Urn Nbn Se Vxu Diva-403-1 Fulltext

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Monomial Dynamical Systems

in the Fields ofp-adic Numbers

and Their Finite Extensions


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Acta WexionensiaNo 62/2005

Mathematics

Monomial Dynamical Systemsin the Fields ofp-adic Numbers

and Their Finite Extensions

Marcus Nilsson

Vxj University Press


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Monomial Dynamical Systems in the Fields ofp-adic Numbers and Their

Finite Extensions. Thesis for the degree of Doctor of Philosophy,

Vxj University, Sweden 2005

Series editors: Tommy Book and Kerstin Brodn

ISSN: 1404-4307

ISBN: 91-7636-458-5

Printed by: Intellecta Docusys, Gteborg 2005


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Abstract

In this thesis we investigate monomial dynamical systems over the fields ofp-adic numbers, Qp, or over finite field extensions of them. A finite extension

of Qp is denoted by Kp. The monomial dynamical systems are describedby iterations of h(x) = xn, where n 2 is an integer. Since the fields Kpare not algebraically closed, the number of r-periodic points, Pr(h(x),Kp),of h(x) in Kp will vary very much with p. We find explicit formulas forPr(h(x),Kp), by using methods from number theory, for example Mobiusinversion. One of the main parts of this thesis is the computation of thelimit

limt

1

(t)

pt

Pr(h,Kp),

where (t) denotes the number of primes p t and where the sum isextended over all prime numbers p t. We interpret this limit as theasymptotic mean value of the number of r-periodic points of h(x) in Kp,when p . For this to make sense, we must assume that the degree ofthe extension Kp/Qp is the same for all p.

We also study the dynamics of balls in Qp under the monomial h(x). Wewill call a cycle of balls a fuzzy cycle. Methods for calculating the numberof fuzzy cycles are presented.

In this thesis we also consider perturbed monomial systems over the fieldof p-adic numbers. This systems are generated by polynomials hq(x) =

xn

+ q(x), where the perturbation q(x) is a polynomial whose coefficientshave a small p-adic absolute value. We find sufficient conditions on theperturbation to guarantee a one to one correspondence of fixed points andcycles between the monomial and the perturbed system.

Keywords: p-adic numbers, discrete dynamical systems, number of cy-cles, perturbation, roots of unity, Mobius inversion, distribution of primenumbers.

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Acknowledgements

First of all, I would like to thank my supervisor Andrei Khrennikov forintroducing me to the subject of p-adic dynamical systems, and for his in-

spiring ideas and advices. I also want to thank my co-supervisor AndersMelin, University of Lund. Scientific contacts with him were very impor-tant during the writing of my master thesis and at the initial stages of mygraduate studies.

I thank Alain Escassut, Bertin Diarra at Universite Blaise Pascal inClermont-Ferrand, and Nicolas Mainetti at Universite dAuvergne for theirhospitality and the fruitful discussions. I would also like to thank FrancoVivaldi at Queen Mary University of London and Robert Benedetto atAmherst College for comments, ideas and suggestions of improvments.

I thank my colleagues, Karl-Olof Lindahl, Robert Nyqvist and Per-Anders

Svensson, in the research group in p-adic dynamics, for many interestingseminars and discussions. I also thank Robert Nyqvist for answering myquestions about unix, emacs and latex. I would also like thank my othercolleagues at the School of Mathematics and Systems Engineering at VaxjoUniversity, for creating such a nice working environtment.

During the writing of this thesis both my parents died. I owe them alarge dept of gratitude. Without their support and encouragement I wouldnever have started my graduate studies.

Finally, I thank my wonderful family, my beloved Malin and our daughterClara, for love, patience and encouragement.

