M. Rypdal, Entropy via multiplicity, Discrete

and Continuous Dynamical Systems 16, 395 (2006)

115

1 Introduction

For a smooth map f of a compact manifold the Ruelle-Margulis inequality togetherwith the variational principle [15, 32] tells us that the topological entropy of f isbounded by the maximal sum over positive Lyapunov exponents. For maps withsingularities this result is no longer true and there are examples of piecewise smoothmaps, where the topological entropy exceeds what can be predicted from the rateof expansion.

In this paper we study the class of piecewise a"ne maps. It follows from [16]that for piecewise a"ne maps the entropy is bounded by the rate of expansion andthe growth in the multiplicity of singularities. The latter was shown by J. Buzzito be zero for piecewise isometries [17], but his proof does not generalize to non-expanding piecewise a"ne maps. In fact, we exhibit an example of a piecewise a"necontracting map with positive topological entropy.

We show that the growth of multiplicity is an e!ect caused by angular expansionthat can be estimated by the expansion rates of the map f . As a corollary we obtainthat the topological entropy of a piecewise conformal map can be estimated by theexpansion rate of the map as in the smooth compact case. It follows that piecewisea"ne non-expanding conformal maps have zero topological entropy.

In the second part of the paper we study the topological entropy of piecewisea"ne maps of skew-product type and obtain a formula which bounds the entropyof the skew products in terms of the entropy and multiplicity growth of the factors.The estimate includes a term which indicates that the entropy of a skew-productsystem may be greater than the sum of the maximal entropy of its factors and wegive an example where this is realized.

Our main results have several corollaries for which one can calculate the topo-logical entropy of various classes of piecewise a"ne maps.

2 Definitions and main results

2.1 Piecewise a"ne maps and topological entropy

Definition 4.1 We say that (X,Z, f) is a piecewise a!ne map if

1. X $ Rn

2. Z = {Z} is a finite collection of open, pairwise disjoint polytopes such thatX # := +Z"ZZ is dense in X.

3. fZ := f |Z : Z ( X is a!ne for each Z % Z

The maps fZ are called the a"ne components of the map f . The linear part offZ is denoted f #Z , and if x % Z we denote dxf = f #Z . Let PA!(X;X) be the set of

116

piecewise a"ne maps on X and let Uf = X # , f!1(X #) , f!2(X #) , . . . be the setof points in X with well-defined infinite orbits. Let Zn be the continuity partitionof the piecewise a"ne map fn. We always assume X to be compact.

Since the maps we consider in this paper have singularities (consult [33, 53]) wemust explain what we mean by topological entropy.

Let Un = ,n!1k=0f!k(X #), then Uf = ,n'0Un. Take a metric d defining the

standard topology on X. Let dfn = max0(k<n(fk)&d, and define S(df

n, *) to be theminimal number of (df

n, *)-balls needed to cover Un. Define

htop(f) = lim"$0

limn$%

1n

log S(dfn, *) .

This number is independent of the choice of metric on X and is finite becauseit is bounded by d · supx log(6dxf6 + |Z|). If we have a finite cover of Uf by(df

n, *)-balls, then concentric (dfn, 2*)-balls cover Un. Therefore htop(f) equals the

(n, *)-entropy htop(f |Uf ), which coincides with the (upper=lower by f -invariance ofthe set Uf ) capacity entropy ChUf (f) [44]. This bounds from above the topologicalentropy hUf (f) of non-compact subsets by Pesin and Pitskel’ [45], so that we havehUf (f) 2 htop(f). Since we estimate htop(f) from above, this bounds the otherentropy too.

Remark 4.1 We observe that even in the presence of singularities the propertyhtop(fT ) = Thtop(fT ) holds for T % N. The proof uses the fact that for all * > 0there is a number $(*) > 0 such that Bd(x, $(*)) , Z $ Bdf

T(x, *) , Z *x % Z $ X #,

where Z % Z is a continuity partition (cf. [32]).

2.2 Singularity entropy

One can also measure the orbit growth of a piecewise a"ne map through the growthof continuity domains. The singularity entropy of f is

Hsing(f) = limn$%

1n

log |Zn| .

For non-expanding piecewise a"ne maps htop(f) 2 Hsing(f) (indeed every (d, *)-ballin an element of Zn is a (df

n, *)-ball). More generally:

Proposition 4.1 For f % PA!(X, X): htop(f) 2 Hsing(f).

Proof. Consider the continuity partition Z of f . Subdivide it by hyperplanesinto the pieces of diameter 2 *. The number of hyperplanes does not exceed K ·d/*,K = diamX. Denote the new partition by Z and its f -iteration by Zn. Elementsof this latter have df

n-diameter 2 *.

117

Let Q be a partition element of Zn. The first n iterations of all x % Q experiencethe same sequence of a"ne maps A1, . . . , An!1. Preimages of the added hyperplanesby Id, A1, A2 · A1, . . . , An!1 · · ·A1 form a subdivision of Q by no more than N =nKd/* hyperplanes. N hyperplanes subdivide Rd into no more than CdNd pieces.Therefore

S(dfn, *) 2 |Zn| 2 Cd

$nKd

*

%d· |Zn|

and the claim follows. !The reverse inequality is not true in general, but holds if all singular points are

unstable ([49] or [35] §4.1):

Proposition 4.2 Let f % PA!(X, X). If fZ(x) -= fZ!(x) for all x % -Z ,-Z # withZ -= Z #, then Hsing(f) 2 htop(f).

Proof. We claim that there exists $0 > 0 such that d(f(x), f(y)) & $0 wheneverd(x, y) 2 $0 and x and y are elements of di!erent domains of continuity. Otherwisethere exist sequences {xm} $ Z and {ym} $ Z # such that d(xm, ym) ( 0 andd(fZ(xm), fZ!(ym)) ( 0. Since X is compact, we can extract a convergent subse-quence xmn ( z. The point z lies in -Z , -Z # and ymn ( z. By the continuity ofthe maps fZ , fZ! and the metric d we get: fZ(z) = fZ!(z), which contradicts ourassumption.

Our claim implies that a small (dfn, *)-ball cannot contain elements from two

di!erent domains of continuity for n > 1. Thus S(dfn, *) & |Zn| for n > 1 and

* < $0/2. !If we enumerate Z = {Z1, . . . , ZN} and let g : Uf ( $+

N be the map associatingto x the code (a0a1 . . . ) of its orbit, i.e. fk(x) % Zak , then Hsing(f) is the topologicalentropy of the symbolic system (g(Uf ),#+

N ).It is also possible to show that the singularity entropy Hsing(f) coincides with

the topological entropy htop(f) for generalized polygon exchanges from [27], whenf is a piece-wise a"ne exchange (this follows from Proposition 2.8 of [27] and ourProposition 4.1).

2.3 Expansion rates and multiplicity entropy

Definition 4.2 The multiplicity of the partition Zn at a point a % X is mult(Zn, a) =|{Z % Zn | Z ; a}|, and the multiplicity of Zn is mult(Zn) = supa"X mult(Zn, a).The multiplicity entropy [16] of f is defined as

Hmult(f) = limn$%

1n

log mult(Zn) .

118

Definition 4.3 For f % PA!(X, X) we define

&+(f) = limn$%

supx"Un

1n

max0(k(d

log 6%kdxfn6 .

We also let (the second quantity can equal )1 for non-invertible maps)

&max(f) = limn$%

supx"Un

1n

log 6dxfn6 and &min(f) = ) limn$%

supx"Un

1n

log 6(dxfn)!16 .

Theorem 4.3 For any f % PA!(X;X) it holds:

htop(f) 2 &+(f) + Hmult(f) .

This result is basically due to Buzzi [16]. However, he only proves it for aspecial class of strictly expanding maps, and he considers the entropy of codingHsing(f) instead of htop(f). Hence we modify his proof. In fact, the proof showsthat Hsing(f) 2 &+(f) + Hmult(f) for any piece-wise a"ne f .

2.4 The spherization and angular expansions

For any k-dimensional submanifold Nk $ X and x % N we define the sphericalbundle STN = {v % TN : 6v6 = 1}, where 6 · 6 is the Euclidian norm on everyTxN $ Rd. Let f be non-degenerate, i.e. each a"ne component is non-degenerate.The spherization of f is defined to be the piecewise smooth map d(s)

x f : STxX (STf(x)X given at x % X # by the formula

d(s)x f(v) =

dxf(v)6dxf(v)6 .

For x % Sing(X) def= X \ X # and v -% Tx Sing(X) (the tangent cone) we defined(s)

x f(v) = lim"$+0

d(s)x+"vf(v) (this is related to the first pseudo-di!erential of Tsujii,

see §3.2). For other (x, v) % STX the map is not defined.The angular expansion of f is exactly the maximal expansion of its spherization.

Definition 4.4 For a non-degenerate map f % PA!(X, X) and i < d we define

+i(f) = limn$%

1n

supx"Un

max0(k(i

supv"Sd"1

log 6%kdvd(s)x fn6.

Note that

+i(f) = limn$%

1n

supNi+1*X

supx"Un,Ni+1

supv"STxNi+1

log 6% dv

Md(s)

x fn|STxNi+1

N6 .

119

(N i+1 runs over all local (i + 1)-dimensional submanifolds of X and we choose thenorm 6%T6 = max 6%kT6).

Informally speaking, the number +1(f) measures the maximal exponential ratewith which angles can increase under the map f . Similarly, the number +i(f)measures the maximal rate of expansion of the i-dimensional spherical volume underf . Clearly +0(f) = 0 for any f % PA!(X, X), and if f is conformal, i.e. all thea"ne components of f are conformal, then +i(f) = 0 for all i.

Theorem 4.4 Hmult(f) 21d!1

i=1 +i(f) for any non-degenerate f % PA!(X;X).

The following corollaries are direct consequences of Theorem 4.4.

Corollary 4.1 If f % PA!(X;X) is conformal, then htop(f) 2 &+(f).

We say that a piecewise a"ne map is non-expanding if all its a"ne componentsare non-expanding, i.e. the eigenvalues of the linear part of each a"ne componenthave absolute values not exceeding 1.

Corollary 4.2 If f % PA!(X;X) is conformal non-expanding, then htop(f) = 0.

It is shown in §4.1 that

+i(f) 2 limn$%

supx"Uf

max0(k(i

1n

log 6%kdxfn6 ) i&min(f) 2 i$&max(f)) &min(f)

%,

This gives the following estimate:

Corollary 4.3 For a non-degenerate map f % PA!(X;X) it holds:

Hmult(f) 2 d(d) 1)2

<&max(f)) &min(f)

=.

Hence we see that the topological entropy of a non-degenerate piecewise a"ne mapf can be bounded using only its expansion rates. In fact, we have

htop(f) 2 &+(f) +d(d) 1)

2<&max(f)) &min(f)

=.

It is not clear if the above estimates are optimal, but they show the nature ofincrease of the topological entropy.

Let us call f % PA!(X;X) asymptotically conformal if &max(f) = &min(f).

Corollary 4.4 For asymptotically conformal f % PA!(X;X): htop(f) 2 &+(f).

120

Remark 4.2 It is essential that in our definition of piece-wise a!ne maps we con-sider a finite number of continuity domains. With countable number of domains(this is related to countable Markov chains) the above theorems become wrong. Infact, according to [2] every aperiodic measure preserving transformation can be rep-resented as an interval exchange with countable number of intervals. In particular,there exist piece-wise isometries with infinite number of continuity domains, whichhave positive topological and metric entropies.

2.5 Piecewise a"ne skew products

Definition 4.5 We say that S#T % PA!(X #Y,X #Y ) is a piecewise a!ne skewproduct if it has the form f(x, y) = (S(x), Tx(y)) for some Tx % PA!(Y, Y ).

The following results hold for piecewise a"ne skew products:

Theorem 4.5 If S#T is a piecewise a!ne skew product, then

htop(S) 2 htop(S#T ) 2 htop(S) + Hmult(S) + supx

<&+(Tx) + Hmult(Tx)

=.

where x = (x0, x1, . . . ) is an orbit of S in X and Tx is the dynamics along thisorbit, i.e. Tn

x = Txn"1 / · · · / Tx0 (see §3.3 for details).

From Theorem 4.5 we can deduce several simple corollaries:

Corollary 4.5 If dim(X) = 1 and Tx % PA!(Y ;Y ) are non-expanding and con-formal for all x % X #, then htop(S#T ) = htop(S).

Corollary 4.6 Let X = [0, 1]d and A % PA!(X;X) be defined by x '( Ax mod Zd

for some A % GLd(R). If Tx % PA!(Y ;Y ) are non-expanding and conformal for allx % X #, then

htop(A#T ) = htop(A) = log Jac+A ,

whereJac+ A =

O

4"Sp(A)

max{|&|, 1} .

(eigenvalues & % Sp(A) are repeated according to their multiplicities).

Corollary 4.7 Let $+N = {1, . . . , N}Z%0 and #+

N be the left shift on $+N . Take

T1, . . . , TN % PA!(Y ;Y ) to be non-expanding and conformal. Then for the mapf : $+

N # Y ( $+N # Y , (t, y) '( (#+

Nt, Tt0(y)), we have: htop(f) = log N .

We can consider more general random maps #+N #T with T depending on more

than one (but finite) entries of t. The claim of the theorem still holds.

121

Remark 4.3 The class of piecewise a!ne skew-products of the form

#+N #T : $+

N # Y ( $+N # Y

is physically relevant and appears in the Zhang sandpile model of Self-OrganizedCriticality [6, 7, 8]. The maps Ti correspond to the avalanches and are contracting[35], though not conformal. So we can get an estimate for the entropy.

In general, we cannot ensure existence of Borel probability invariant measures forpiecewise a"ne maps. In fact, there are examples with no invariant measure at all.In this case the variational principle htop(f) = suphµ(f) fails (it can therefore faileven in the presence of invariant measures — take a disjoint union of such a systemand an identity). However we can give an estimate for the metric entropy hµ(f)whenever an invariant measure exists (we should suppose that it is not supportedon singularities, because f is not defined there).

Theorem 4.6 Let S#T be a piecewise a!ne skew product and Tx % PA!(Y ;Y ) benon-expanding for all x % X #. If µ is a S#T -invariant Borel probability measureon X # Y , then

h5#µ(S) 2 hµ(S#T ) 2 h5#µ(S) + Hmult(S) ,

where % : X # Y ( X is the projection to X.

Corollary 4.8 Let A#T be as in Corollary 4.6 with A expanding. If µ is a measureof maximal entropy for A#T on X # Y , then %&µ is absolutely continuous withrespect to the Lebesgue measure on X.

3 Proof of the theorems

3.1 Proof of Theorem 4.3

Fix * > 0 and let T = T (*) % N be such that mult(Zn) 2 exp((Hmult(f) + *)n) forall n & T and

*x % U, n & T, 0 2 k 2 d : 6%kdxfn6 2 exp((&+(f) + *)n)

We suppose that4

d + 1 2 exp(*T/d). Take r = r(*) to be compatible with thepartition ZT , i.e. any r-ball intersects maximally mult(ZT ) partition elements.

We will prove inductively on l that each Z % Z lT can be covered by a familyQZ = {W} satisfying the following properties:

1.1

Z"ZlT card QZ 2 C0 exp((&+(f) + Hmult(f) + 3*)lT )

122

2. diam(f lT (W )) 2 r.

3. *x, y % W : dfT

l (x, y) < * and dflT (x, y) < $(*) with $(*) from Remark 4.1.

The base of induction l = 0 is obvious and C0 2 |Z|(diamX/min{r, *})d.Take a partition element W % QZ that is used to cover the set Z % Z lT . By the

induction hypothesis it can be continued to cover an element of length Z(l+1)T inat most mult(ZT ) ways. So to cover the cylinders Z % Z(l+1)T we make a divisionof W :

W =),

i=1

W #i , 1 2 mult(ZT ) .

Let W ##i = fTl!1(W #

i ) and W ###i = fT (W ##

i ). By the assumption diam(W ##) 2 r, butthe set W ### may have greater diameter than r. Thus we need to divide the setsW ###

i and pull this refinement back to the partition of sets W #i .

The image fT (W ##i ) is the image of one a"ne component of fT . Let LT denote

the linear part of this a"ne component. We can assume that LT is symmetric andtake {ek} to be a basis of eigenvectors corresponding to eigenvalues &T

1 , . . . ,&Td . Let

{vk} be a basis in the vector subspace corresponding to W ###i . We can choose this

basis to be orthonormal and triangular with respect to {ek}. Divide W ###i by the

hyperplanes

,j(x) def= !vj , x" = pmin{r, *}4

d, p % Z, j = 1, . . . , d .

This defines cells W of diameter less than min{r, *}. Since ,j(W ###) = ,j(LT (W ##))has diam 2 |&T

i |min{r, *}, the number of cells W needed to cover W ### is less thanor equal to

(4

d + 1)d|&T1 |+ . . . |&T

d |+ 2 (4

d + 1)d supx"UT

max1(k(d

6%kdxfT 6

2 exp<(&+(f) + 2*)T

=,

where |&|+ = max{|&|, 1}.Therefore the total cardinality of the new partition is less than or equal to

mult(ZT ) exp<(&+(f) + 2*)T

=exp

<(&+(f) + Hmult(f) + 3*)lT

=

2 exp<(&+(f) + Hmult(f) + 3*)(l + 1)T

=.

The elements of the partition QZ have diameter less than * in the metric dfT

l . ByRemark 4.1 each partition element has diameter less than some number $(*) > 0 inthe metric df

lT . This proves the statement.

123


Define the bundles

S(k)TX = {(x, v1, . . . , vk) |x % X, v1 % STxX, vi % STxX , !v1, . . . , vi!1"+}.