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Contents

Abstract v

Acknowlegements viGeneral Introduction 11 Introduction 32 Fields ofp-adic Numbers 4

2.1 Non-Archimedean fields . . . . . . . . . . . . . . . . . . . . . 42.2 The field ofp-adic numbers . . . . . . . . . . . . . . . . . . . 52.3 Extensions of the field of p-adic numbers . . . . . . . . . . . . 72.4 Hensels lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Discrete Dynamical Systems 113.1 Periodic points and their character . . . . . . . . . . . . . . . 11

4 Summary of the Thesis 13References 15

I Cycles of Monomial and Perturbed Monomial p-adic Dy-namical Systems 191 Introduction 212 Properties of Monomial Systems 213 Number of Cycles 234 Distribution of Cycles 27

5 Existence of Fixed Points of a Perturbed System 326 Cycles of Perturbed Systems 37References 43

II Distribution of Cycles of Monomial p-adic Dynamical Sys-tems 451 Introduction 472 Notation and Earlier Results 473 Cycles of Monomial Systems 484 Distribution of Cycles 50

5 Expectation and Variance of 546 Acknowlegement 56References 56

III Asymptotic Behavior of Periodic Points of Monomials inthe Fields of p-adic Numbers 591 Introduction 612 Roots of Unity 61

2.1 Roots of unity in Fp . . . . . . . . . . . . . . . . . . . . . . . 622.2 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . 63

3 Periodic Points and Cyclotomic Polynomials 644 Group Action on Finite Sets 675 Periodic Points in Finite Fields 68

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6 Up to Qp 696.1 The p-adic fields . . . . . . . . . . . . . . . . . . . . . . . . . 706.2 Roots of unity in Qp . . . . . . . . . . . . . . . . . . . . . . . 716.3 Periodic points in p-adic fields . . . . . . . . . . . . . . . . . . 72

References 73IV Fuzzy Cycles of p-adic Monomial Dynamical Systems 751 Introduction 772 Global Dynamics 793 Local Dynamics 82

3.1 Dynamics around neutral points . . . . . . . . . . . . . . . . 853.2 Dynamics around attractors . . . . . . . . . . . . . . . . . . . 92

4 Distribution of Fuzzy Cycles 93References 94

V Monomial Dynamics in Finite Extensions of the Fields ofp-adic Numbers. 951 Introduction 972 Finite Field Extensions of the Field ofp-adic Numbers 973 Roots of Unity 984 Monomial Dynamics 1005 Counting Periodic Points 1016 Asymptotic Behavior 1037 Periodicity 106References 109

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General Introduction


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1 Introduction

1 Introduction

Almost everything in nature and society that evolves in time can be de-scribed as a dynamical system, the solar system, the weather, the stock

exchange, the flow of minds in our brains, et cetera.We essentially have two kinds of dynamical systems, continuous and dis-

crete. A continuous dynamical system describes phenomena that evolvecontinuously in time, while a discrete dynamical system describes phenom-ena that evolve only at certain moments. In the latter case the dynamicalsystem is described by iterations of a function. In this thesis we will considerdiscrete dynamical systems.

The purpose of studying dynamical systems is to predict the future of agiven phenomenon. Even if we could not do this exactly we are often ableto get some information about the long-time behavior of the system. One

way to classify this behavior is to find fixed points and cycles of the system,and determine if these are attractive, repulsive or neutral.

When constructing a model of a phenomenon in nature or in society we,almost without exception, use some set of numbers to represent a measur-able quantity of the phenomenon. The by far most used set of numbersis the field of rational numbers, Q, or some field that contains them (forexample the real numbers or the complex numbers). We also need a way ofmeasuring distances between different numbers, that is, we need a metric ora topology on the set of numbers we use. On the rational numbers we veryoften use the absolute value of the difference of two numbers to measure thedistance between them. But there are many other possibilities, for examplethe so called p-adic absolute value.

We know that Q is not complete as a metric space with the metric inducedby the ordinary absolute value. We have the same situation for Q endowedwith the metric induced by the p-adic absolute value. The completionis the field of p-adic numbers, denoted by Qp. The metric satisfies thestrong triangle inequality

(x, z) max((x, y), (y, z)), (1.1)

and Qp is therefore an example of a so called ultrametric space.The fact is that the ordinary absolute value and the p-adic absolute values

(for each prime number p) are essentially the only absolute values on Q.This result is described in Ostrovskis Theorem, see for example [14, 17, 32].This theorem gives the p-adic numbers a special position, when it comes tomodeling. Which way of measuring distances on Q we choose must dependon the phenomenon we are making a model of. Of course, the most commonway of measuring distances is by the ordinary absolute value, but there arephenomena that need the p-adic distance, see for example [35, 16, 17, 36].