They form the spherical towers:

S(d)TX5d)( S(d!1)TX

5d"1)( · · · 52)( S(1)TX = STX51)( X

with fibers S0, S1, . . . , Sd!1 respectively.The spherization d(s)f : STX ( STX induces the maps S(k)f : S(k)TX (

S(k)TX. Although defined on S(k)TX # they extend over the strata of Sing(X) asin §2.4 (modulo spherization this corresponds to the di!erential fx,v1,...,vk"1(vk) ofTsujii and Buzzi [55, 17]):

S(k)f(x, v1, . . . , vk) = lim"1$+0

. . . lim"k$+0

d(s)x+"1v1+···+"kvk

f(x, v1, . . . , vk) =

=7

lim"1$+0

. . . lim"k$+0

f$x +

"k

1*ivi

%, lim"1$+0

. . . lim"k$+0

d(s)

x+Pk

1 "ivif(v1), . . .

8.

In particular, S(d)f is defined everywhere on S(d)TX (but is not continuous).Let Zn

(x,v1,...,vk"1)(S(k)f |S(k!1)f) be the continuity partition of the piecewise

smooth map (S(k)f)n restricted to the fiber %!1k (x, v1, . . . , vk!1). Define

Hsing(S(k)f |S(k!1)f) = limn$%

sup(x,v1,...,vk"1)

1n

log |Zn(x,v1,...,vk"1)

(S(k)f |S(k!1)f)| .

Define mult(Zn(x,v1,...,vk"1)

(S(k)f |S(k!1)f), vk) as the multiplicity of the parti-tion Zn

(x,v1,...,vk"1)(S(k)f |S(k!1)f) at the point (x, v1, . . . , vk) % S(k)TX. Clearly

mult(Zn(x,v1,...,vk"1)

(S(k)f |S(k!1)f), vk) = |Zn(x,v1,...,vk)(S

(k+1)f |S(k)f)|. So if we de-fine the conditional multiplicity entropy Hmult(S(k)f |S(k!1)f) as the growth rate ofthe above multiplicity (as in definition 4.2) we get:

Hmult(S(k)f |S(k!1)f) = Hsing(S(k+1)f |S(k)f).

In particular, we have: Hmult(f) = Hsing(S(1)f |f). In Theorem 4.3 we dealt withpiece-wise a"ne maps. The spherizations S(k)f are not piece-wise a"ne, but theyare spherically piece-wise a"ne along fibers, meaning that they map spherical poly-gons into spherical polygons. In addition, the di!erentials of S(k)f are uniformlycontinuous on continuity domains. Thus the arguments from the proof of Theorem4.3 apply and we can estimate the conditional entropy via the fiber expansion rateand the conditional multiplicity entropy:

Hsing(S(1)f |f) 2 +d!1(f) + Hmult(S(1)f |f) .

124

Doing its once more for Hmult(S(1)f |f) = Hsing(S(2)f |S(1)f) we get:

Hsing(S(2)f |f) 2 +d!2(f) + Hmult(S(2)f |f) .

Applying the same argument d) 1 times yields: Hmult(f) 2 +d!1(f) + · · ·+ +1(f).


The inequality htop(S) 2 htop(S#T ) is obvious since S is a quotient of S#T . Wewill now prove the upper bound for htop(S#T ). Denote dX = dimX.

Let ( be the set of all sequences ((x0, z0), (x1, z1), (x2, z2), . . . ) % (X #RdX )Z%0

satisfying S(xi + *zi) ( xi+1, dxi+"ziS(zi) ( zi+1, xi + *zi % X # as * ( +0. DefineT(x,z) = lim"$+0 Tx+"z and let Yn,x be the collection of non-empty sets

Y(x0,z0) , T!1(x0,z0)

(Yx1,z1) , · · · , (T(xn"1,zn"1) / . . . T(x0,z0))!1(Y(xn,zn)) ,

for x = ((x0, z0), (x1, z1), (x2, z2), . . . ) % (, where the sets Y(xi,zi) are the continuitydomains of the maps T(xi,zi). Let

Pn(x) = {Y % Yn,x |x = ((x0, z0), (x1, z1), . . . ) % (, x0 = x}

be the continuity partition of T on the fiber %!1(x) iterated for n steps along allpossible S-orbits starting from x % X. Clearly

|Pn(x)| 2 mult(Sn, x) supx"$

|Yn,x| .

To simplify notations we will write elements of ( as x = (x0, x1, x2, . . . ), where xi

consists of a point in X and a vector in RdX . If xi % X # the vector zi is not essential.Denote Tn

x = Txn"1 / · · · / Tx0 for x = (x0, x1, x2, . . . ) % ( and let

Hmult(Tx) = limn$%

1n

log mult(Tnx ) .

Lemma 4.7 For a piecewise a!ne skew product S#T it holds:

htop(S#T ) 2 htop(S) + Hfibersing (T |S) ,

whereHfiber

sing (T |S) = supx"X

limn$%

1n

log |Pn(x)| .

Remark 4.4 The statement of the lemma is similar to Bowen’s Theorem 17 [12],but the direct generalization fails, see Example 2.

125

Proof. Assume at first that the maps Tx are non-expanding for all x % X #.This assumption is not crucial, but it simplifies the proof.

Let * > 0 be arbitrary. Denote a = Hfibersing (T |S), and fix / > 0 and m% = [1//] %

N. For all x % X we let

n%(x) = min{n & m% |1n

log |Pn(x)| 2 a + /} .

Let Z be the continuity partition for the map S and Zn correspond to Sn. Themultiplicity functions mult(Z, x) and |Pn(x)| are upper semi-continuous. Conse-quently, the same is true for the function n%(x). So n% := supx"X n%(x) is finite.Then we have 0 < m% 2 n%(x) 2 n% < +1 for all x % X.

Let r% > 0 be compatible with all the partitions Zn, m% 2 n 2 n%, i.e. any ballof radius r% can intersect at most mult(Zn) di!erent elements of the partition Zn

for m% 2 n 2 n%. We can assume that r% < *.Let En denote a (n, r%)-spanning set of minimal cardinality for S in X. For each

x % X consider the singularity partition {Z , ({x} # Y ) |Z % Pn&(x)(x)}. Eachelement of this partition is an open polytope in the fiber {x} # Y , and hence wecan subdivide the partition so that each element has diameter no greater than *.Denote the resulting partition of {x} # Y by Fx. The refinement can be done insuch a way that |Fx| 2 C0/*d · |Pn&(x)| for some constant C0 % R+.

For x % X we let t0(x) = 0 and define recursively

tk+1(x) = tk(x) + |FStk(x)(x)| .

Let q(x) = min{k > 0 | tk+1(x) & n}. We will denote q = q(x). For x % En,z0 % Fx, z1 % FSt1(x)(x), . . . , zq % FStq(x)(x) denote

V (x; z0, . . . , zq) =>w % X # Y

PP d((S#T )t+tk(x)(w), (S#T )t(zk)) < 2*

* 0 2 t 2 |FStk"1(x)(x)|, 0 2 k 2 q(x)?

.

Then +x,z0,...,zqV (x; z0, . . . zq) = Un # Y and for any (n, 4*)-separating set K $X # Y for S#T we have |K , V (x; z0, . . . zq)| 2 1. Thus if K is a maximal (n, 4*)-set, then the cardinality of K is bounded by the number of ways we can choosex, z0, . . . , zq modulo the partitions specified above. For fixed x % X the number 'x

of such admissible combinations satisfies

'x 2q(x)O

k=0

|FStk(x)(x)| .

126

Since q(x) 2 n/m% we have:

log 'x 2 (q(x) + 1) logC0

*d+

q(x)"

k=0

log |Pn(Stk(x)(x))(Stk(x)(x))|

2 n + m%

m%log

C0

*d+ (a + /)

q(x)"

k=0

n(Stk(x)(x)))

2 n + m%

m%log

C0

*d+ (a + /)(n + n%)

Let Q(S#T, n, 4*) denote the cardinality of a maximal (n, 4*)-separating set forS#T in X # Y . We have that

1n

log Q(S#T, n, 4*) 2 1n

log |En|+$ 1

m%+

1n

%log

C0

*d+ (a + /)(1 +

n%

n) .

This yields

limn$%

1n

log Q(S#T, n, 4*) 2 htop(S) +1

m%log

C0

*d+ a + / .

Since the left hand side does not depend on / we can let / ( 0. Then m% ( 1and we get

limn$%

1n

log Q(S#T, n, 4*) 2 htop(S) + a .

Finally let * ( 0.If the maps Tx have expansions, we may need to make a better refinement of

the partitions Fx. But this gives a sub-exponential increase of 'x (see proof ofProposition 4.1) and so does not change the inequality. !

Lemma 4.8 *x % ( : limn$%

1n log |Yn,x| 2 &+(Tx) + Hmult(Tx) .

The proof of Lemma 4.8 is similar to the proof of Theorem 4.3 and will be omitted.Combining Lemmata 4.7 and 4.8 we obtain:

htop(S#T ) 2 htop(S) + supx"$

limn$%

1n

log3mult(Sn, x0)|Yn,x|

4

2 htop(S) + Hmult(S) + supx"$

limn$%

1n

log |Yn,x|

2 htop(S) + Hmult(S) + supx"$

$&+(Tx) + Hmult(Tx)

%.

127


Let µ be an f -invariant Borel probability measure on X # Y . Denote the projec-tion of µ to X by µX = %&µ and let {.x} be the canonical family of conditionalmeasures on the fibers %!1(x). By the generalized Abramov-Rokhlin formula [11](Bogenschutz and Crauel removed restrictions on the the maps S and Tx in theoriginal formula [1]) we have:

hµ(S#T ) = hµX (S) + hµ(T |S) ,

where

hµ(T |S, () = limn$%

1n

2H&x

$ n!1C

k=0

(TSk"1(x) / · · · / Tx)!1(()%

dµX(x) ,

for a measurable partition ( of Y and hµ(T |S) = sup! hµ(T |S, (), where the supre-mum is taken over all finite measurable partitions ( with finite entropy and therefinement

Qn!1k=0(TSk"1(x) / · · · / Tx)!1(() is understood with respect to all orbits

(x, Sx, . . . , Sk!1x) starting at x (as in §3.3).For * > 0 we choose ( of diam(() 2 * such that hµ(T |S) 2 hµ(T |S, ()+*. Clearly

1n

H&x

$ n!1C

k=0

(TSk"1(x) / · · · / Tx)!1(()%2 1

nlog mult(Sn, x)

+ supx"$x

1n

H&x

$ n!1C

k=0

(T(xn"1,zn"1) / · · · / T(x0,z0))!1(()

%.

where (x $ ( is the set of sequences ((x0, z0), (x1, z1) . . . ) with x0 = x. If themaps Tx are non-expanding, then the last term µX -uniformly tends to zero, sinceby Ruelle-Margulis inequality [32] h&x(Tx) = inf! h&x(Tx, () vanishes, where

h&x(Tx, () = limn$%

1n

H&x

$ n!1C

k=0

(T(xn"1,zn"1) / · · · / T(x0,z0))!1(()

%.

In fact, our framework is more general, but the standard proof of Ruelle inequality ase.g. in [39] works here. We should only indicate why singularities do not contributeto the entropy. Since the maps Tx are non-expanding, the image Tn

x ((%) of apartition element (% meets < C0 (independent of *) elements of the partition (unless the iteration enters *-neighborhood of the singularities, in which case weshould subdivide the partition element into no more than K = maxx |P (x)| pieces,P (x) being the continuity partition of Tx, and iterate them. Therefore there are< C0Ks(n) elements of (Tn

x )!1(() meeting (%, where s(n) is the number of entrancesof (% into the *-neighborhood of singularities.

128

By Kac’s theorem this happens with frequency lim s(n)n equal to the measure

<(*) of this *-neighborhood, which tends to zero as * ( 0. Thus the contributionto the entropy h&x(Tx, () does not exceed <(*) log K.

Therefore hµ(T |S) 2 Hmult(S) + * + <(*) log K. Let * ( 0.

4 Proof of the corollaries

Corollaries 4.1, 4.2 and 4.7 are obvious. Corollaries 4.3, 4.4 follow from the followingsection. Corollary 4.5 is implied by the fact that Hmult(S) = 0 if dim(X) = 1 andCorollary 4.8 follows from our Theorem 4.6 and Theorem 3 of Buzzi [16].

4.1 Estimate for the angles expansion rates

Define&+

[i](f) = limn$%

supx"Un

max0(k(i

1n

log 6%kdxfn6.

Obviously &+[i](f) 2 i&+

[1](f) = i · &max(f).

Proposition 4.9 The following estimate holds: +i(f) 2 &+[i](f)) i&min(f).

Proof. Let us first calculate the di!erential of the spherical transformation(s)A : Sd!1 ( Sd!1, (s)A(x) =

Ax

6Ax6 , corresponding to linear A : Rn ( Rn.

Lemma 4.10 If A % GL(Rd), then d<(s)Ax

=(v) = P+

w

7Av

6Ax6

8, where P+

w is the

orthogonal projection along w = (s)A(x).

In fact, d6Ax6 = !(s)Ax, d(Ax)" and so

d<(s)Ax

=(v) =

Av

6Ax6 )(s)A(x) ·

R(s)A(x),

Av

6Ax6

S.

From this lemma we get: 6d<(s)Ax

=(v)6 2 6Av6

6Ax6 2 C · max |Sp(A)|min |Sp(A)| for 6x6 =

6v6 = 1, where the constant C depends on the eigenbasis of A (in non semi-simplecase – normal basis) only. Since this eigenbasis is the same for all iterates of A and we

have a finite number of pieces in f , we get for all v: 6dvd(s)x fn6 2 Cn·

max |Sp(dxfn)|min |Sp(dxfn)|

(with sub-exponentially growing Cn).Therefore the maximal vertical (spherical) Lyapunov exponent for the map d(s)f

at the point (x, v) % STX, i.e. limn$%

1n log 6dvd(s)

x fn6, does not exceed the di!erence

129

'max(x) ) 'min

(x) ('i(x),'i(x) are usual upper resp. lower Lyapunov exponents

of f). This latter quantity does not exceed the di!erence of uniform Lyapunovexponents &max(f)) &min(f).

Similarly, we have: 6%kd<(s)Ax

=6 2 6%kA6

6Ax6kand

6%kdvd(s)x fn6 2 Cn ·

max{|&1 · · ·&k| : &j % Sp(dxfn)}min |Sp(dxfn)|k

with limn$%

1n log Cn = 0 (we take repeated eigenvalues, so the numerator equals

|&1 · · ·&k| if |&1| & . . . |&k| & . . . |&d|}), whence the claim. !Note that +i(f) is conformally invariant in the cocycle sense: The cocycle A :

X ( GL(Rn), x '( dxf , can be changed by any cocycle / : X ( R, A '( / · A.Then the Lyapunov-type characteristics +i(f) do not change.

However if the cocycle / has di!erent upper and lower Lyapunov exponents,then the quantity &max(f)) &min(f) used in the bound is not invariant.

4.2 Proof of Corollary 4.6

It was shown by Buzzi [16] that Hmult(A) = 0. This was proven for strictly ex-panding maps, but the proof extends literally for any non-degenerate A. Thus thebound from above follows from Theorem 4.3.

Let’s prove that htop(A) & Jac+ A. Assume A is semi-simple. Let & be aneigenvalue with |&| > 1 and v the corresponding unit eigenvector. We divide [0, 1]dinto domains k*/|&|n 2 !x, v" 2 (k+1)*/|&|n. A (df

n, *)-ball intersects no more thantwo such domains, the total number of which is less than

4d · |&|n/2*.

The same holds for other & % Sp(A), so the number of (dfn, *)-balls to cover

[0, 1]d is at least C0(4

d/2*)m(Jac+ A)n, where m is the number of eigenvalues withabsolute value greater than 1 and C0 some n-independent constant.

If A % GLd(R) is not semi-simple, the estimates change sub-exponentially, im-plying the same result. Note that the formula of the corollary holds true even inthe case, when A is degenerate, though arguments should be modified.

5 Examples

Let us demonstrate some of the angular expansion e!ects.

Example 1: Let X be a triangle with vertices in a = ()1, 0), c = (1, 0) andf = (0, 1). Divide this triangle in two by taking X1 to be the left triangle withvertices a = ()1, 0), b = (0, 0) and f = (0, 1), see Figure 4.1.

130

b ca

f

ed

Figure 4.1: Shows the domains of continuity and their image in Example 1.

Let X2 be the right triangle with vertices b = (0, 0), c = (1, 0) and f = (0, 1).Let X be compact, i.e. the sides are contained in X, and let X1 and X2 be open.Then Z = {X1, X2} is a finite collection of open disjoined polytopes in X, andX1 +X2 is dense in X.

We define a map S on X # = X1 +X2 by the formula

S(x) =

#A1x + B1 if x % X1

A2x + B2 if x % X2where A1 =

12

@2 )10 1

A, A2 =

12

@2 10 1

A

and B1 = (1/2, 1/2), B2 = ()1/2, 1/2). This maps both X1 and X2 to the trianglewith vertices d, e and f . Observe that Sn(x) tends to the point (0, 1) with exponen-tial speed for all x % Uf . This implies that htop(S) = 0. However the multiplicityof Zn at the point f = (0, 1) is 2n, whence Hmult(S) = log 2.

It is also easy to see that Hsing(S) = log 2. The eigenvalues of A are 12 , 1, so the

map is non-expanding and &+(f) = 0. Changing S(x) '( 12 (S(x) + f) we obtain a

strictly contracting piecewise a"ne map with positive Hsing and Hmult.The growth of multiplicity is produced by angular expansion: on STfX ? S1

the spherization is conjugated to the map 0 '( 20, whence +(S) = log 2.

Example 2: The map in Example 1 can be modified to obtain positive topologicalentropy. We give here an example of a piecewise a"ne non-expanding map withpositive topological entropy in dimension 3, but it is also possible to construct suchan example in dimension 2, see [37].

131

Let Y = [0, 1] and let S : X1+X2 ( X be as in Example 1. For each x % X1+X2

we take a piecewise a"ne map Tx % PA!(Y, Y ). For x % X1 we let Tx = IdY andfor x % X2 we let Tx be the interval exchange

Tx(y) =

#y + 1/2 if y % (0, 1/2),y ) 1/2 if y % (1/2, 1).

The map f = S#T % PA!(X # Y, X # Y ) has three domains of continuity Z1, Z2

and Z3. These domains and the images are shown in Figure 2.As in Example 1 the cardinality of the continuity partition of S grows like 2n,

but in this example we see that if x1, x2 % X are elements of di!erent continuitydomains for S, then d(f(x1, y), f(x2, y)) & 1/2 for all y % Y . Hence the numberof (df

n, *)-balls needed to cover X # Y is at least Zn, where Zn is the continuitypartition of fn. This implies htop(f) & log 2. The opposite inequality follows fromTheorem 4.4, whence htop(f) = log 2.

This example has several interesting aspects. First of all it is an example of anon-expanding piecewise a"ne map with positive entropy. We can easily modify fto make it strictly contracting without changing its topological entropy.

A second important point is that htop(S) = htop(Tx) = 0 for all x % X #, so thethe entropy of the skew product S#T exceeds the combined entropy of its factors.This shows that the term Hmult(S) on the right hand side of the inequality in The-orem 4.5 cannot be removed. Also this justifies Remark 4.4.

Example 3: Let us consider a system with continuity domains as in Figure 4.3.The dynamics f is piece-wise a"ne and is given by the rules:

ABC )( AEF, ADC )( AFE, BDFE )( BDFE,

Figure 4.2: The left figure shows the three domains of continuity for the skew-product S#T in Example 2. Two right figures show the images of the continuitydomains.

132

! !

! ! !

!

!!!!!!!!!!

""

""

""

""

""A

BC

D

E F

Figure 4.3: Example of a system with zero Lyapunov exponents and positive en-tropy.

! !

! !

Figure 4.4: A countable piece-wise isometry consisting of random vertical shifts.

the last map being the identity. Since every point eventually comes into the domainBDFE, all the Lyapunov exponents vanish. But the multiplicity of the point A islog 2 and this easily yields htop(f) = log 2.

This shows that in the estimates of the main theorems we cannot change the dif-ference &max(f)) &min(f) to the maximal di!erence of upper and lower Lyapunovexponents supx"Uf

maxi,j('i(x) ) 'j(x)). However we suggest that the estimate

in Theorem 4.3 can be refined by changing &+(f) to the maximal sum of positiveupper Lyapunov exponents supx"Uf

1'+

i (x) 2 &+(f).

Example 4: Consider the following map f of T 2 to itself. We represent thetorus as a glued square, which is partitioned into countable number of rectangles'i = Ii # [0, 1]. We define f(x, y) = (x + /, y + 2i) if x % Ii. Thus the map is apiece-wise isometry with countable number of continuity domains 'i, see figure 4.4.

The number / is chosen irrational. The intervals Ii (with their lengths li = |Ii|)and the shift lengths 2i are supposed to be su"ciently generic. The value of htop(f)

133

(note that singularity entropy Hsing(f) does not have sense here) depends on thespeed of convergence of the series

1%i=1 li.

Consider, for instance, the case of rapid convergence, when li decrease expo-nentially or at least polynomially, namely li 2 Ci!r for some r > 1 and C % R+.Then the frequency with which an interval of length * meets a singularity of thebase +-Ii $ S1 under rotation by the angle / is at most p" 3 *

r"1r (for exponential

convergence p" 3 * log 1" ). The number of (df

n, *)-balls to cover the torus satisfies:S(df

n, *) 2 c · ( 1" )p'·(n+2). Consequently htop(f) = 0.

On the other hand, if the series1%

i=1 li converges slowly, then the entropymay become positive and even infinite. For example, if li 3 1/i log i(log log i)2,then the frequency with which an *-interval meets [1/*] di!erent intervals Ii underrotation by the angle / has asymptotic #" 3 1/ log log 1

" . Thus choosing the shifts2i and geometry of the decomposition I = +Ii appropriately we may arrive toS(df

n, *) 3 ( 1" )/'·(n+2), which yields htop(f) = 1 (note that f is a skew-product

with vanishing entropies of the base and the fibers).Note that htop(f) = 0 for a piecewise isometry f with finite number of continu-

ity domains [17] and the same holds for conformal non-expanding maps [35] (andCorollary 4.2). For infinite number of domains this fails.

Example 5: In Corollary 4.4 we stated that if a piecewise a"ne map f is asymp-totically conformal, i.e. &max(f) = &min(f), then Hmult(f) = 0. This result doesnot generalize to piecewise smooth maps. This is shown by the following exampledue to Buzzi [17]:

Let X = [)1, 1]2 and define f : X ( X by

(x, y) '( (x/2, y/2) sgn(y)x2) .

We observe that

Jac(f) =@

1/2 0)2x sgn(y) 1/2

Afor y -= 0 ,

so &max(f) = &min(f) = 1/2. However it is easy to verify that multiplicity of theorigin grows like 2n, whence Hmult(f) = log 2.

134

Chapter 5

A piecewise a"necontracting map withpositive entropy


Abstract: We construct the simplest chaotic system with a two-point attractor onthe plane.

Reference for this paper: B. Kruglikov, M. Rypdal, A piece-wise a!ne contracting

map with positive entropy, Discrete and Continuous Dynamical Systems 16, 393 (2006)

135

If f : X ( X is an isometry of the metric space (X, d), then the topologicalentropy vanishes: htop(f) = 0 (for definitions and notations, consult e.g. [32]).

This follows from the fact that the iterated distance dfn = max

0(i<n(f i)&(d) equals

d. If f is distance non-increasing, the same equality holds, and again htop(f) = 0.Whenever f can have discontinuities of some tame nature so that f is piece-

wise continuous, even the isometry result becomes di"cult. In dimension two forinvertible maps, it was proven by Gutkin and Haydn [27]. In arbitrary dimension,Buzzi proved that piece-wise a"ne isometries have zero topological entropy [17].

In the same paper after the theorem (remark 4), it is claimed that the resultholds for arbitrary piece-wise (non-strictly) contracting maps. This latter is wrong,however, and the goal of this note is to present a counter-example.

Example: Let X be a rhombus ADBC with vertices (±1, 0), (0,±1); see the figurebelow. Let O be its center and P,Q,R, S be on the sides as is shown.

!! !

!

!! !

! !#

##

##

#

$$

$$

$$

$$

$$

$$

##

##

##

A

B

C D

P Q

R S

O

Let f be partially defined on the rhombus, namely let it be defined on the interiorof four big triangles forming the rhombus. These triangles are bijectively mappedby f as follows:

ACO )( APQ, ADO )( BRS, BCO )( AQP, BDO )( BSR.

Thus the piece-wise a"ne map is defined.If P,Q and R,S are middle-points of the intervals AC,AD and BC, BD, then

the map is not strictly contracting. But if they are closer to the vertices A andB respectively than to C and D, then f is strictly contracting. In any case, theattractor of the system is the two-point set {A,B}. Notice that the points belongto the singularity set, where the map f is not (uniquely) defined.

Taking 3 = 12 , we observe that the cardinality of minimal (n, 3)-spanning set

satisfies: 2n+2 2 N(f, n, 3) 2 2n+3. In fact, if we partition CD into 2n equal

136

intervals ZiZi+1, then every dfn 3-ball is contained in some triangle AZiZi+1 or

BZiZi+1, and every such a triangle is covered by two dfn 3-balls.

Therefore the topological entropy htop(f) = log 2 is positive. In addition, theLyapunov spectrum is strictly negative at each point (for strict contractions), andno invariant measure exists, so the variational principle breaks.

The result of Buzzi [17] generalizes, however, in the following fashion:

Theorem. Let f be a piece-wise a!ne map with restriction to each continuitycomponent being conformal (non-strict) contraction. Then htop(f) = 0.

Now we can repeat Buzzi’s remark 4 [17]: the proof of his theorem 3 appliesalmost literally to the above case of piece-wise a"ne conformal contracting maps.Therefore we omit the proof.

Remark 5.1 It is obvious that if the attractor consists of one point only, thenhtop(f) = 0. If the phase space X $ R1 is one-dimensional and the map is(non-strictly) contracting, then again htop(f) = 0. We don’t even need to requirepiece-wise a!ne property. This follows from the Buzzi proposition 4 [16], yieldinghtop(f) 2 Hmult(f), where Hmult(f) is the multiplicity entropy because the latteralways vanishes in dimension one.

Thus, our example with two-points attractor and two-dimensional phase-space Xis the simplest possible example of a contracting chaotic system.

137

Part 2:

Stochastic modeling oftemporal fluctuations

139

Chapter 6

Introduction to thestochastic model

Abstract: This chapter is an introduction to the second part of the thesis. Weexplain how to derive an approximate stochastic model of the toppling activityin sandpile models and discuss some fundamental concepts related to self-similarstochastic processes and stochastic di!erential equations. Numerical verificationand applications of the stochastic model are provided in chapters 7 and 8.

141

1 Introduction

In the first part of the thesis we presented descriptions of the BTW and Zhangmodels within the framework of dynamical systems theory, and the results andconclusions where written in terms of mathematical quantities specifically designedto study chaotic maps acting in low dimensions. These results are useful for thegeneral understanding of SOC, but it is di"cult to compare them directly with ex-perimental/observational data, and they do not answer any of the classical questionsregarding SOC.

Traditionally, investigations of SOC focus on the scaling properties of the sta-tistical distributions of certain avalanche observables. For instance, it is commonlybelieved that the probability distributions1 of avalanche size and avalanche durationare power laws:

Psize(s) 3 s!& and Pdur(") 3 "!% .

The exponents . and / vary from model to model, and this has lead to the notionof universality classes, which simply are collections of SOC models with the sameset of critical exponents. However, as computer simulations of sandpiles have be-come more sophisticated it has become evident that the determination of criticalexponents is more di"cult than what was first believed.

One of the problems is the lack of finite-size scaling for avalanche observables.In the BTW model for instance, the shape of the probability distributions dependon the system size a non-trivial way, making it di"cult to extrapolate numericalresults to the thermodynamic/continum limit.

Remark 6.1 Consider a sandpile on a lattice with N#N sites and let Psize(s;N) =s!&GN (s) denote the distribution of avalanche sizes. Simple finite size scalingmeans that we can write GN (s) = g(s/ND), where D > 0 is some exponent andg(r) is some (N -independent) function which decays rapidly to zero for r > 1. Thiscan be tested numerically by plotting s&Psize(s;N) versus s/ND for various N . Forsystems with simple finite-size scaling these curves should coincide.

The second problem is that the probability distribution of some avalanche ob-servables (examples are durations in the BTW and Zhang models) are not scalingunless their definitions are modified. The classical definition of an avalanche is a setof subsequent time steps for which the configuration is non-stable. This definitioncan be expressed in terms of the toppling activity signal as follows:

If x(k) % {0, 1, . . . } is the number of overcritical sites in the lattice at time stepk, then an avalanche is a sequence of subsequent time steps for which x(k) > 0.In other words, the duration of an avalanche is the return time of x(k) to the line

1We define these as the probability of having an avalanche of size (or duration) larger than s(or !), i.e. Psize = Prob[size of avalanche > s]

142

x = 0, and the size of an avalanche is the area under the graph of x(k) betweensubsequent zeros.

The problem with this definition is that it cannot be applied directly to experi-mental/observational data. Such data always includes noise, and it is impossible todetermine if the value of x is exactly equal to zero or not. In this case we are forcedto introduce a small threshold xth > 0, and change the definition so that avalanchesare subsequent times with x(k) > xth. The durations of avalanches then becomereturn times to the set x < xth, and the definition of sizes is modified accordingly.

Definition of avalanches with respect to a positive threshold on the toppling ac-tivity was first introduced in [48], where it was demonstrated that the waiting-timestatistics for the BTW sandpile undergoes a significant change with this modifica-tion. Another important feature is that the distributions for durations in the BTWand Zhang models now become good power laws. (See chapters 7 and 8.)

A third advantage of using a threshold is that, when combined with a stochasticdi!erential equation for the toppling activity, it allows us to describe the essentialcharacteristics of (height-type) sandpile models by a single parameter H % (0, 1). Inthe next two chapters we will show how the toppling process (in the the continumlimit) can be described by the equation

dX(t) = f(X) dt + #!

X(t) dBH(t) , (6.1)

where BH(t) is fractional Brownian motion with Hurst exponent H. For a thresholdXth > 0 we can change variable to Y (t) = X(t))Xth to get

dY (t) = f(Y + Xth) dt + #!

Xth + Y (t) dBH(t)

=$f(Xth) +O(Y )

%dt + #

$!Xth +O(Y )

%dBH(t) .

If the threshold is chosen such that f(Xth) = 0, then Y (t) is essentially proportionalto BH(t) on short time scales, and the corresponding duration statistics is simplythe return time (to zero) of fractional Brownian motion.

In what follows we give an introduction to derivation of equation 6.1.

2 From a dynamical to a stochastic point of view

In the first part of the thesis our basic structure was a map * ( * (where * =$+

N #M), and we made general characterizations of its orbits !0,!1,!2, · · · % *.We will now change our approach by fixing an observable ) : * ( R. The sequencesx(k) = )(!k) depend only on the initial condition !0 and we can write x(k) : * ( R,where

x(k)(!) = )(!k) given !0 = ! .

143

If we equip * with the probability measure m = µBer # µLeb we obtain a familyof random variables {x(k) | k % N}. By definition this is a discrete-time stochasticprocess.

Remark 6.2 In the following we will drop the brackets and denote the process byx(k). For each ! % *, the sequence k '( x(k)(!) is called a realization of theprocess.

It is often convenient to describe the process in terms of the family {Pk1···kn |ki %N, n > 0} of finite-dimensional marginals. These can be constructed from theprocess by defining random vectors xk1···kn = (x(k1), . . . , x(kn)), and letting Pk1···kn

be the distribution of the probability measure (xk1···kn)&m on Rn, i.e.

Pk1···kn(y1, . . . , yn) = m$x!1

k1···kn

<()1, y1] , · · · , ()1, yn]

=%

= Prob[x(k1) 2 y1, . . . , x(kn) 2 yn] .

Two stochastic processes are said to be equal in distribution (denoted by @) if thefamilies of finite dimensional marginals coincide. Moreover, there is a theorem ofKolmogorov that ensures that for any consistent2 family of probability distributionsP = {Pk1···kn |ki % N, n & 0} there is a real valued stochastic process, whose familyof finite dimensional marginal distributions equals P.

Remark 6.3 If the random variables x(k) and y(k) have the same distribution (forfixed k) we write x(k) 3 y(k). Clearly, x(k) @ y(k) implies that x(k) 3 y(k) for allk. The inverse statement is in general false.

3 The branching model

Let )(!) be the number of overcritical sites in the configuration corresponding to! % *. Then the corresponding process x(k) is the toppling activity of the sandpilemodel. We will derive an approximation of x(k) using stochastic di!erential equa-tions. The model will be valid for the BTW and Zhang models, but has a simplerderivation for the so-called random neighbor model [20]. This a modified version ofthe two-dimensional BTW model, where the toppling rule is such that when site itopples, then the grains are distributed to 2d random sites j1(i), . . . , j2d(i). Siteson the boundary of the lattice distribute grains to less than 2d random sites, butthe details of this implementation is not essential for the following.

2This means that if {ks1 , . . . , ksm} !{ k1, . . . , kn}, then Pks1 ···ksnis the marginal of Pk1···kn

corresponding to the index set {s1, . . . , sm}.

144

Let OC(k) denote the set of overcritical sites at time step k and let x(k) be thetoppling activity. If OC(k) = {i1(k), . . . , ix(k)(k)} we define

(n,k := ((in(k)) :=2d"

m=1

&$<

z(k + 1)=jm(in(k))

) 2d%

,

where z(k + 1) is the configuration at the time step k + 1. This defines a family ofrandom variables {(n,k | k % N, n = 1, . . . , x(k)}.

Note that (n,k can be interpreted as the number of overcritical sites at time k+1which are produced by the n-th overcritical site at time k. In the random neighbormodel, site i distributes grains to sites which typically are far away from i and fromeach other. Thus it is reasonable to make the following mean-field approximations:

1. All random variables (n,k are independent and identically distributed.

2. E[(n,k] = 1 for all n % N and all k = 1, . . . , x(k).

Let #2 = Var[(n,k] and notice that since

x(k + 1)) x(k) =x(k)"

n=1

((n,k ) 1)

we have

x(k + 1) = x(k) + #

x(k)"

i=n

wi ,

where wi are independent random variables on {)1, . . . , 2d} with unit variance andzero mean.

A realization of this process will exist as long as long as x(k) & 0, but thisproblem can be avoided by defining

x(k + 1) = max5

x(k) + #

x(k)"

i=n

wi, 06

.

We now have a stochastic model for the toppling activity x(k). Notice thatit does not include the random dropping of sand grains onto the lattice, and newavalanches are not initiated after the first avalanche has terminated. Nevertheless, ifwe assume that the toppling activity in one avalanche is independent of the previousavalanches, then the stochastic model describes the statistical distribution of theavalanche size and duration. For instance, if we choose x(0) = 1, then the durationof an avalanche is a random variable defined by " = min{k |x(k) = 0}.

145

It is not di"cult to derive an analytic expression for the distribution functionof avalanche duration based on this stochastic model, but it is easier if we firstproceed to a continuous time limit. Before we do this we will use the next section todescribe some of the fundamental properties of Brownian motion and Ito stochasticdi!erential equations.

4 Brownian motion

A stochastic process can be defined for continuous times t % R or t % [0,1) inthe same way as we defined discrete processes. This means that we can think of aprocess as a family {X(t)|t % R} of random variables of the same probability space,or as a family {Pt1,...,tn |t1, . . . , tn % R, n > 0} of finite-dimensional marginals. Asbefore, two processes are said to be equal in distribution if their families of finite-dimensional marginals coincide.

4.1 Self-similar processes

A process X(t) is self-similar if there for all a > 0 exist b > 0 such that

X(at) @ bX(t) .