In this thesis we will study discrete dynamical systems over the Qp andover finite field extensions ofQp. During the last couple of dekades therehas been an increasing interest for p-adic dynamical systems, induced by

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p-adic mathematical physics, see [35, 36, 4, 13, 16] 1. The p-adic dynamicalsystems have for example been studied in [17, 21, 23, 33, 34] and in [5, 8,9, 15, 22, 29, 30]. There were studies not only on dynamical systems overQp but also on extensions ofQp and more general non-Archimedean fields.

There are also several articles, [6, 7, 12, 20, 19], that propose p-adic modelsfor cognitive processes. Recently, in [1], multidimensional non-Archimedeandynamical systems have been investigated.

The aim of this thesis is to investigate and describe a dynamical systemsgiven by iterations of the monomial

h(x) = xn, n 2. (1.2)

Even though this is function of a very simple type, we will see that thedynamics will have a rich structure. A detailed analysis of monomial systems

over non-Archimedean fields (in particular Qp) was first provided in [17].Monomial systems over the field Qp and its finite exensions have then beenstudied in for example [24], [25], [26], [27] and in [28]. These are the articlesthat this thesis is based upon.

There are essentially two things that make the monomial dynamics moreinteresting in Qp and its finite field extensions, than in the field C of complexnumbers. First, Qp and its finite extensions are not algebraically closed.For exampel, the number of periodic points of a fixed period will vary withp. Second, Qp and its finite extensions are totally disconnected, and theypossess a tree structure. Hopefully the structures described in this thesis

will imply new areas of applications of p-adic dynamical systems in thefuture.

2 Fields ofp-adic Numbers

In this chapter we will give a short introduction to the fields of p-adicnumbers and their extensions.

2.1 Non-Archimedean fields

Definition 2.1. Let K be a field. An absolute value on K is a function|.| : K R such that

|x| 0 for all x K,

|x| = 0 if and only if x = 0,

|xy| = |x||y|, for all x, y K,

|x + y| |x| + |y|, for all x, y K.

1p-adic mathematical physics also stimulated studies in non-Archimedean functional

analysis, see [10, 11, 2, 3].

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If |.| in addition satisfies the strong triangle inequality,

|x + y| max(|x|, |y|), (2.1)

for all x, y K then we say that |.| is non-Archimedean.

If |x| = 1 for all non-zero x K we call |.| the trivial absolute value. It iseasy to see that the trivial absolute value is non-Archimedean.

Proposition 2.2. LetK be a field and let|.| be a non-Archimedean absolutevalue on K. Let x, y K such that |x| = |y|. Then

|x + y| = max(|x|, |y|). (2.2)

Every non-Archimedean field can be regarded as an ultrametric space

with the metric (x, y) = |x y| induced by the absolute value. Let a Kand let r R+. The open ball of radius r with center a is the set

Br (a) = {x K : |x a| < r}.

The closed ball of radius r with center a is the set

Br(a) = {x K : |x a| r}.

The set

Sr(a) = {x K : |x a| = r}is called the sphere of radius r with center a.

It is sometimes important to underline in which field a ball or a sphereis included. We then use the symbols Br (a, K), Br(a, K) and Sr(a, K).The strong triangle inequality and Proposition 2.2 have some remarkableconsequences for the balls in K.

Every element of a ball can be regarded as a center of it.

Each open ball is both open and closed as sets.

Each closed ball of positive radius is both open and closed.

Let B1 and B2 be balls in X. Then either B1 and B2 are orderd byinclusion (B1 B2 or B2 B1) or B1 and B2 are disjoint.

An ultrametric space is totally disconnected.