The simplest kind of self-similar processes is when X(t) is independent of t almostsurely. We call such a process trivial. For any non-trivial process there exists a realnumber H > 0 such that b = aH . We call H the Hurst exponent of the process,and we say that X(t) is H-self-similar. If X(t) is a non-trivial self-similar process,then X(0) 3 aHX(0), and letting a ( 0 we see that X(0) = 0 almost surely.

In general we say that a process X(t) has stationary increments if all the jointdistributions of X(t + #t) )X(t) are independent of t. Increments are said to beindependent if the random variables X(t2))X(t1), . . . , X(tn))X(tn!1) are inde-pendent for any n % N and t1, . . . , tn % R'0. A Gaussian self-similar process withstationary and independent increments is called Brownian motion. Equivalently wecan define Brownian motion as a process B(t) with the following properties:

1. B(0) = 0 almost surely

2. Realizations are almost surely continuous curves

3. Independent increments

4. X(t + #t))X(t) 3 N (0,#t) for all #t > 0

It is easy to see that if B(t) satisfies proporties 1-4, then so does a!1/2B(at) for alla > 0. Hence Brownian motion is self-similar with H = 1/2.

146

4.2 Construction of Brownian motion

It is instructive to see how we can construct Brownian motion as a limit of discreterandom walks: Fix a parameter $x > 0, let x(k) = 0 and

x(k + 1) = x(k) + $xw(k) ,

where w(k) are independently and identically distributed (i.i.d.) random vari-ables with finite variance and zero mean. Assume with no loss of generality thatVar[w(k)] = 1. Denote #x(k) = x(k + 1)) x(k). Then

x(k) =k"

i=1

#x(k) = $xk"

i=1

w(k) .

This is a sum of i.i.d. random variables, so by the central limit

Prob3a <

x(k)$x4

k< b

4( 14

2%

2 b

ae!y2/2 dy as k (1 .

Re-scale the time variable by t = k · $t and let B(t) = x(k). Then

Prob3a <

B(t)$x$t!1/2

4t4

k< b

4( 14

2%

2 b

ae!y2/2 dy as k (1 .

Choose $x depending on $t such that $x 3 $t1/2 as $t ( 0, and let # = lim't$0

$x$t!1/2.Holding t fixed and letting $t, $x ( 0 we get

Prob[a < B(t) < b] =14

2% #2 t

2 b

ae!

y2

2(2t dy .

The process B(t) in the limit $x, $t ( 0 is the Brownian motion.

Denote #XB(t) = B(t+#t))B(t) and let P (B, t) and +(#B, t) denote the prob-ability density functions of B(t) and #B(t) respectively. From the construction it isclear that +(B, t) = +(B) is independent of t (as it should be when the increment pro-cess is stationary). Notice also that + depends on #t. In fact, +(#B) = P (#B,#t).

4.3 The di!usion equation

Let P and + the probability density functions for Brownian motion as defined above.Then

P (X, t + #t) =2

RP (X )#X, t)+(#X)d#X .

147

Expanding in #t and #X we get

P (X, t) +-P

-t(X, t)#t +O(#t2) =

2

R

$P (X, t)) -P

-X(X, t)#X +

12

-2P

-X2(X, t)#X2 +O(#X3)

%+(#X) d#X .

Since 2#X+(#X)d#X = 0 ,

2#X2+(#X)d#X = #t ,

and higher moments of #X grow faster than linearly in #t, we get the di!usionequation

-P

-t=

12

-2P

-X2

in the limit #t ( 0. This is an example of a (Kolmogorov) Fokker-Planck equation.

5 Stochastic di!erential equations

The Brownian motion is a special case of a solution to the Ito stochastic di!erentialequation (SDE)

dX(t) = µ(X, t) dt + #(X, t) dB(t) , (6.2)

which is a notation representing the integral equation

X(t) = X(0) +2 t

0µ(X(t#), t#)dt# +

2 t

0#(X(t#), t#) dB(t#) .

The second integral on the right hand side of the equation is called the Ito inte-gral. For a real valued stochastic process H(t) this integral is defined as the limit(convergence is in probability)

2 t

0H(s) dB(s) = lim

"$0

"

i

H(si!1)$B(si))B(si!1)

%,

where {[si!1, si]} is a partition of [0, t] with diameter 2 *.Ito’s lemma tells us that Ito’s equation transforms in a non-trivial way under a

change of variable Y (t) = f(X(t), t):

Theorem 6.1 (Ito’s lemma) Assume that the stochastic process X(t) is a solu-tion of the stochastic di"erential equation dX(t) = µ(X, t) dt + #(X, t) dB(t) andthat Y (t) = f(X, t), where f(X, t) is a smooth function of two real variables. ThenY (t) is a solution of the stochastic di"erential equation

dY (t) =$µ

-f

-X+

-f

-t+

#2

2-2f

-X2

%dt + #

-f

-XdB(t) .

148

Remark 6.4 The classical example of use of the Ito lemma is in finance. Thefamous Black-Scholes equation describes a price by the equation

dX(t) = µX dt + #XdB(t) .

Preforming the substitution Y (t) = log X(t) and using Ito’s lemma we get

dY (t) = (µ) #2/2) dt + #dB(t) .

In other words, the logarithmic price is a simple Brownian motion with a drift term.

By using the Ito lemma one can derive the Fokker-Planck equation for processessatisfying equation 6.2: If P (X, t) is the probability density function of X(t), then

-P

-t= ) -

-X(µP ) +

12

-2

-X2(#2P ) . (6.3)

6 SDE formulation of the branching process

We now return to the discrete branching model:

x(k + 1)) x(k) = #

x(k)"

i=1

wi .

By the central limit theorem, the sum on the right hand side of the equation can beapproximated by a Gaussian random variable with zero mean and variance #2x(k).Since B(t + $t))B(t) 3 N (0, $t) we can write

#

x(k)"

i=1

wi =#!

x(k)4$t

$B(t + $t))B(t)

%.

Similar to the construction of Brownian motion from a random walk we wantto take a continuous limit of this discrete process. Let $x, $t > 0, define t = $t · kand let X(t) = x(k) $x. Then

X(t + $t))X(t)$x

= x(k + 1)) x(k) = #14$t

!x(k)

$B(t + $t))B(t)

%

= #

TX(t)$x · $t

$B(t + $t))B(t)

%.

This gives the equation

X(t + $t))X(t) =T

$x

$t#

!X(t)

$B(t + $t))B(t)

%.

149

Let $t depend on $x in such a way that lim'x$0 $x/$t = 1. When $x and $t go tozero we get the Ito stochastic di!erential equation

dX(t) = #!

X(t) dB(t) .

Remark 6.5 For a standard random walk we can obtain a Brownian motion byletting $x 3 $t1/2, whereas here we must choose $x 3 $t. Although the processX(t) is not self-similar in a strict sense, the relation between $x and $t indicatesself-similar properties with H = 1.

Notice that use of the central limit theorem requires x(k) 5 1. This meansthat the model is not valid for small activities. In particular, the initial conditionx(k) = 1 corresponds to X(0) = $x ( 0, and then X(t) is trivial. Hence we haveto consider initial conditions X(0) = Y , where Y > 0 is chosen in the rescaledcoordinates.

Another important property is that the process only is defined for positive X,and even for a positive initial condition X(0) = Y > 0, the solution X(t) will ceaseto exist after finite time (because X(t) crosses the line X = 0). We can see thisby considering fixed $t, $x > 0. Then X(t + $t) = X(t) + #

!X(t) w(t; $t), where

w(t; $t) are independent Gaussian random variables with zero mean and variance $t.Since conditional probability density function of X(t + $t))X(t) given X(t) = Xis

+(#X;X, $t) =14

2%#2$t Xe!

"X2

2(2%t X ,

the probability of “jumping” from a positive value X = X(t) > 0 to a negativevalue X(t + $t) < 0 is equal to

2 X

!%+(#X, X, $t) d#X =

12) 1

2Erf

$ 4X4

2 #4

$t

%=

12) 1

2Erf

$!x(k)42 #

T$x

$t

%.

This quantity is bounded away from zero in the limit $x, $t ( 0.When a solution X(t) ceases to exist, we can interpret this as the termination

of an avalanche. This allows us to calculate the distribution function for avalanchedurations analytically. Let P (X, t) be defined by

Prob[a < X(t) < b and X(t#) > 0 *t# % [0, t] |X(0) = Y ] =2 b

aP (X, t) dX .

Then the probability that an avalanche is still running after a time t is

Prob[" > t] =2 %

0P (X, t) dX .

150

In particular, the probability density function for avalanche duration is

pdur(t) = ) d

dtProb[" > t] = )

2 %

0

-P

-t(X, t) dX . (6.4)

We now apply the Fokker-Planck equation (see equation 6.3)

-P

-t=

#2

2-2

-X2(X P ) . (6.5)

This gives

pdur(t) = )#2

2

2 %

0

-2

-X2(XP ) dX =

#2

2lim

X$0

-

-X(XP )

=#2

2lim

X$0P (X, t) .

The termination of realizations when they reach the line X = 0 corresponds tothe Fokker-Planck equation (Eq.6.5) with an absorbing boundary in X = 0 andinitial condition P (X, 0) = $Y (X). Using separation of variables and superpositionvia Hankel transforms it is not di"cult to establish the solution

P (X, t) =12

TY

X

2 %

0J1(s

4Y ) J1(s

4X) exp

7)#2 s2

8t

8s ds ,

where J1 denotes the Bessel function of first kind. Then

limX$0

P (X, t) =4

Y

4

2 %

0s2 J1(s

4Y ) exp

7)#2 s2

8t

8ds

=4 X0

#4t2exp

7) 2 Y

#2 t

8,

andpdur(") =

2 Y

#2"2exp

7) 2 Y

#2 "

8.

For " 5 2Y/#2 we have pdur(") 3 "!%, with / = 2.

7 Extension to the BTW and Zhang models

The result pdur(") 3 "!2 is known to agree well with numerical simulation of therandom neighbor sandpile model. However, the exponent / = 2 does not agreewith simulations of the BTW and Zhang models. We will now generalize the modelin order to obtain good descriptions of the toppling activities in the BTW andZhang models as well. The central element is this generalization is introduction offractional noise.

151

7.1 Fractional noise terms

Recall that the main assumption leading to the stochastic di!erential equationdX(t) =

!X(t)dB(t) is that the random variables (n,k are identically and inde-

pendently distributed. Our first generalization involves relaxing this requirement byonly assuming that we for a fixed k % N have independence between (1,k, . . . , (x(k),k.This is all we need to apply the central limit theorem and obtain the stochastic dif-ference equation

x(k + 1)) x(k) = #!

x(k) w(k) ,

where w(k) are Gaussian.Previously we assumed that w(k) where independent for di!erent k, and then

we could write w(k) = B(k+1))B(k). A generalization of this would be to replaceB(t) with a Gaussian process with dependent increments. The obvious candidatefor such a process is a fractional Brownian motion BH(t), which can be defined asan H-self-similar Gaussian process with stationary increments.

We can also define BH(t) explicitly: For H % (0, 1) a fractional Brownian motioncan be defined from a Brownian motion by the following formula [41]:

BH(t) = C

2 t

0K(t, t#) dB(t#)

where

K(t, t#) =$ t

t#

%H! 12(t) t#)H! 1

2 )$H ) 1

2

%t#

12!H

2 t

t!sH! 3

2 (s) t#)H! 12 ds

An important property of BH(t) is that

Prob[a < BH(t + #t))BH(t) < b] =14

2%#tH

2 b

ae!

"X2

2"t2H d#X .

Moreover B1/2(t) is Brownian motion, but for H -= 1/2 increments are not inde-pendent. Inversely, any H-self-similar Gaussian process with stationary incrementsis equal to BH(t) in distribution.

A generalization of the branching process can be obtained by writing

x(k + 1)) x(k) = #!

x(k)$BH(k + 1))BH(k)

%.

As before we obtain a stochastic di!erential equation by letting t = k $t and X(t) =x(k) $x. If we let relate $x and $t such that lim't$0 $x/$t2H = 1, then we get thestochastic di!erential equation

dX(t) = #!

X(t) dBH(t)

in the limit $x, $t ( 0:

152

Remark 6.6 There is no Fokker-Planck formulation for stochastic di"erential equa-tions driven by fractional noise, so the method we used to calculate the distributionof avalanche durations in the case H = 1/2, no longer applies for H -= 1/2.

Once we have generalized the model to include the possibility of fractional Brow-nian noise terms we can start to compare it with the numerical simulations of theBTW and Zhang models. The fist step in this analysis is to verify that the con-ditional probability distribution of x(k + 1) ) x(k) are Gaussians with varianceproportional to x(k). The results of this analysis is presented in the next twochapters.

The second step of the analysis is to verify that the noise term is produced bya self-similar process with stationary increments. From numerical simulation of thesandpiles we obtain time series representing the signal x(k). From this signal wecan construct a signal representing w(k) by taking

w(k) =x(k + 1)) x(k)

#!

x(k).

We have modeled w(k) as a di!erence BH(k + 1) ) BH(k), so we can obtain adiscrete approximation to BH(t) by taking the sums

k"

k!=0

x(k# + 1)) x(k#)#!

x(k#).

If our model is correct, then this process should have the characteristics of a frac-tional Brownian motion. This means that we should check that it is Gaussian, thatthe increments are stationary and that the process is H-self-similar.

It is easy to check that the process is Gaussian, and weak stationarity of in-crements can be tested by constructing various realizations of starting from zero,and verifying that the moments of the increment process are time independent. Amore di"cult problem is to verify self-similarity. From a strict mathematical pointof view we are forced to calculate all finite dimensional marginals of the processto verify self-similarity. Obviously, this is not a practical method of analysis fornumerical data, and instead we resort to the analysis of structure functions.

For a fractional Brownian motion BH(t) we have

+(#B) =14

2%#t2He!

"B2"t2H .

This implies that the q-th moment satisfies

Sq(#t) = EMPPB(t + #t))B(t)

PPqN =2

R|#B|q+(#B) d#B

=14

2%#t2H

2

R|#B|qe!

|"B|2"t2H d#B A #tHq .

153

The functions Sq(#t) are called the structure functions of the process. In particularthe double-logarithmic plot of S2(#t) versus #t is called the variogram of the pro-cess. In the physical literature (if this curve is a straight line) the slope divided bytwo is called the Hurst exponent of the process. Of course, this coincides with ourdefinition of the Hurst exponent in the case of fractional Brownian motion, sincewe have Sq(#t) A #t2H . However, there are processes for which Sq(#t) A #t2H

but where, for instance, S4(#t) 3 #tr for r < 4H. If increments of such a processare stationary, then we know that the process is not a self-similar process. Actuallythere exist a generic class of processes, which are omnipresent in nature, where theexponents in the di!erent structure functions depend on q non-lienarly. These pro-cesses are called multi-fractals, and they can be characterized by the zeta-function4(q) which is defined through the relation Sq(#t) 3 #t6(q). We know that forfractional Brownian motion the zeta-funciton is of the form 4(q) = Hq, but formulti-fractal processes the zeta-functions are concave.

Since multi-fractal processes are as generic as self-similar processes, it is very im-portant to test for multi-fractality before modeling a signal as a self-similar process.The results from this test are presented in chapter 7.

Remark 6.7 The stochastic model dX(t) = #!

X(t) dBH(t) has two essentialweaknesses when compared with the toppling activity in real sandpiles. First ofall, the model does not reflect the toppling process correctly for very small values ofx(k). Secondly, there is no information about the system size in the model. How-ever, both of these problems can be solved by heuristically introducing a drift termf(X) dt to the model. The shape of this term, and its interpretation is discussed inchapters 7 and 8.

With the drift term our stochastic model has the final form

dX(t) = f(X) dt + #!

X(t) dBH(t) .

154

Appendix

A Stochastic Integrals

A stochastic di!erential equation dX(t) = µ(X(t), t)dt + #(X(t), t)dB(t) is a nota-tion for the integral equation

X(t) = X(0) +2 t

0µ(X(t#), t#) dt# +

2 t

0#(X(t#), t#) dB(t#) .

The second integral on the right hand side is a so-called Ito integral, and it canbe defined, not only for Brownian motion, but for all semi-martingales. However,fractional Brownian motions are not semi-martingales, and thus it is not completelytrivial to extend this integral to include fractional Brownian motion BH(t). In fact,there exist several definitions of the integral

2 b

aX(t) dBH(t)

for H -= 1/2, and today it is not generally agreed which of these definition is“correct”. In this section we present an approach given by Øksendal [58].

Remark 6.8 Brownian motion B(t) is a martingale. This means that the condi-tional expectation of B(t) given the trajectory {B(t#) | t# 2 s} up to a time s 2 t, isequal to B(s). That is,

E3B(t)

PPP{B(t#)|t# 2 s}4

= B(s) *s 2 t .

There is also a notion of local martingales, which loosely defined, are processes X(t)for which the stopped processes

X(min{t, tk})&(tk ) t)

are martingales for all k in an infinite almost surely increasing and divergent se-quence {tk} of stopping times. A semi-martingale is a process X(t) which can be

155

decomposed into a sum M(t) + V (t), where M(t) is a local martingale and V (t) isa cadlag (right continuous with left limits) process of bounded variation.3

A.1 Brownian motion and white noise

We have so far defined Brownian motion B(t) in terms of its family of finite-dimensional marginals and we have constructed a discrete random walk processwhich approximates B(t) as $t, $x ( 0. However, it is also possible to definethe process B(t) explicitly by specifying a probability space (*,F , µ) and a family{B(t) : * ( R|t % R} of random variables. The standard way of doing this isto choose * to be the set of tempered distributions on R, i.e. * = S(R)&, whereS(R) $ C%(R) is the Schwartz space of rapidly decreasing functions on R. In otherwords, the elements of ! % * are functionals taking functions f % S(R) to realnumbers !!, f" := !(f) % R. Let F be the Borel #-algebra on *. It can be shownthat there exists a unique Borel probability measure µ on * satisfying the equation

2

"ei-.,f.dµ(!) = e

! 12 ||f ||

2L2(R) . (6.6)

This defines our probability space (*,F , µ).With this set-up every function f % S(R) defines a random variable by ! '(

!!, f". Hence a family {ft|t % R} of functions will define a stochastic process! ( !!, ft". Brownian motion can be defined simply by specifying

ft(s) =

#1 for s % [0, t]0 otherwise

when t > 0 and

ft(s) =

#)1 for s % [t, 0]0 otherwise

for t < 0.Two important properties of the measure defined by equation 6.6 is that for all

f % S(R):

E[!!, f"] =2!!, f" dµ(!) = 0

andE[!!, f"2] = ||f ||2L2(R) .

3Bounded variation on an interval means that supPP

n |V (ti) " V (ti!1)| < +#, where thesupremum is taken over all finite partitions {[ti, ti+1]} of the interval. Intuitively, this means thatV (t) has only jump singularities.

156

Using these properties it is not di"cult to verify that the family of random variabledefined by {ft} satisfies the defining properties of Brownian motion.

A more elegant description of Brownian motion is possible if we choose an or-thonormal basis of Hermite functions {(n} in L2(R) and expand the functions ft:

ft ="

n

((n, ft)L2(R) (n ="

n

$ 2

R(n(s) ft(s) ds

%(n =

"

n

$ 2 t

0(n(s) ds

%(n .

Then Brownian motion can be written as

B(t)(!) ="

n

$ 2 t

0(n(s) ds

%!!, (n" .

This shows that Brownian motion can be written as an expansion over randomvariables !·, (n" with time-dependent coe"cients. The result suggests that oneshould try to find a basis of in which all random variables on the form !·, f" (withf % L2(R)) can be expanded. Such a basic set exists, and its elements can be givenexplicitly by the formula

Hk(!) = hk1(!!, (1")hk2(!!, (2") · · ·hkn(!!, (n") ,

where hi are the Hermite polynomials and k = (k1, . . . , kn) is a multi-index. It isthen natural to define the space of random variables as the set (S)& of all formalexpressions on the form "

k"I

ckHk ,

with some conditions on ck being satisfied. Here I is the set of all finite-lengthmulti-indices k = (k1, . . . , kn) with ki % {0, 1, 2, . . . }.

Note that if en = (0, 0, . . . , 0, 1) (where the 1 is in the n-th entry), then we canwrite

B(t) ="

n

$ 2 t

0(n(s)ds

%Hen .

We observe that the map B(·) : R ( (S)& is di!erentiable, and we can define theprocess

W (t) =dB(t)

dt=

"

n

(n(t)Hek .

This process is called white noise.The Wick product of two elements F1, F2 % (S)& is defined by the formula

F1 B F2 ="

k,l"I

c(1)k c(2)

l Hk+l ,

157

whereF1 =

"

k

c(1)k Hk and F2 =

"

k

c(2)k Hk .

Using this product we can formulate Ito integration with respect to Brownian mo-tion in terms of white noise. In fact, if X(t)(!) is an Ito integrable process, thenits integral with respect to B(t) can be written as

2 t

0X(t#) dB(t#) =

2 t

0X(t) BW (t) dt . (6.7)

A.2 Fractional Brownian noise

With the white-noise formalism presented above, the generalization to fractionalBrownian noise is straight forward. It is not di"cult to see that fractional Brownianmotion can be defined through the family of functions {MHft}, where the operatorMH : L2(R) ( L2(R) is given by

(MH))(s) =

-../

..0

CH

BR

$(s!s!)!$(s)|s!|3/2H

ds# H % (0, 1/2)

)(s) H = 1/2CH

BR

f(s!)|s!!s|3/2"H ds# H % (1/2, 1)

.

Here CH is a normalization constant only depending on H.We define an inner product (f, g)L2(R) = (MHf,MHg)L2(R) and let L2

H(R) bethe space of functions f with ||f ||L2

H(R) < 1. In this space we have an orthonormalbasis 5n = M!1

H (n. It is easy to verify that (MHf, g)L2(R) = (f,MHg)L2(R) forf, g % L2(R) , L2

H(R), and this gives

BH(t)(!) = !!,MHft" = !!,"

n

(ft, 5n)L2H(R)MH5n"

= !!,"

n

(MHft, (n)L2(R)(n" ="

n

(ft,MH(n)L2(R)!!, (n"

="

n

$ 2 t

0MH(n(s) ds

%Hen(!) .

This shows that BH(·) : R ( (S)& is a di!erential map and the di!erential is

WH(t) ="

n

MH(n(t)Hen .

The process WH(t) is called fractional Brownian noise with Hurst exponent H. Byanalogy with equation 6.7 we define

2 t

0X(t#)dBH(t#) =

2 t

0X(t#) BWH(t#) dt# .

158

Also note that the stochastic di!erential equation

dX(t) = µ(X(t), t) dt + #(X(t), t) dB(t)

can be written asdX

dt= µ(X(t), t) + #(X(t), t) BWH(t) ,

where the di!erentiation of X(t) is with respect to the map X(·) : R ( (S)&. Inparticular, the equation dX(t) = #

!X(t) dB(t) can be written as

dX

dt=

!X(t) BWH(t) . (6.8)

Remark 6.9 Linear stochastic di"erential equations on the form

dX

dt= µX + # BWH(t) ,

can be solved in terms of the Wick exponential which is formally defined as

exp/(F ) =%"

n=0

1n!

F /n .

This can be used to derive a generalized Ito-lemma which works for these linearequations. However, this is approach does not work for equation 6.8. The maindi!culty here is that the square root is not a Wick-type square root, and we havea combination of two di"erent products. This makes it di!cult to write a formalsolution.

159

Chapter 7

A stochastic theory for thetoppling activity in SOC

Joint with K. Rypdal

Abstract: A stochastic theory for the toppling activity in sandpile models is de-veloped, based on a simple mean-field assumption about the toppling process. Thetheory describes the process as an anti-persistent Gaussian walk, where the di!u-sion coe"cient is proportional to the activity. It is formulated as a generalization ofthe Ito stochastic di!erential equation with an anti-persistent fractional Gaussiannoise source and a deterministic drift term. An essential element of the theory isre-scaling to obtain a proper thermodynamic limit. When subjected to the mostrelevant statistical tests, the signal generated by the stochastic equation is indistin-guishable from the temporal features of the toppling process obtained by numericalsimulation of the Bak-Tang-Wiesenfeld sandpile.

Reference for this paper: M. Rypdal, K. Rypdal, A stochastic model of the toppling

activity in self-organized critical systems, Preprint: arXiv:0710.4010v2 (2008), (Accepted

by the New Journal of Physics).

161

1 Introduction

The existence of self-organized critical dynamics in complex systems has tradition-ally been demonstrated through numerical simulation of certain classes of cellularautomata referred to as sandpile models [30, 19]. Non-linear, spatio-temporal dy-namics is always essential for the emergence of SOC behavior, but the details ofthis dynamics for a specific natural system is often poorly understood and/or notaccessible to observation. In many cases the information available is in the form oftime-series of spatially averaged data like stock-price indices, geomagnetic indices,or global temperature data. For scientists who deal with such data a natural ques-tion to ask is: are there specific signatures of SOC dynamics that can be detectedfrom such data?

In this letter we shall report some results which provide a partial answer to sucha question. Some important statistical features of the toppling activity are commonto most weakly driven sandpile models described in the literature, and these areused to formulate a stochastic model for the toppling activity signal. A benchmarkcase against which our results are tested, is a numerical study of the Bak-Tang-Wiesenfeld (BTW) sandpile [3]. A crucial step in our work is a re-scaling of thedynamical variables which allows a natural passage to the thermodynamic (contin-uum) limit. We demonstrate that this leads to new results concerning SOC scalinglaws. We find that the probability density function (pdf) for the toppling activity isa stretched exponential, close to the Bramwell-Holdsworth-Pinton distribution [13],or close to a gaussian, depending on whether the sandpile is so slowly driven thatavalanches are well separated, or it is driven so hard that several avalanches runsimultaneously. The pdf for avalanche durations is unique in the thermodynamiclimit, but is not a power law, unless we redefine the meaning of an avalanche tobe the activity burst between successive times for which the activity rises abovea positive threshold. Implementing such a threshold yields an exponent for theavalanche duration pdf of 1.63, in agreement with [48], but in contradiction to [40].It also gives power-law quiet-time statistics as in [48] and thus refuting the claimin [9] that SOC implies power-law distributed avalanche durations, but Poisson-distributed quiet times.

The sandpile models considered in this short paper deal with a d & 2-dimensionallattice of Nd sites each of which are occupied by a certain integer number of quantawhich we conveniently can think of as sand grains. The dynamics on the latticeis given by a toppling rule which implies that if the number of grains on a siteexceeds a prescribed threshold, the grains on that site are distributed to its nearestneighbors. If the occupation number of some of these neighbors exceed the topplingthreshold these sites will topple in the next time step, and the dynamics continuesas an avalanche until all sites are stable. The details of this toppling rule can vary,but a useful theory for a broad class of natural phenomena should not be very

162

sensitive to such detail.In natural systems the SOC dynamics is usually driven by some weak random

external forcing. In sandpile models this can be modeled by dropping of sand grainsat randomly selected sites at widely separated times. In numerical algorithms thisis often done by dropping sand grains only at those times when no avalanche isrunning. This ensures that the drive does not interfere with the avalanching process.Usually it will then only take a few time steps from one avalanche has stopped untila new starts, so for a large system the quiet times between avalanches will appearinsignificant compared to their durations.

A more physical drive would be to drop sand also during avalanches. If thedropping rate is slower than the typical duration of a system-size avalanche thedrive would still not interfere with the avalanche dynamics, but the quiet timeswould depend on the statistical distribution of dropping times, which is typicallya Poisson distribution. In many natural systems, however, avalanching occurs allthe time, corresponding to a higher driving rate. In such cases, and also becausethere will always be noise in time-series data, we cannot identify the start andtermination of an avalanche from a zero condition of our observable. In practice wehave to define avalanches as bursts in the time series identified by a threshold onthe signal [48]. In a sandpile simulation such bursts are correlated and therefore thequiet times between the bursts are power-law distributed even if the dropping ofsand grains is chosen to be a Poisson process. Hence if focus is on modeling featuresthat can be detected in observational data we shall think of avalanches as activitybursts starting and terminating at a non-zero threshold value. Moreover, one of themain results of this work is that power-law shape of the pdf for avalanche durationis true only if one defines avalanches in this way.

2 The stochastic model

The BTW model was first developed in the seminal paper [3], and is described inmany monographs like [30, 19]. If zi is the number of sand grains occupying thei’th site, the toppling rule is zi ( 0 and zj ( zi + zi/4 if zi is overcritical and j is anearest neighbor of i. Whenever the configuration has no overcritical sites a grainis added to a random site with uniform probability on the lattice. In a continuouslydriven system, grains are added at random times at a preset average rate, evenwhen avalanches are running.

We shall assume that the lattice has linear extent L = 1 with Nd sites, sothe thermodynamic limit N ( 1 can be thought of as a continuum limit. Thesandpile evolves in discrete time steps labeled by k = 1, 2, 3 . . ., and the number ofsites whose occupation number exceeds the toppling threshold at time k is called thetoppling activity xN (k). The toppling increment is #xN (k) def= xN (k + 1)) xN (k).Let us define two active sites as dynamically connected if they have at least one

163

Figure 7.1: The blue squares mark the active (toppling) sites at a randomly choseninstant during an avalanche in a BTW lattice. Note that all the marked sites belongto one avalanche only, i.e. they all ere the result of one site going unstable sometime in the past due to the feeding of a grain at a marginally stable site. The figureillustrates that a large avalanche in a large sandpile at a given time will consist ofa large number of small disconnected clusters.

common nearest neighbor, and define a connected cluster as a collection of activesites which are linked trough such connections. From numerical simulations ofsandpiles we observe that such clusters never consist of more than a few elementsand that the instantaneous number of clusters nN increases in proportion to xN

(see figure 7.1). This implies that at each time k we can label the clusters byi = 1, . . . , cxN (k), where c < 1 is a constant depending on the specific toppling ruleand the dimension d of the sandpile. We can then decompose the increment #xN (k)into a sum of local increment contributions (N,i(k) produced by each of the clusters,i.e. #xN (k) =

1cxN (k)i=1 (N,i(k). We think of the local increment contributions

as random variables which take values in a finite sample space. Indeed, if eachcluster i only consists of a single overcritical site, then (i,N takes values in the set

164

0 1000 2000 3000 40000

100

200

300

0 1000 2000 3000 4000-30-20-100102030

(a)

(a)(a)

(b)

(c)P[xN+!xN | xN]

time t

xN (t)

!xN (t)

0 2 4 6 8 10 120246810

0 2 4 6 8 10 12

0.0.20.40.6

Var(!x|x)E(!x|x)

E(!x|x)

activity xN

(d)

Figure 7.2: a): A realization of the toppling activity xN (t) in the BTW sandpile. b):The increments #xN (t) = xN (t+1))xN (t) of the trace in (a), showing that #xN (t)is large when xN (t) is large. c): Conditional pdfs of xN + #xN for xN = 10, 20, 30respectively. d) The conditional mean and variance of #xN versus xN .

{)1, 0, . . . , 2d) 1}.As a first step to a stochastic model we make a mean-field assumption [54,

29], which impiles that (N,i(k) and (N,j(k) are statistically independent for i -=j. Then the central limit theorem states that in the limit N ( 1, xN (k) (1 the conditional probability density P [#xN (k)|xN (k)] of an increment #xN (k),given xN (k), is Gaussian with variance #2 xN (k), where #2 = c2(E[(2

N,i|xN ] )(E[(N,i|xN ])2). This has been verified numerically in the two-dimensional BTW-model as shown in figure 7.1. The figure demonstrates the need to introduce aconditional probability: The conditional variance of the increments is proportionalto xN and the conditional mean is not zero.

In fact, numerical simulations show that the the conditional mean increment,E[#xN |xN ], is positive for small xN , reflecting the natural tendency for the activityto grow when it is small. On the other hand the mean increment decays exponen-tially to zero for moderate xN , and becomes negative when xN is comparable tothe activity of a system-size avalanche, reflecting the limiting influence of the finitesystem size. These e!ects will be incorporated as a drift-term correction to the

165

model, but for now we consider for simplicity of argument a Gaussian process withnon-stationary increments and no drift term:

#xN (k) = #!

xN (k) w(k) , (7.1)

where w(k) is a stationary Gaussian stochastic process with unit variance. In Sec. 3we demonstrate from the numerical sandpile data that the normalized topplingprocess

W (k) def=k"

k!=0

w(k#) =k"

k!=0

#xN (k#)#!

xN (k#)

has the characteristics of a fractional Brownian walk with Hurst exponent H =0.37 on time scales shorter than the characteristic growth time for a system-sizeavalanche, consistent with a power spectrum which scales like f!1.74. Thus wemodel the normalized increment process as w(k) = WH(k + 1) ) WH(k), whereWH(k) is a fractional Brownian walk with Hurst exponent H. For the transitionto the thermodynamic limit, where time will become a continuous variable, wecan think about WH(k) as the result of a discrete sampling of the (continuous-time) fractional Brownian motion (fBm) WH(t). This process has the property!|WH(t + "))WH(t)|2" = "2H . We now have a stochastic di!erence equation

#xN (k) = #!

xN (k) (WH(k + 1))WH(k)). (7.2)

Numerical simulations show that xN 3 ND1 1, where 0 < D1 2 d can be interpretedas a fractal dimension of the set of active sites imbedded in the d-dimensional latticespace. This property is used to re-scale xN (k) such that it has a well-defined limitas N ( 1. We also have to re-scale the time variable by letting t = k#t, where#t = N!D2 . The value of D2 will become apparent if we define the normalizedactivity variable XN (t) = N!D1xN (t/#t), such that the corresponding incrementbecomes

#XN (t) = NHD2!D1 #!

XN (t) #WH(t) , (7.3)

where #WH(t) = WH(t+#t))WH(t). A well-defined thermodynamic limit N (1requires D2 = D1/2H, for which Eq. (7.3), by introduction of the limit functionX(t) = limN$%XN (t), reduces to the stochastic di!erential equation

dX(t) = f(X) dt + #!

X(t) dWH(t), (7.4)

1The BTW model does not exhibit perfect finite-size scaling [19] and hence the scaling xN $ND1 is not valid for very large activity. The e!ect of imperfect scaling with increasing N can bebuilt into Eq. 7.4 through an N -dependent drift term. However, the distributions of duration andsize of sub-system size avalanches (defined by a threshold Xc > 0) is not sensitive to this featureof the BWT model. We have given a detailed treatment of this problem in [51].

166

where we have heuristically added a drift term f(X) dt to account for the non-zeromean of the conditional increment. We take f(X) to be an exponentially decayingfunction based on the numerical results from the sandpile. In the 2-dimensionalBTW model we find that D1 = 0.86 and hence D2 = 1.16. This defines re-scaledcoordinates XN = xN/N0.86 and tN = k/N1.16.

3 The normalized toppling process is a fractionalBrownian walk

A fractional Brownian walk is a self-a"ne stochastic process with gaussian in-crements and self-a"nity (Hurst) exponent H. In strict mathematical terms astochastic process W (k) is self-a"ne only if the rescaled process c!HW (ck) isequal in distribution to W (k) for any positive stretching factor c. This meansthat for any sequence of time points k1, . . . , kn and any positive c, the random vari-ables c!HW (ck1, . . . , ckn) have the same joint distribution as the random variablesW (k1, . . . , kn). This definition can be used for theoretical purposes, but not asa practical tool to verify the self-a"nity of actual time-series. This can be done,however, by means of multifractal analysis. The simplest method is to computestructure functions,

S(l, q) = E[|W (k + l))W (k)|q]. (7.5)

For a multifractal process S(l, q) 3 l6(q), where 4(q) may be a non-linear functionof q, while for a monofractal (self-a"ne) process 4(q) = qH, where H is the self-a"nity exponent (see for instance the review [10]. This means that for a self-a"neprocess we have linear relations between log(S(l, q)) and log(l) and between 4(q)and q. These linear relationships are shown for the normalized toppling activity ofnumerical sandpile simulations in figure 7.3 , and demonstrates that this processis a fractional Brownian walk with Hurst exponent H = 0.37.