2.2 The field of p-adic numbers

Let p be a fixed prime number. By the fundamental theorem of arithmetics,

each non-zero integer n can be written uniquely as

n = pvp(n)n,

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where n is a non-zero integer, p n, and vp(n) is a unique non-negativeinteger. The function vp : Z \ {0} N is called the p-adic valuation. Ifa, b Z+ then we define the p-adic valuation of x = a/b as

vp(x) = vp(a) vp(b). (2.3)

One can easily show that the valuation is well defined. The valuation ofx does not depend on the fractional representration of x. By using the p-adic valuation we will define a new absolute value on the field of rationalnumbers.

Definition 2.3. The p-adic absolute value of x Q \ {0} is given by

|x|p = pvp(x) (2.4)

and |0|p = 0. When it is clear from the context which absolute value we usewe denote the p-adic absolute value by |.|.

It is easy to prove that the p-adic absolute value is non-Archimedean,and that the metric (x, y) = |x y|p induced by it, is an ultrametric. Twoabsolute values on a field K are said to be equivalent if they generate thesame topology on K. Essentially there are only two types of non-trivialabsolute values on Q. This is the essence of the following theorem.

Theorem 2.4 (Ostrovski). Every non-trivial absolute value onQ is eitherequivalent to the real absolute value or to one of the p-adic absolute values.

For a proof of Ostrovskis theorem see, for example, [32] or [14].

Let Q be endowed with the ultrametric induced by the p-adic absolutevalue. However, this space is not complete. The completion ofQ will bea field, the field of p-adic numbers, Qp . The p-adic absolute value can beextended to Qp and Q is dense in Qp. It is worth noting that

{|x|p : x Qp} = {|x|p : x Q} = {pm : m Z} {0}.

The set B1(0) = {x Qp; |x|p 1} is sometimes called the p-adic integers.It is denoted by Zp. In fact, Zp is a subring of Qp and B

1 (0) = {x

Zp; |x|p < 1} is a maximal ideal ofZp. The quotient ring Zp/B1 (0) is then

a field, called the residue class field ofQp. The residue class field ofQp isisomorphic to the finite field Fp of p elements.

Theorem 2.5. Every x Qp can be expanded in the base p in the followingway

x =

jjmin

yjpj , (2.5)

where jmin = vp(x) Z and 0 yj p 1 for j jmin.

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2.3 Extensions of the field of p-adic numbers

Everywhere below we denote by Kp a finite extension of the p-adic numbers.Let m = [Kp : Qp] denote the dimension ofKp as a vector space over Qp.The p-adic absolute value |.|p can be extended to Kp, in the unique way. See

[14], [32] or [31] for details. But, how can we evaluate the p-adic valuationon elements in Kp? We need a function

NKp/Qp : Kp Qp,

which satisfies the equality

NKp/Qp(xy) = NKp/Qp(x) NKp/Qp(y).

This function is the so called norm from Kp to Qp. There exists several

ways to define NKp/Qp , all equivalent. Below, three of them are listed.

Let Kp and consider Kp as a finite dimensional Qp-vector space.The map from Kp to Kp defined by multiplication by is a Qp-linearmap. Since it is linear it corresponds to a matrix. We define NKp/Qpto be the determinant of this matrix.

Let Kp and consider the subfield Qp(). Let r = [Kp : Qp()]and let T(,Qp) be the minimal polynomial of over Qp and letn = deg(T(,Qp)). Then the norm is defined as

NKp/Qp() = (1)nrar0,

where T(,Qp) = anxn + an1x

n1 + + a1x + a0.

Suppose that Kp is a normal extension ofQp. Let G(Kp/Qp) be theGalois group of this extension. Then, for Kp, the norm is definedas

NKp/Qp() =

(), for all G(Kp/Qp).

Observe that |G(Kp/Qp)| = [Kp :Qp], because Kp is a normal exten-sion ofQp and Qp is of characteristic zero.

Since NKp/Qp() Qp for each Kp it has a p-adic absolute value. Wecan now use this to extend the p-adic absolute value to Kp.

Theorem 2.6. LetKp be a finite extension ofQp and m = [Kp :Qp]. Thenthe function |.| : Kp R+ defined by

|x| =

m| NKp/Qp(x)|p

is a non-Archimedean valuation onKp that extends |.|p.