4 Analysis of avalanches

A time series X(t) & 0, representing a succession of avalanches with zero quiettimes, can be constructed numerically from the discrete-time version of Eq. (7.4) byintegrating the equation using realizations of the fractional Gaussian noise process#WH(t). At those times when X(t) drops below zero we consider the avalanche asterminated, and a new, independent realization of #WH(t) is generated and usedto produce the next avalanche. From long, stationary time-series generated fromthe stochastic model and from the sandpile model this way, we can construct pdfsP(X) which turn out to give almost identical results for the two models (see figure7.4). The shape of this pdf is universal in the thermodynamic limit: a stretched

167

1 10 1021

10

102

103

l1 2 3 4 5

0.5

1.0

1. 5

q

S(q,l)

!(q)

q=1

q=2

q=3

q=4

q=5(a) (b)

Figure 7.3: a) Structure functions S(q, l) plotted versus l in a log-log plot for q =1, . . . , 5. b) The scaling exponent for the structure functions (the slopes of the linesin (a)) plotted against q.

exponential P(X) 3 exp ()aXµ) with µ = 0.5. A di!erent pdf appears if thetime-series are constructed by launching the avalanches at random times (Poisson-distributed) with characteristic time between launches shorter than the growth timeof a system size avalanche. In this case several avalanches may run simultaneously,and for moderate strength of the drive P(X) from both models are close to theBramwell-Holdsworth-Pinton distribution, which was claimed to be valid for thetoppling-activity in the BTW-model in [13]. However, for stronger drive the pdftends more towards a gaussian. These results are merely empirical, based on thenumerical sandpile simulations. So far, there are no hard analytical results on theshape of the activity pdf.

Consider a solution of Eq. (7.4) with initial condition X(0) = Y > 0, and letP (X, t) be the evolution of the density distribution in X-space of an ensemble ofrealizations of the stochastic process X(t) all launched at activity X = Y at timet = 0. Every realization X(t) will sooner or later terminate at a finite time t = " forwhich X(")1) > 0 and X(") 2 0, and then we remove it from the ensemble. P (X, t)contains information about all commonly considered avalanche characteristics. Forexample, it is easily found from from Eq. (7.4) that, on time scales shorter thanthe growth time of a system-size avalanche, X(t) is a self-similar process with non-stationary increments and self-similarity exponent h = 2H [51]. Hence the varianceof X(t) with respect to P (X, t) will scale as 3 t2h. That this relation holds forthe 2-dimensional BTW model can easily be verified through numerical simulation(figure 7.5(a)).

168

10 -1

1

10

SDEBTW

proba

bility

dens

ity fu

nctio

n P(X

)

toppling activity X10 -2

0 0.5 1.0 1.5 2.0

Figure 7.4: Logarithmic plots of P(X) from simulations of the 2-dimensional BTWsandpile for N = 1024. Also shown is P(X) found from simulations of Eq. (7.4),and a stretched exponential fit (dashed curve, vertically shifted for visibility). Allpdfs are scaled to unit variance.

We can also compute the survival probability +(") =B%0 P (X, ") dX, which is

the probability that a realization of an avalanche has not terminated at the time " .This function is related to the pdf for avalanche durations by pdur(") = )+#("), sothat pdur(") is a power law if and only if +(") is a power law. figure 7.5(b) showsthe function +(") for numerical simulations of the BTW sandpile in the re-scaledcoordinates XN and tN , demonstrating that the pdf for avalanche durations doesnot represent a power law. The power-law form +(") 3 "0.5 proposed in [40] canonly be obtained as a tangent to the log-log plot of +(") at a given duration time" , and the slope of this tangent depends crucially on the duration time " for whichthis tangent is drawn.

The situation changes if we let all avalanches terminate when X drops below asmall threshold Xc > 0 as proposed in [48]. In this case avalanche durations arethe return times to the line X = Xc, and by changing coordinates to Y = X )Xc

we see that this corresponds to the return times to the time axis of the processgiven by the stochastic di!erential equation dY (t) = #

!Xc + Y (t) dWH(t). For

small avalanches where X(t) )Xc < Xc we can approximate this expression with

169

dY (t) = #4

Xc dWH(t), i.e. can approximate Y (t) by a fractional Brownian motionwith Hurst exponent H. Using the result of Ding and Yang [24] on the return timesof a fractional Brownian motion we get pdur(") 3 "2!H = "!1.63.

Numerical simulations of the BTW model verify this result: The survival func-tion +(") becomes a power law on time scales shorter than a system-size avalanche(see figure 7.6(a)), and the slope of the graph in a log-log plot is approximately)0.63, which corresponds to a scaling of the pdf for duration times on the formpdur(") 3 "!1.63. The result is also reproduced by simulations of Eq. (7.4) with anexponentially decaying drift term. figure 7.6(b) shows the log-log plot of the pdf forduration times in the stochastic di!erential equation and a line with slope )1.63,demonstrating that the avalanche statistics in the BTW sandpiles is captured bythe stochastic di!erential equation. From the scaling +(") 3 "!% we can deduce anexponent for the pdf of avalanche size as well. On the time scales where the topplingactivity can be approximated by a fractional Brownian motion WH(t), the signaldisperses with time as X 3 tH , the size of an avalanche of duration " scales likeS(") 3

B (0 tH dt 3 "H+1. Assuming that the pdf for avalanche size is on the form

psize(S) 3 S!& , the relation psize(S) dS = pdur(") d" yields "!&(H+1)+H 3 "!%!1,so

. =H + / + 1

H + 1=

2H + 1

. (7.6)

With H = 0.37 we obtain . = 1.46. The dependence of / and . on H is the sameas obtained in [56] and [18].

We also remark that if we omit the drift term and let H = 1/2 and Xc = 0we obtain the so-called mean-field theory of sandpiles. In this case the stochasticdi!erential equation has a corresponding Fokker-Planck equation

-P

-t=

#2

2-2

-X2(XP ) .

If we solve this equation on the interval [0,1) with absorbing boundary condi-tions in X = 0 we can obtain an analytical expression for P (X, t), and from somestraightforward algebra we find for large " that pdur(") 3 "!2 [51]. Since Xc = 0 wecan not approximate the toppling activity by a Brownian motion on any scale andthus X(t) diverges like 3 th, where h = 2H. By replacing H with h = 2H in Eq.(7.6) we get psize(S) 3 S!3/2, in agreement with previous mean field approaches[54, 29].

5 Concluding remarks

We point out that the validity of Eq. 7.4 is not restricted to the BTW model.For instance, the equation has been verified for the Zhang model [51, 57], though

170

with a di!erent Hurst exponent H. Time series of global quantities derived fromnumerical simulation of di!erent sandpile and turbulent fluid systems can be shownto be adequately described by generalizations of Eq. 4, where H, the specific formof the “di!usion coe"cient” in the stochastic term, and the drift term, all dependon the system at hand [51].

171

1

1

N 64N 256N 1024N 2048

1

1

10 -4 10 -3 10 -2 10 -1 1 10

10 -4

10 -3

10 -2

10 -1

1

N 64N 256N 1024N 2048

varia

nce o

f X(t)

10 -1

10 -2

10 -3

10 -4

10 -5

10 -3 10 -2 10 -1time t

survi

val fu

nctio

n !("

)

10 -1

10 -4 10 -3 10 -2 10 -1

duration "

(a)

(b)

====

====

Figure 7.5: a) Double-logarithmic plots of the variance of X(t) with respect to thepdf P (X, t). The variance grows like t2h, with h = 2H = 0.74 for times less thanthe duration of a system size avalanche. b) Double-logarithmic plots of the survivalfunction +(") in the re-scaled coordinates XN and tN , demonstrating that the pdfof avalanche durations is not a power-law. The dotted line has slope )0.5.

172

10 10 10 1 10

10

10

10-210- 1

1

10

duration !

Survivalfunction "(!)

N • 64N • 256N • 1024N • 2048

1 10 102 103

PDFforduration

-3

-4

-3 -2 -1

10

10

10-2

10- 1

-3

-4

duration !

(a)

(b)

Figure 7.6: a) The survival function for the BTW sandpile in re-scaled coordinatesXN and tN for N = 64, 256, 1024, 2048 when the durations are defined by puttinga small threshold Xc on the toppling activity. The function shows power-law be-havior with exponent )0.63 for avalanches smaller than system size. b) The pdffor duration times from simulations of Eq. (7.4) when avalanches are defined in thesame way as for the sandpiles. The dotted line has slope )1.63.

173

Chapter 8

Modeling temporalfluctuations in sandpiles

Joint with K. Rypdal

Abstract: We demonstrate how to model the toppling activity in avalanching sys-tems by stochastic di!erential equations (SDEs). The theory is developed as a gen-eralization of the classical mean field approach to sandpile dynamics by formulatingit as a generalization of Itoh’s SDE. This equation contains a fractional Gaussiannoise term representing the branching of an avalanche into small active clusters,and a drift term reflecting the tendency for small avalanches to grow and largeavalanches to be constricted by the finite system size. If one defines avalanchingto take place when the toppling activity exceeds a certain threshold the stochasticmodel allows us to compute the avalanche exponents in the continum limit as func-tions of the Hurst exponent of the noise. The results are found to agree well withnumerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. Thestochastic model also provides a method for computing the probability density func-tions of the fluctuations in the toppling activity itself. We show that the sandpilesdo not belong to the class of phenomena giving rise to universal non-Gaussian prob-ability density functions for the global activity. Moreover, we demonstrate essentialdi!erences between the fluctuations of total kinetic energy in a two-dimensionalturbulence simulation and the toppling activity in sandpiles.

Reference for this paper: M. Rypdal, K. Rypdal, Modeling of temporal fluctuations in

sandpiles, Preprint: arXiv:08073416v1 (2008), (Accepted by Physical Review E).

175

1 Introduction

The aim of this paper is to present a consistent framework for the modelling oftemporal fluctuations, including definition and computation of avalanche exponents,in sandpile (height) models such as the Bak-Tang-Wiesenfeld (BTW) and Zhangmodels [3, 57]. One of the defining properties of self-organized criticality is thatavalanche duration and size are quantities subject to scaling, i.e. pdur(") 3 "!%

and psize(s) 3 s!& [30, 19]. The calculation of the exponents / and . in thethermodynamic limit N ( 1 (Nd is the number of sites in the d-dimensionallattice) has proven to be a di"cult task in dimensions d = 2 and d = 3. Thisis partly due to the lack of simple finite-size scaling in models such as the BTWmodel [19]. Moreover, it has recently been pointed out [50] that the di"culty mayoriginate from the fact that the " and s do not scale when defined in the traditionalsense, which is to consider the duration of an avalanche as the time interval wheretoppling takes place between two successive zeroes in the toppling activity. In thispaper we shall denote avalanches defined this way as type-I avalanches.

The scaling property can be restored, however, if one defines the duration of anavalanche as the time interval when the toppling activity x(t) exceeds a prescribedthreshold xth. We shall use the term type-II avalanches when the start and end of anavalanche is determined via such a threshold criterion. The idea of using a thresholdon the toppling activity in the definition of avalanches was first introduced in [48],where it was argued that any avalanche analysis of real-world activity time seriesmust define avalanches from a threshold, since there is no way to uniquely determinewhether a non-negative continuous-valued experimental quantity is actually zero,or just small. For type-II avalanches it can be shown by numerical simulation thatthe quiet times between avalanches in the BTW model are power-law distributed.For type-I avalanches the quiet times only depend on the statistics of the driver,which is usually assumed to be Poisson distributed.

The present work represents the first systematic investigation of type-II avalanchestatistics for the BTW and Zhang sandpiles. We are particularly interested in thecontinuum limit where the system size L = 1 is considered fixed and the spatialresolution increases as N ( 1. The power-law statistics of avalanche observableshave cuto!s for large avalanches due to the finiteness of the system, but as we in-crease the resolution we see an increasing range of scaling for smaller avalanches.As N ( 1 we keep the threshold fixed relative to the time-averaged activity !x",which scales as !x" 3 ND1 , where 0 < D1 < 2. For the BTW sandpile, numericalsimulations yields D1 = 0.86 [50]. This means that the number of overcritical sitescorresponding to the threshold value diverges like xc 3 ND1 in the limit N (1.

We model the toppling activity in the continum limit by a stochastic di!erentialequation [58] for a normalized toppling activity X(t). In its simplest form this

176

equation is on the formdX(t) = #

!X(t) dW (t) , (8.1)

where W (t) is the Wiener process. Without the factor!

X(t) on the right handside we would simply have that X(t) = #W (t) + X0 is a Brownian motion withdi!usion coe"cient D = #2/2. This factor, however, gives rise to a non-uniform(X-dependent) di!usion coe"cient D = #2X(t)/2, and the stochastic process X(t)will have non-stationary increments. This model can be perceived as a continuousversion of the classical mean field theory of sandpiles [29]. We use its correspondingFokker-Planck equation to derive that / = 2 and . = 3/2 when avalanches aredefined in the type-I sense. This is the same results obtained by mean field theory[29, 19, 30]. For type-II avalanches the e!ect of the non-uniform di!usion coe"-cient vanishes for avalanches of durations short compared to that of a system-sizeavalanche, and for the purposes of calculating / and . we can assume that thethe toppling activity is a standard Brownian motion. Solving the Fokker-Planckequation for Brownian motion (or equivalently using the known distribution of firstreturn times in Brownian motion) we obtain / = 3/2 and . = 4/3.

For the modeling of non-trivial sandpile models (BTW and Zhang) the stochasticdi!erential equation takes the form

dX(t) = f(X) dt + #!

X(t) dWH(t) . (8.2)

Two important generalizations of the stochastic model are included here. First,from numerical simulations of sandpiles we find that for small activities X(t) thereis an e!ective positive drift term. This term is dominant for very small activity,since the di!usion term is negligible for very small X(t) due to the X-factor inthe di!usion coe"cient. The positive drift term is X-dependent and quickly de-creases as X increases, but strongly influences the avalanche statistics because itcontributes to prevent avalanches from terminating when X(t) approaches zero.We believe that this e!ect is responsible for destroying the scaling of avalancheduration and size (when these are defined in the type-I sense). However, if thedrift term is small compared to the di!usion term for X > Xth, the drift term willnot a!ect the avalanche statistics if one employs a threshold Xth to define type-IIavalanches. This explains why scaling of size and duration is restored when whentype-II avalanches are introduced. Another generalization, which is essential forthe avalanche statistics, is the Hurst exponent H of the noise term. The mean-fieldapproach to sandpiles implicitly assumes that H = 1/2. However, this is not thecase for the BTW and Zhang models. Actually, analysis of numerical simulations ofthe sandpiles show that H = 0.37 for the BTW model and H = 0.75 for the Zhangmodel.

As in the mean-field model, the e!ect of the non-uniform di!usion coe"cientvanishes as the threshold increases, and hence keeping the threshold Xth fixed andincreasing N we can, for the purposes of computing / and . for avalanches where

177

X never grows much greater than Xth, consider the toppling activity as a fractionalBrowian motion. Using the result of Ding and Yang [24], that the first return timein fractional Brownian motion scales like 3 "H!2 , we obtain the general results/ = 2)H and . = 2/(1 + H).

Although the drift term and the non-uniformity of the di!usion coe"cient arenot important to calculate the avalanche exponents for type-II avalanches whoseduration are short enough not to be limited by the finite system size, they areimportant on the time scales where the toppling activity is a stationary process.These are scales su"ciently long that the toppling activity of avalanches is limitedby the boundaries. The stochastic equation (8.2) is fully equipped to handle thesetime scales. A good example of the applicability of these aspects of the stochasticmodel is the computation of the probability density function (PDF) of the temporalfluctuations in the activity signal itself.

For weakly driven sandpiles the PDFs of the fluctuations in the toppling activityare stretched exponentials. This result is reproduced by simulation of the stochasticmodel [50]. For stronger driving the activity exhibits fluctuations which are moreconfined around a mean value where the drive and dissipation balance each other. Ithas been claimed that the PDFs of the toppling activity in sandpiles are examples ofuniversal Bramwell-Holdsworth-Pinton (BHP) distributions [13, 14], a certain classof asymmetric PDFs commonly seen in complex systems. Our sandpile simulationsshow that the BHP distributions can only be seen if one fine tunes the driving rateto a certain value, and for other driving rates the PDFs belong to a much widerclass of distributions. For su"ciently strong drive Gaussian PDFs are observed.These can also be obtained from the stochastic model if one correctly models thedrift term in this parameter range.

The rest of the paper is structured as follows: In Sec. 2 we explain and derivethe stochastic model for the toppling activity. In Sec. 3 we compute the avalancheexponents for type-I and type-II avalanches for the mean field case (H = 1/2)by solving a Fokker-Planck equation, and for H -= 1/2 we compute the avalancheexponents for type-II avalanches as a function of H. The results allow us to predictthe avalanche exponents for sandpile models by computation of the Hurst exponents.These results are then tested against numerical simulations of the BTW and Zhangmodels and are shown to agree well. In Sec. 4 we present results which indicate thattype-II avalanches exhibit so-called finite-size scaling, even though type-I avalanchesdo not.

In Sec. 5 we use the stochastic theory to calculate the PDFs of the toppling ac-tivity signal, both for strongly driven sandpiles and in the weak driving limit. Themethod is finally applied to the fluctuations in kinetic energy in a two-dimensionalturbulence simulation. In this case the process is given by a di!erent kind of stochas-tic di!erential equation:

dX(t) = b dt + c ea X dW (t) . (8.3)

178

0 5000 10000 15000 20000 25000 30000

0

100

200

300

400

500

0 10 20 30 40 500.0

0.1

0.2

40

20

0

20

40

0 2 4 6 8 10 12

0

2

4

6

8

10

0 2 4 6 8 10 12

0.20.00.20.40.6

activity xN

xN (

t)

time t

0 5000 10000 15000 20000 25000 30000

time t

xN

(t)

!

activity xN

P[x

N+

xN |

xN]

!

E( x|x)!

Var ( x|x)

E( x|x)

!

!

xN

xN

(t)

!

-

(a) (c)

(b) (d)

x=10

x=20x=30

Figure 8.1: a) A realization of the toppling activity x(k) in the BTW sandpile. b) The

increments #x(k) = x(k + 1)! x(k) of the trace in (a), showing that #x(k) is large when

x(k) is large. c) Conditional PDFs of x + #x for x = 10, 20, 30 respectively. d) The

conditional mean and variance of #x versus x.

This equation gives rise to a Fischer-Tippet-Gumbel (FTG) distribution [26, 21],which is very close to the BHP. The di!erences between the stochastic models forsandpiles activity and kinetic energy fluctuations in 2D turbulence may representan essential distinguishing feature between 2D turbulent dynamics and the kind ofavalanching dynamics which are observed in the classical sandpile models.

In Sec. 6 we summarize and conclude the work.

2 The stochastic model

Let x(k) denote the number of overcritical sites at time step k in a sandpile. Acommon feature of (height-type) sandpile models is that the typical size of incre-ments #x is proportional to the square root of the toppling activity x. To be moreprecise, the conditional probability of an increment #x(k) = x(k + 1)) x(k), givenx = x(k), is

P (#x|x) =14

2%#2xe!

"x2

2(2x . (8.4)

In figure 8.1 this property is verified for the BTW model ( it holds in the Zhangmodel as well). This result can be explained as follows: At a given time k there

179

are x = x(k) overcritical sites, which we can enumerate i = 1, 2, . . . , x. In thenext time step k ( k + 1 each site i distributes energy to its neighbors, and willusually (always in the BTW model) become subcritical. If none of the neighborsreceive su"cient energy to become overcritical, the contribution to #x from site iis (i = )1. If exactly one of the neighbors become overcritical, then (i = 0, and soon. For the two-dimensional models the maximal value of (i is 4 (3 for the BTWmodel) since a site maximally can excite 4 neighboring sites. Hence, for this case,we consider (i to be random variables with realizations in {)1, 0, 1, 2, 3, 4}. Therandomness originates from the local configuration in the vicinity of the overcriticalsite, which for these purposes is considered to be random. In other words, we thinkof the configuration on the lattice as a random background.

As an approximation we consider the di!erent realizations of (i (at a fixed time k)as independent of each other. We also consider the distribution of (i to be identicalfor all overcritical sites and for all times. In this approximation the total increment#x can be written as a sum of independent, identically distributed, random variables#x = (1 + · · · + (x and by the central limit theorem we have (8.4) provided thatthe local means !(" are zero. Then #2 = !(2" = ()1)2 p!1 + · · · + 42 p4, wherep = (p!1, . . . , p4) is the probability vector for the local increment processes.

If the local processes at di!erent times k are independent of each other, thenx(k) is a Markov process which satisfies a stochastic di!erence equation

#x(k) = #!

x(k) w(k) ,

where w(k) is a stationary, normalized, and uncorrelated Gaussian process, i.e.w(k) = W (k + 1))W (k), where W (t) is the Wiener process. Under a rescaling oftime t = k $t and X(t) = x(k) $x we have

#X(t) = X(t + $t))X(t) = $x$x(k + 1)) x(k)

%

= $x#!

x(k)w(k) = $x1/2 #!

X(t)$W (k + 1))W (k)

%

=$$x

$t

%1/2#

!X(t)

$W (t + $t))W (t)

%.

In the last step we used the self-a"nity of the Wiener process, W (t/$t) d=$t!1/2W (t).For $x = $t we have a well defined model in the limit $t, $x ( 0, namely the Itostochastic di!erential equation Eq. (8.1).

The first generalization of this model is obtained if we relax the requirement thatthe local increment processes (i(k) at time k are independent of the local incrementprocesses (j(k#), j = 1, . . . , x(k#) at previous times k# < k. In this case we need tomodel memory e!ects in the stationary Gaussian process

w(k) =#x(k)

#!

x(k).

180

From the power spectrum or the variogram of the activity signal from numeri-cal simulation of the BTW and Zhang models we find that w(k) can be accu-rately modelled as a colored noise characterized by a Hurst exponent H. That is,w(k) = WH(k + 1) ) WH(k), with WH being a normalized (di!usion coe"cient= 1) fractional Brownian motion (fBm). If we perform the rescaling t = k $t andX(t) = x(k) $x in the case H -= 1/2 we obtain

#X(t) =$ $x

$t2H

%1/2#

!X(t)

$WH(t + $t))WH(t)

%,

and by requiring that $x = $th, with h = 2H, we obtain the stochastic di!erentialequation

dX(t) = #!

X(t) dWH(t) . (8.5)

If we assume that the the stochastic process X(t) is self-a"ne with self-a"nityexponent h, i.e. X(st) d= shX(t), it is easy to verify that Eq. (8.5) is invariantwith respect to the transformation t ( st if h = 2H. Thus, the exponent h = 2His the self-a"nity exponent of the process X(t), where H is the Hurst exponentdetermining the color of the noise process driving the stochastic di!erential equa-tion. Observe that the reason why h -= H is the non-stationarity of the incrementprocess due to the the factor

!X(t) in Eq. (8.5). Note also that the case H = 1/2

corresponds to h = 1.The self-a"nity of X(t) described by Eq. (8.5) implies that there is no upper

bound on the fluctuations on increasing time scales, i.e. Eq. (8.5) describes theactivity of an infinite sandpile where the activity is never influenced by the systemboundaries. From a physical viewpoint, however, it is more interesting to con-sider the dynamics of a finite sandpile in the continuum (thermodynamic) limit,and for this purpose it is natural to let the scaling factor $x depend on N suchthat XN = xN$x(N) is bounded in the limit N ( 1, for instance such thatlimN$%max(XN ) = 1. Such a bound on X(t) can be obtained by the introductionof a drift term f(X) dt leaving the stochastic equation in the form of Eq. (8.2),where f(X) is negative for large X. The form of f(X) can be found from sandpilesimulations by computing the conditional mean E(#x|x) of the increments, and isshown in figure 8.1d. It appears that f(X) is a decreasing function, positive forsmall X and negative for large X, and f(X) = 0 for a characteristic activity Xc 3 1.Without the stochastic term the drift term establishes Xc as a stable fixed pointfor the dynamics.

Since f(X = 0) > 0 solutions of Eq. (8.2) with initial condition X(0) > 0 existand are positive for all t > 0. This means that while Eq. (8.5) has solutions forwhich X(t) = 0 after a finite time (avalanches terminate), avalanches described byEq. (8.2) will never terminate in the meaning X(t) = 0 (type-I avalanches). Thissignifies that type-I termination never occurs in the the continuum limit. If sandpiles

181

of increasing N are simulated, and the type-I avalanche durations are computed inthe rescaled coordinates, the durations generally grow without bounds for increasingN . The reason is that the e!ective threshold for type-I termination in a discretesandpile is xN = 1, but in rescaled coordinates this threshold XN = xN $x(N) goesto zero as N (1. As this rescaled threshold vanishes the duration in rescaled timegoes to infinity. Since the type-I termination is a discreteness e!ect, the resultingPDFs of avalanche durations depends on N (system discreteness) and are not power-laws. As we shall demonstrate later, the introduction of activity thresholds whichare defined in the rescaled coordinates, and hence remain finite in the continuumlimit, will give rise to PDFs of durations of type-II avalanches which converge to aspecific power-law in this limit.

Eq. (8.2) remains valid also for sandpiles which are driven by continuous feedingof sand during avalanches. The drive adds a positive contribution to the driftfunction f(X) for X < Xc, but a negative contribution for X > Xc, because forlarge activites f(X) mainly accounts for the increased boundary losses. The resultis a steeper f(X), which tends to confine the activity closer to the fixed point Xc.On the other hand the increased drive also increases the di!usion coe"cient (byincreasing #) due to a larger number of new active clusters initiated per unit time.The net e!ect is a positive shift of Xc and that the fluctuations in X are confinedto a smaller region around Xc. For su"ciently strong drive the range of variationin X(t) becomes so small that the di!usion coe"cient does not vary much. It isnevertheless important to model it correctly in order to calculate the PDFs of thefluctuations in toppling activity.

3 Calculation of avalanche exponents

We denote the stochastic model Eq. (8.1) (which is Eq. (8.2) with H = 1/2 andf(X) = 0) the mean field model of sandpiles. This is because its underlying assump-tions and the results derived from it coincide with what is known as the mean fieldsolution of sandpiles in the literature [29]. Eq. (8.1), together with its correspondingFokker-Plack formulation can be used to calculate the avalanche exponents / and.. The idea is that each avalanche corresponds to a realization X(t) with someinitial condition X(0) = X0 (X0 < Xmax). The avalanche propagates until therealization X(t) terminates at t = t1 in the meaning that X(t) > 0 for t < t1 andX(t1) = 0. Calculating the ratio of surviving realizations at di!erent times t inan ensemble will provide information about the distribution of avalanche durations.The avalanche size distribution can then be obtained by using a general relationshipbetween the self-a"nity exponent h = 2H = 1 and the duration statistics.

The scenario outlined above can be mathematically formulated as follows: Let

182

P (X, t) be the probability density function such that

Prob[X(t) exists and b < X(t) < a] =2 b

aP (X, t) dX .

Then the the probability +(t) that an avalanche still runs after a time t (we call itthe survival function) is given by

+(t) =2 %

0P (X, t) dX , (8.6)

and the probability density function for durations is pdur(") = )+#("). The densityP (X, t) can be calculated by solving the Fokker-Planck equation

-P

-t=

#2

2-2

-X2(X P ) (8.7)

on X % [0,1), t % [0,1), subject to an absorbing boundary condition limX$0 XP (X, t) =0 and an initial condition P (X, 0) = $(X )X0).

To correctly incorporate the absorbing boundary condition we let U = XP ,and solve the corresponding Fokker-Planck equation for U with boundary conditionU(0) = 0 to get

P (X, t) =2 %

0G(X, Y, t) P (Y, 0) dY ,

where

G(X, Y, t) =12

TY

X

2 %

0J1(s

4Y ) J1(s

4X) exp

7)#2 s2

8t

8s ds .

Remark 8.1 The most elegant way to obtain the solution is to use the integraltransform pair

F (s) =12

2 %

0F (X) J1(s

4X)

4X dX

F (X) =14X

2 %

0F (s)J1(s

4X) s ds .

Taking the transform of the Fokker-Planck equation we obtain the ODEs

-P

-t(s, t) = )#2 s2

4P (s, t) ,

which we can solve and take the inverse transform. It is also easy to solve theFokker-Plack equation for U = XP by separation of variables.

183

If P (X, 0) = $(X ) X0) the solution of the absorbing boundary problem isP (X, t) = G(X, X0, t). Moreover

limX$0

P (X, t) =4

X0

4

2 %

0s2 J1(s

!X0) exp

7)#2 s2

8t

8ds =

4 X0

#4t2exp

7)2 X0

#2 t

8.

(8.8)From Eqs. (8.6), (8.7), (8.8), and the absorbing boundary condition at X = 0 wefind that

d+

dt= )#2

2lim

X$0P (X, t) = )2 X0

#2t2exp

7)2 X0

#2 t

8.

Hence the PDF for avalanche durations " is

pdur(") =2 X0

#2"2exp

7)2 X0

#2 "

8,

and for " >> 2X0/#2 we have pdur(") 3 "!%, with / = 2.This result crucially depends on the correct formulation of the Fokker-Planck

equation. If the stochastic process X(t) were a classical Brownan motion, theFokker-Planck equation would have the form of the standard heat equation, andby performing the analogous calculations for this equation we get pdur(") 3 "!3/2.The same scaling relation (3 "3/2) is obtained if one (incorrectly) employs theStratonovich formulation of the Fokker-Planck equation rather than the Itoh form[58].

From the derivation of the stochastic model we have seen that the process X(t)has a self-a"nity exponent h = 2H (for H = 1/2 we have h = 1). This meansthat X(t) disperses like 3 th. For long avalanches this implies that the size of theavalanche scales as

s =2 (

0X(t) dt 3

2 (

th dt 3 "h+1 . (8.9)

This property is easy to check directly by studying the relation between durationand sizes of avalanches in sandpiles. We find that the relation holds for largeavalanches, whereas for very small avalanches s 3 "1. If the survival function scalesas +(") 3 "!' = "!%+1, then using (8.9) together with psize(s) ds = pdur(") d"yields that if psize(s) 3 s!& , then

. =1 + h + $

1 + h=

/ + h

1 + h. (8.10)

In the case H = 1/2 (h = 1) and / = 2 this gives . = 3/2.This mean-field solution is known to be quite correct for the random neighbor

sandpile model [20], but numerical simulation shows that is fails for type-I as wellas type-II avalanches in the BTW and Zhang models. The computation of theseexponents are shown for these models in figures 8.2, 8.3, and 8.4. The computation

184

1

10

102

103

104

105

106

varianceofX(t)

1

10

102

103

104

105

varianceofX(t)

10 102 103 104 105

time t

10 102 103

time t1

N=256

N=1024

N=2048

N=256

N=64

N=512(a) (b)

Figure 8.2: a) Determining h = 2H in the Zhang model through the calculation of the

variance of the activity X(k) in avalanches still running at time t. On shorter time scales

there is a Hurst exponent H = 0.5 (the first dashed line has slope 2h = 4H = 2) on

longer time scales the Hurst exponent is H = 0.75 (the second dashed line has slope

2h = 4H = 3). b) Determining h = 2H in the BTW model by the same method as in (a).

In this case we have H = 0.37.

of h presented in figure 8.2 for type-I avalanches yield H = h/2 = 0.37 for theBTW model, and H = 0.75 for the Zhang model. This invalidates the Fokker-Planck formulation, which can be strictly justified only for a white noise sourceterm (H = 1/2). This is one reason for the failure of the mean-field approach, butthere are also others.

For avalanche duration and size type-I avalanches do not yield good power-lawPDFs. The reason for this was discussed in the previous section: letting avalanchesterminate when xN (t) = 0 in a sandpile with Nd sites corresponds to using ane!ective threshold for termination in the rescaled coordinates XN (t) which goes tozero as N (1. The termination process depends on the discreteness of the systemand one cannot expect convergence to scale-invariant behavior in the continuumlimit.

For type-II avalanches Figs. 8.3 and 8.4 show good scaling for duration and sizein BTW and Zhang models, but the scaling exponents di!er from those of the mean-field approach. The discrepancy is partly due to the fact that the sandpile modelshave H -= 1/2, but is also related to the observation that the introduction of a finitetermination threshold Xth modifies the exponent h as it appears in Eqs. (8.9) and(8.10). In the following we shall demonstrate that it is possible to obtain analyticalresults for type-II avalanches from the stochastic model which are in agreement withthe corresponding sandpile simulations.

First we observe that omission of the drift term will only have e!ect on avalanches

185

1

10

102

103

104

105

surv

ivalfu

ncti

on

1 10 102 103 104 105

duration !

1

10

102

103

104

105

# o

favala

nches

siz

e >

s

10 102 103 104 105 106 107

size s

(a) (b)

N = 256N = 1024

N = 2048N = 2048

N = 1024

N = 256

Figure 8.3: a) The survival function of avalanches in the BTW model computed with

thresholds "X#/3 (type-II) and without thresholds (type-I, the dotted curves). The dashed

line corresponds to " = 2 ! H = 1.63. b) The probability density function of avalanches

with size > s in the BTW model computed with thresholds "X#/3 and without thresholds

(the dotted curves). The dashed line corresponds to # = 2/(1 + H) = 1.49.

which are so large that they are strongly limited by boundary dissipation. Next,we notice that the e!ect of the positive drift term for small activities is eliminatedfor type-II avalanches if Xth > Xc. In other words, it is only the cut-o!s of thepower-law PDFs due to finite system size which are lost by this omission. Thus, byconsidering a threshold Xth which is not much smaller than the mean activity !X",and considering only avalanches which are su"ciently small not to be influenced bythe boundary dissipation, we have X(t) 3 Xth and thus can justify the substitutionX(t) ( Xth(t) on the right hand side in Eq. (8.5). This equation then reducesto the equation for an fBm with Hurst exponent H. For an fBm the duration ofavalanches is given by the return time statistics for WH(t), which is known to scalelike "H!2 [24], i.e. we have

/ = 2)H . (8.11)

Now we need to address a slightly subtle point: the exponent h, as defined inEq. (8.9), is h = 2H for times " so large that X(") 5 X(0). Only on such timescales will the e!ect of the factor

!X(t) in the stochastic term show up in the

scaling. However, this makes sense only for type-I avalanches where we can chooseX(0) < !X". For type-II avalanches we have X(0) = Xth, and it is more natural toconsider the opposite limit where " is so small that X(") @ X("))Xth < X(0) =Xth. On these time scales the activity measured relative to the threshold levelscales like an fBm with Hurst exponent H, i.e. X(t) 3 tH , because X(t) = Xth.The avalanche size of type-II avalanches defined as s(") @

B (0 X(t) dt then scale as

3 "1+H . Thus, for type-II avalanches we have h = H, and hence Eq. (8.10) for this

186

1

10

102

103

104

105

su

rviv

alfu

ncti

on

102

103

104

# o

fav

ala

nch

es

siz

e >

s

1 10 102 103 104

duration !10 102 103 104 105 106

size s

N = 512

N = 256N = 64N = 64 N = 256

N = 512

(a) (b)

Figure 8.4: a) The probability of having avalanches of duration > $ in the Zhang model

computed with thresholds "X#/3 and without thresholds (the dotted curves). The first

dashed line corresponds to " = 2 ! H = 1.50 obtained with H = 0.50, and the second

dashed line corresponds to " = 2!H = 1.25 obtained with H = 0.75. b) The probability

of having avalanches with size > s in the Zhang model computed with thresholds "X#/3

and without thresholds (the dotted curves). The first dashed line corresponds to # =

2/(1 + H) = 1.33 obtained with H = 0.50, and the second dashed line corresponds to

# = 2/(1 + H) = 1.14 obtained with H = 0.75.

case reduces to

. =2

1 + H. (8.12)

In the mean field limit H = 1/2 these exponents reduce to those given in the righthand column in Table 8.1. The results obtained analytically by approximatingthe coe"cient on right hand side in the stochastic equation by its threshold valuehas been verified by numerical solutions of the full equation. Similar results forfractional Brownian motion have been obtained in [56] and [18].