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Since |.| is unique, |.|p can also be used to denote the extended p-adicvaluation. From algebra we know that for each finite extension Kp ofQpthere exists a finite normal extension ofQp which contains Kp. The smallestsuch normal extension ofQp is called the normal closure ofQp over Kp. If

Kp is not a normal extension ofQp and we want to define a norm by usingQp-automorphisms, then we consider the normal closure ofQp over Kp anduse the third definition of the norm.

Let Kp be a finite field extension ofQp and m = [Kp :Qp]. For x Kpset y = NKp/Qp(x). Then we have by Theorem 2.6 that

|x|p =m

|y|p =

m

pvp(y) = pvp(y)/m = pvp(x),

where vp(x) = vp(y)/m, that is, vp(x) 1mZ, because vp(y) Z.

If a, b Kp then vp(ab) = vp(a) + vp(b). This gives that vp is a homo-morphism from the multiplicative group Kp to the additive group Q. Then

the image Im(vp) is an additive subgroup of Q, and Im(vp) 1mZ. Let

d/e be in Im(vp), where d and e are relatively prime, chosen so that thedenominator e is the largest possible. This choice can be done because ehas to be a divisor of m, and the set of possible divisors is bounded. Sinced and e is relatively prime, there must be a multiple of d which is congruentto 1 modulo e, that is, we can find r and s such that rd = 1 + se. But then

rd

e=

1 + se

e=

1

e+ s

is in Im(vp). Since s Z 1mZ, it follows that 1/e Im(vp). Since e

was chosen to be the largest possible denominator in Im(vp), it follows thatIm(vp) =

1eZ. This unique positive integer e is called the ramification index

of Kp over Qp. The extension Kp over Qp is called unramified if e = 1,ramified if e > 1 and totally ramified if e = m.

Definition 2.7. We say that an element K is a uniformizer ifvp() = 1/e.

The unit ball B1(0,Kp) = {x Kp; |x|

1} is a subring of Kp andB1 (0,Kp) is a unique maximal ideal of B1(0,Kp). The quotient ring isthen a field, the residue class field ofKp.

We state a few facts about the extention Kp:

Kp is locally compact and complete, but it is not algebraically closed

Each x Kp can be written as x = uv(x), where u S1(0) and

v(x) = vp(x)e.

The residue class field of Kp is isomorphic to Fpf , the field of pf

elements, where f = m/e. This number is called the residue classfield degree. The residue class field ofQp is Fp so f is the degree ofthe residue class field as an extension of the residue class field ofQp.

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Let C = {c0, c1, . . . , cpf1} be a fixed complete set of representativesof the cosets of B1 (0) in B1(0). Then every x Kp has a unique-adic expansion of the form

x =ii0

= aii

,

where i0 Z and ai C for every i i0.

The union of all finite extentions ofQp is a field and an algebraic closureof Qp. We denote this field by Qp. It is possible to extended the p-adic

absolute value to Qp. The possible positive values are pr, where r Q.

The algebraic closure Qp ofQp is an infinite extension and it is not com-plete with respect to the metric induced by the p-adic absolute value. The

completion ofQp is however also algebraically closed and we call it the fieldof complex p-adic numbers. We denote this field by Cp.

2.4 Hensels lemma

Let Kp be a finite extension ofQp and let be a uniformizer. Let , Kp.We say that (mod ) if||p ||

p . The following theorem and its

corollary are important tools for finding solutions of polynomial equations.

Theorem 2.8. Let F(x) be a polynomial with coefficients in B1(0,Kp).

Assume that there exists 0 B1(0,Kp) and N such that

F(0) 0(mod 2+1)

F(0) 0(mod )

F(0) 0(mod +1).

Then there exists B1(0,Kp) such thatF() = 0 and 0 (mod +1).

Corollary 2.9 (Hensels lemma). Let F be a polynomial with coefficentsin B1(0,Kp) and suppose that there exits 0 B1(0,Kp) such that F(0)

0(mod ) and F(0) 0(mod ). Then there exists B1(0,Kp) suchthat F() = 0 and 0 (mod ).

2.5 Roots of unity

The roots of unity in Kp will be essential for our investigations of themonomial dynamical systems. Let Kp be a finite extension of Qp of de-gree m = e f, where e is the ramification index and f is the residue classdegree. The residue class field is isomorphic to Fpf .

Definition 2.10. We say that x Kp is an n-th root of unity f xn

= 1. Ifxn = 1 and xm = 1 for every m < n then we say that x is a root of unity oforder n or a primitive n-th root of unity.

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The multiplicative group ofFpf is cyclic and has pf 1 elements. Since

a cyclic group has a cyclic subgroup of order d for each divisor d of pf 1,for every d | pf 1 there exists x F

pfthat generates the subgroup of d

elements and we also have xd = 1. The element x generates a group of d

roots of the polynomial xd 1 in Fpf . Let us denote the d roots x1, . . . , xd.Take now d elements y1, . . . yd ofS1(0,Kp) such that yj belongs to the cosetthat corresponds to xj . Then there are d approximate roots of F(x) =xd 1 = 0 in B1(0,Kp) because F(yj) 0(mod ) and F

(yj) 0(mod ).Of course, the d different yj are located in d different cosets ofB

1 (0,Kp).

Hence they are noncongruent modulo . By Hensels lemma, for each d |pf 1, the equation xd 1 = 0 has d solutions in Kp. We have proved thefollowing proposition.

Proposition 2.11. The fieldKp contains the (pf 1)-roots of unity.

We also have the following results, see for exemple [14] or [31] for proofs.

Proposition 2.12. Let n be an integer that is relatively prime to pf 1.Let xn = 1. Then x 1(mod ) or in other words x B1 (1).

Proposition 2.13. If x B1 (1) such that xn = 1 then n is divisible by a

power of p and x is a root of unity for that power of p.

We know even more about the p-power roots of unity.

Theorem 2.14. Let be a pt

th root of unity in Kp. Then | 1|p =|p|

1/(pt)p , where (pt) = pt1(p 1) (Eulers -function).

See [31] for a proof. We have the following immediate consequences ofthis theorem.

Corollary 2.15. Let e be the ramification index ofKp as an extension ofQp. Let Cp be a root of unity of order p

t, where t 1. A neccesarycondition for Kp is that (p

t) | e.

Corollary 2.16. Assume thatKp has ramification index e as an extension

ofQp. We then have at most e/(1 1/p) p-power roots of unity.

Corollary 2.17. Let m be the degree ofKp as an extension ofQp. Thenthere is only a finite number of primes p, such thatKp possesses a p-powerroot of unity.

Proof. Let e be the ramification index of the extension Kp/Qp. Ifp 1 > mthen p 1 > e and (ps) e for any s 1.

Let t be the largest integer for which there exists a root of unity of order

pt

in Kp. Recall that if is a root of unity of order pt

, then generates acyclic group of order pt. The elements of this group are of course p-powerroots of unity of order ps, where 0 t t.

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3 Discrete Dynamical Systems

Theorem 2.18. Let 1 < t t On the sphere Sp1/(pt)(1) there are (pt)

different roots of unity, all of order pt. Moreover, Kp contains pt, p-power

roots of unity.

Proof. Let be a root of unity of order pt. Since the group generated by

is cyclic, where are cyclic groups of order ps for every 1 < s t. Every suchgroup has (ps) generators, which all are roots of unity of order ps. Sincet

t=0 (pt) = pt, Kp contains p

t p-power roots of unity. The rest followsfrom Theorem 2.14.

Theorem 2.19. LetKp be a finite extension ofQp with residue class degree

f. Lett be the p-power root of unity of highest order. We then have (pf1)pt

roots of unity inKp.

3 Discrete Dynamical Systems

This chapter is devoted to discrete p-adic dynamical systems, namely iter-ation

xn+1 = h(xn) (3.1)

of functions h : K K on a complete non-Archimedean field K. Mostly, wewill let K be Qp or a finite extension, Kp, ofQp. Below, we will sometimes

write the dynamical system h(x) when referring to the dynamical systemthat is described by iterations of h.