Table 8.1: Exponents in mean-field solutions (H = 1/2) of the stochastic equationwith zero threshold (type-I) and large threshold (type-II).

type-I avalanches type-II avalanchesH 1/2 1/2h 1 1/2/ 2 3/2. 3/2 5/4

187

Eqs. (8.11) and (8.12) show that the calculations of / and . reduces to deter-mining the Hurst exponent of the normalized increment process

w(k) =#x(k)!

x(k),

which can easily be constructed from the toppling activity signal. The correspondingHurst exponent of the motion

W (k) =k"

i=0

w(k)

can be calculated using standard techniques such as taking the power spectrum orconstructing variograms. Alternatively we can find H by computing the varianceof X(t) in an ensemble of realizations starting with small initial values of X. Thisvariance scales like !X2(t)" 3 t2h = t4H . figure 8.2 shows this computation forthe BTW and Zhang models. Since we find H = 0.37 for the BTW model thetype-II scaling exponents are / = 1.63 and . = 1.49. This is verified by numericalcalculation of / and . using a threshold !x"/3, as shown in figure 8.3.

In the Zhang model the process w(k) is more complicated. figure 8.2 shows thatW (k) has a Hurst exponent H = 0.5 on short time scales and a di!erent Hurstexponent H = 0.75 on longer timescales. This means that short avalanches shouldsatisfy the mean-field solution / = 1.5 and . = 1.33, whereas longer avalanchesshould have exponents / = 1.25 and . = 1.14. All of these predictions are verifiedby the direct calculation of / and . as shown in figure 8.3.

For increasing N the long-avalanche scaling dominates an increasing portionof the graph, so in the continuum limit the mean-field solution only prevails atinfinitely small scales in rescaled coordinates. It is however a nice verification ofour method to see that the relation between the avalanche exponents and the Hurstexponent correctly predicts the avalanche exponents on short time scales as well.

These results on the BTW and Zhang models are summarized in table 8.2.

Remark 8.2 The numerical simulations of the Zhang model are run using the stan-dard toppling rule zi ( 0 and zj ( zi + zi/4 if zi is overcritical and j is a nearestneighbor of i. Whenever the configuration has no overcritical sites a random sitei is chosen with respect to uniform probability and a mass * is added to this site:zi = zi+*. In the simulations presented in this paper we use * = 0.1. In the stronglydriven Zhang models presented in Sec. 5 the feeding times (which can now be duringavalanches) are Poisson distributed and we have used * = 0.25.

For the BTW model we have used the standard toppling rule zi ( zi ) 4 andzj = zj + 1 through out the paper. As usual a mass * = 1 is added to a random sitewhenever the configuration is stable.

188

Table 8.2: Exponents for type-II avalanches computed from H = 0.37 in the BTW model,

and H = 0.50 (short time scales) and H = 0.75 (long time scales) in the Zhang model.

BTW Zhang Zhang(short times) (long times)

H 0.37 0.50 0.75/ 1.63 1.50 1.25. 1.49 1.33 1.14

4 Finite-size scaling for type-II avalanches

A systematic technique for the determination of the avalanche exponents / and. is the so called moment analysis [19]. We illustrate how this works for the sizedistribution of the BTW model. The analysis confirms our result . = 0.49 for theBTW model for type-II avalanches.

Let psize(s;N) be the PDF of avalanche size s in the two-dimensional BTWmodel with N2 sites and a threshold !x"/3. Let us assume finite-size scaling (FSS).This means that for N, s 5 1 we have

psize(s;N) A s!& G(s/ND) , (8.13)

for some exponent D > 0, where G(r) falls o! quickly for r > 1. From Eq. (8.13 )we observe that

!sq" A ND(1+q!&)

2 %

1/ND

rq!& G(r) dr .

The integral tends to a constant as N (1 so we have !sq" 3 ND(1+q!&). Thus, ifwe plot !sq" versus N in a log-log plot the slope of a fitted straight line will give usan estimate of the exponent 4(q) = D(1+q).) for q = 1, 2, 3, . . . . figure 8.5a showsthe computation of moments of s as a function of N for type-II avalanches obtainedfrom simulation of the BTW model, and figure 8.5b shows the exponent 4(q) versusq. We find that 4(q) 3 q2.81, hence that D = 2.81. Moreover, when plotted ina log-log plot, the intersection of 4(q) with the first axis corresponds to the value.)1. Hence, based on our predictions this intersection should be in the point 0.49.This value is plotted as a dotted horizontal line in figure 8.5b, confirming our resultwith good accuracy.

Strictly, this method requires that we have data collapse when re-scaling theavalanche sizes by s/ND. The shape of the scaling function G(r) can be seenby plotting s&psize(s;N) versus s/ND. This is shown in figure 8.6. The datacollapse for the available system sizes is not perfect, but it is better than for type-I

189

102 103

105

1010

1015

1020

1025

-1 0 1 2 3 4 5

0

2

4

6

8

10q’t

h m

om

ent

E(s

)

q

D(q

-1-!

)

qN

q = 1

q = 2

q = 3

q = 4

(a) (b)

Figure 8.5: a) The moments "sq# plotted as functions of N for the size distribution of the

BTW model ith respect to a threshold "X#/3. b) The shape of the structure function %(q).

The slope of the line is D = 2.81 and the intersection with the first axis is # ! 1 = 0.49.

This value is indicated by the dotted vertical line.

avalanches. In fact, it seems that we might have convergence to a N -independentscaling function as N ( 1, and that the lack of data collapse observed here issimply due to the fact that we are not able to simulate su"ciently large sandpiles.Thus the general lack of scaling for for type-I avalanches, including the finite-sizescaling, seems to be restored for type-II avalanches.

5 The PDFs of the toppling activity

When calculating the PDFs of the toppling activity signal we have to distinguishbetween the weakly and strongly driven sandpiles. For the weakly driven sandpilesthe toppling activity covers a large range and the high-activity tail of the PDFdecays like a stretched exponential. This property can be reproduced by simulationsof Eq. (8.2) where f(X) decays exponentially to zero as X increases. figure 8.7compares the PDF obtained from such a simulation of the stochastic di!erentialequation (run with H = 0.37) with the PDF of the weakly driven BTW model.The stochastic model is run by initiating new avalanches (realizations) wheneverthe previous avalanche terminates, thus representing the classical slow drive of thesandpile model. Since the Hurst exponents of the Zhang and BTW models aredi!erent from 1/2, the application of the Fokker-Planck equation can not be used to

190

N = 2048

N= 1024

N = 256 N= 64

10-9 10-7 10-5 10-3 10-1103

104

105

s /ND

s P

(s;N

)!

Figure 8.6: Attempted data collapse for the avalanches in the BTW model with respect

to a threshold "X#/3. We have plotted s"!1R"

spsize(s

#) ds# as a function of s/LD with

D = 2.81 for N = 64, 256, 1024, 2048.

calculate the shape of the PDFs, and thus we have to rely on numerical simulations.It is also interesting to study the PDFs of toppling activity in strongly driven

sandpiles. In particular in the light of the recent claims that this toppling activ-ity belongs to a class of fluctuating global quantities with a universal non-gaussianshape [13, 14]. These PDFs are reminiscent of distributions derived from the ex-treme value theory of statistics [26, 21], which deals with a sequence of n identicallydistributed, independent random variables y1, . . . , yn. If the tail of the PDF ofthese variables decays faster than any power-law, the PDF of the m’th largest valuedrawn from each of M realizations of this sequence converge to the Gumbel classof stable distributions in the limit M (1:

Gk(y) = K(e!!(y!s)e!e")(y"s))m. (8.14)

The distribution for the largest value in each realization (m = 1) is often calledthe Fischer-Tippet-Gumbel distribution (FTG). The FTG with zero mean and unitvariance requires K = ( = %/

46 and s = 1

46/%, where 1 = 0.58 is the so-called

Euler constant.A distribution obtained from the spin-wave approximation to the 2D XY model

for equilibrium crititical fluctuations in a finite-size magnetized system is the so-called Bramwell-Holdsworth-Pinton (BHP) distribution [14], which corresponds toEq. (8.14) with k having the non-integer value m = %/2 = 1.57. With K = 2.16,/ = 1.58, ( = 0.93 and s = 0.37 this distribution has zero mean and unit variance.

191

0.0 0.5 1.0 1.5 2.010-2

10-1

1

10

activity X

PD

F o

f ac

tivit

y

Figure 8.7: The PDF of the fluctuations in the toppling activity of the slowly driven

BTW sandpile together with the corresponding PDF produced from the stochastic model.

The dashed line is a fitted stretched exponential.

The di!erence between the normalized FTG and BHP distributions is rather small,and it is di"cult to distinguish between the two based in experimental and numericaldata.

Analysis of our simulations show the claim that the toppling activity in stronglydriven sandpiles has a PDF similar to the BHP or FTG distributions is wrong, unlessthe driving rate of the system is fine tuned to some particular value. Actually, thenormalized PDFs of the toppling activity is only insensitive to variation of thedriving rate in the limits of weak and strong drive. In the limit of weak drive thePDFs are stretched exponentials as shown in figure 8.7 and in the limit of strongdrive the PDFs are close to Gaussian.

figure 8.8 shows the PDFs of the toppling activity in the strongly driven Zhangmodel where the feeding rate is a Poisson process with characteristic time scales& = 0 (feeding one unit * of mass in every time step), & = 2.3, and & = 3.0. For& = 0 and & = 2.3 the sandpile is running, in the meaning that avalanches neverterminate. For & = 3, the sandpile is no longer running, and we see a deviationfrom the Gaussian shape in the left tail of the PDF.

The Gaussian PDFs for strongly driven sandpiles can actually be explained interms of the stochastic model in Eq. (8.2). The idea is that the range of fluctuationsis confined by the drift term f(X), which now has a di!erent shape than in theslowly driven sandpile. figure 8.9 shows the shape of both the di!usion coe"cientD(X) = #2X/2 and the drift term for the strongly driven Zhang model. The

192

-3 -2 -1 0 1 2 3

10-2

10-1

activity X

! = 0

! =2.3

! = 3

PD

F P

(X)

Figure 8.8: The PDF of the fluctuations in the toppling activity of the strongly driven

Zhang sandpile driven BTW sandpile for di!erent values of & (see text). The dashed line

is a fitted Gaussian.

di!usion coe"cient has the same form as for the slowly driven sandpile, whereasthe drift term is well approximated by a parabola: f(X) = )aX2 + bX + c. Asexplained in Sec. 1 this drift term confines the fluctuations in toppling activity toa bounded region around the positive root Xc of f(X).

Due to the higher rate of random feeding, the memory e!ects described bythe Hurst exponent H -= 1/2 becomes inessential in the strongly driven sandpile,allowing us to give an approximate description of the time dependent PDF throughthe Fokker-Planck equation

-P

-t= ) -

-X

$f(X) P

%+

#2

2-2

-X2(X P ) . (8.15)

Stationary solutions of this equation must satisfy

) d

dX

$f(X)P

%+

#2

2d2

dX2(X P ) = 0 ,

and substituting )aX2 + bX + c for f(X), we have the Gaussian solution

P (X) =142% $

exp7) (X ) µ)2

2 $2

8,

withµ =

b

aand $ =

142

#

a.

193

((

(a)

(b)

(c)

(d)

Figure 8.9: a) Part of a time series for the toppling activity in a strongly driven Zhang

sandpile. b) The increments #x(t) = x(t + #t) ! x(t) of the signal in (a). As for the

slowly driven case. c) The conditional variance of the increments #x given the value of x.

d) The conditional mean of increments. This curve is fitted to a parabola.

To further substantiate that the toppling dynamics of sandpiles is fundamentallydi!erent from the fluctuating quantities that give rise to BHP and FTG distribu-tions we apply the above analysis to the fluctuations in total kinetic energy in asimulation of two-dimensional Navier-Stokes turbulence. The two-dimensional ge-ometry is chosen because of the inverse energy cascade in k-space caused by mergingof small vortices and formation of large structures, reminiscent of formation of largeavalanches from emerging from localized random perturbation in sandpile models.In the simulation energy is injected via a source term on a characteristic, smallspatial scale throughout the simulation area, and is dissipated as loss through theopen boundary.

We compute the conditional mean and conditional variance of the incrementprocess. These results are presented in figure 8.10. We observe that contrary tothe sandpile models, the di!usion coe"cient for this process grows exponentiallywith X. Moreover, the drift term can be approximated by a small positive con-stant except when the kinetic energy is very large. This leads us to the stochastic

194

0 1000 2000 3000 4000 5000

10

15

20

25

10 12 14 16 18 20

0.1

0.2

0.3

0.4

0.5

0 1000 2000 3000 4000 5000-4

-2

0

2

4

-0.2

-0.1

0

0.1

0.2

time t

time t

kin

etic

en

erg

y X

(t)

!X

(t)

Var

(!X

|X)

Lo

g V

ar(!

X|X

)

X

X

E(!

X|X

)

10 12 14 16 18 20

X

(a)

(b)

(c)

(d)

Figure 8.10: a) Part of a time series for the kinetic energy in the 2D turbulence simulation.

b) The increments #X(t) = X(t + #t) ! X(t) of the signal in (a). As for the toppling

activity in sandpiles we see that the increments are large when the kinetic energy itself is

large. c) The conditional variance of the increments #X given the value of X. The inset

shows the logarithm of this variance versus X. The dashed curves corresponds to a fitted

exponential function. d) The conditional mean of increments. We see that the mean is

approximately constant for a large range of X.

di!erential equation Eq. (8.3), and the corresponding Fokker-Planck equation is

-P

-t= )b

-P

-X+

c2

2-2

-X2

$e2aXP

%. (8.16)

Stationary solutions of this equation must satisfy

)bdP

dX+

c2

2d2

dX2

$e2aXP

%= 0 .

We can put c = 1 without loss of generality, and obtain the solution

P (X) =12

e!(X!µ)/+e!e"(X"µ)/*

,

where 2 = 1/2a and µ = (1/2a) log (1/2a). This is the standard FTG distribution.figure 8.11 shows the normalized PDF for the fluctuations in total kinetic energy

195

-2 -1 0 1 2 3 4

10-2

10 1

kinetic energy

SDE

Fluid

Dashed : Gumbel

Solid : BHP

PD

F

-

Figure 8.11: The normalized PDF for the fluctuation of kinetic energy in the 2D turbu-

lence simulation together with a normalized FTG PDF (smooth solid curve), the normal-

ized BHP PDF (smooth dashed curve) and a PDF obtained from the simulation of Eq. 8.3

(dotted curve).

obtained from the fluid simulation along with a normalized FTG distribution andthe PDF obtained from the simulation of Eq. (8.3).

6 Conclusions

When output from simple models for avalanching systems are compared to obser-vational data of real-world systems one has to deal with the problem of establishinga correspondence between the model variables and the observables of the naturalsystem. In general, this is usually not an obvious task, since the model is usuallynot derived from first physical principles. The observation may be spatiotemporal,or just temporal. Likewise we may choose to analyze the spatiotemporal outputfrom an avalanche model, or just some spatially integrated quantity like the totalactivity variable in a sandpile, presented as a time series. In this paper we havefocused on the latter, and in particular on reproducing the statistical propertiesof such time-series by modeling the stochastic process by means of a stochasticdi!erential equation.

Since our interest is avalanching dynamics we focus on avalanche statistics, andwe have to face the problem of how to define what an avalanche is when our ob-servational data is in the form of a time series. In general we cannot expect thatsuch a definition is necessarily equivalent with a definition based on spatiotemporal

196

data, at least not in those cases where feeding of new “sand grains” occurs whileavalanches are running. In those cases several spatially separated avalanches mayrun simultaneously, but these cannot be separated in a pure temporal analysis. Forsuch a continuously driven model system the temporal signal may never be zero,and in an observational time series noise also makes it impossible to use a zerosignal condition to separate an avalanching state from a quiet state. Thus, thenatural way out is to define the avalanching state by means of a threshold level onthe activity signal, giving rise to the concept of type-II avalanches.

In this paper we have shown that the toppling activity in sandpiles, and alsoglobal kinetic energy in a 2D fluid simulation, can be modeled by stochastic dif-ferential equations. The modeling clarifies that the main discrepancy between themean-field approach and the actual BTW and Zhang models is related to the Hurstexponent of the activity process, which is di!erent for the two sandpile models. Italso clarifies the origin of the di!erences between the scaling exponents for type-Iand type-II avalanches, and why type-II avalanches exhibit clearer scaling charac-teristics than type-I avalanches.

It follows from the theory how to rescale coordinates to approach the thermo-dynamic limit, and the results obtained for finite-size scaling in the BTW model inSec. 4 gives a strong indication that this limit actually exists.

For continuously driven sandpiles the stochastic equation can be cast into aFokker-Planck equation due to the loss of memory caused by the random feeding.This allows an analytic solution for the activity PDF, which is a Gaussian for thesandpile models, but gives rise to the FTG distribution for the 2D fluid simulation.These results show that the non-Gaussian universal PDF described in [13, 14] is notrelevant for strongly driven sandpiles, but may be so for certain turbulence models.The stochastic theory relates the di!erence between Gaussian and FTG distributedactivity signal to a the di!erence between a linear and exponential X dependenceof the di!usion coe"cient in the stochastic equation.

197

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