3.1 Periodic points and their character

For a given point x0 the set of points {hm(x0); m N} is called the trajectory

or orbit through x0. Some orbits of a dynamical system are of particularinterest:

Definition 3.1. A point x0 X is said to be a periodic point if there exists

r N such that hr

(x0) = x0. The least r with this property is called theperiod ofx0. Ifx0 has period r, it is called an r-periodic point. A 1-periodicpoint is called a fixed point. The orbit of an r-periodic point x0 is

{x0, x1, . . . , xr1},

where xj = hj(x0), 0 j r 1. This orbit is called an r-cycle.

An r-cycle consists of r different r-periodic points. See Figure 3.1. Eachelement of the cycle has the cycle as its orbit. As a simple consequence we

have that the number of r-periodic point of a discrete dynamical system isalways divisible by r.

The periodic points have different charcters.

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x0

x1

x2

x3

x4

f

f

f

f

f

Figure 3.1: A 5-cycle contains five different 5-periodic points.

Definition 3.2. Let x0 be an r-periodic point and let g(x) = hr(x). If

there exists a ball B (x0) such that for every x B (x0) we have

lims

gs(x) = x0

then we say that x0 is an attractor. The set

A(x0) = {x X; lims

gs(x) = x0}

is called the basin of attraction of x0.

Definition 3.3. Let x0 be an r-periodic point and let g(x) = hr(x). If there exists a ball B (x0) such that |x x0| < |g(x) x0| for every x B (x0), x = x0 then x0 is said to be a repeller.

Definition 3.4. See [17]. Let x0 be an r-periodic point. If there existsan open ball B (x0) such that for every

< the spheres S(x0) areinvariant under the map g = hr then B (x0) is said to be a Siegel disk andx0 is said to be a center of a Siegel disk. The union of all Siegel disks withcenter x0 is the Siegel disk of maximal radius of x0. It is denoted by SI(x0).

Definition 3.5. An r-periodic point x0 is said to be attractive if |g(x0)| 1.

Theorem 3.6. Let a be a fixed point of a dynamical system given by apolynomial function h(x) over a non-Archimedean valued field.

(i) Ifa is an attracting point of h then it is an attractor of the dynamicalsystem.

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where (t) denotes the number of primes p t and the sum is extendedover prime numbers less than or equal to the real number t. We interpretthis limit as the asymptotic mean value of the number of r-periodic pointsin Kp, when p .

The limit (4.3) is calculated for Qp in Paper I, Paper II and Paper III byuse of different methods. We find that

limt

1

(t)

pt

Pr(h,Qp) =d|r

(r/d)(nd 1), (4.4)

where (m) for m Z+ denotes the number of positive divisors of m. InPaper I, we perform an elementary proof by use of some well known technicsfrom number theory.

In Paper II we instead use probabilistic methods for proving (4.4). We

run into some difficulties when defining a probability measure on the setof prime numbers, but solve this by considering a more general probabilityconcept, see [18]. There is no uniform Kolmogorov probability measureon the set of all prime numbers. Therefore we have to consider a finiteadditive generalized probability. We consider the number of cycles as ageneralized random variable and calculate the expectation and the variancewith respect to a finite-additive probability measure. In Paper III we useGalois theory and methods from algebraic number theory, especially theTheorem of Cebotarev to compute the limit (4.4).

In Paper V we instead consider the limit (4.3) for finite extensions Kpoch Qp, with fixed residue class field degree f. By generalizing the technicfrom Paper I we find that

limt

1

(t)

pt

Pr(xn,Kp) =

d|r

(r/d)

l|nd1

f(l), (4.5)

where f(l) denotes the number of solutions of xf 1(mod l). It turns out

that the limit (4.5) is a periodic in f.In paper IV we study the dynamics of balls in Qp under the monomial

x xn

. Following Khrennikov in [17] a cycle of balls is called a fuzzycycle1. There is a one-to-one correspondence between the fuzzy cycles ofballs of radius 1/p and the cycles in Qp. However, the structure of fuzzycycle of balls of radius r 1/p2 is non-trivial. Some numerical experimentsto clearify the structure were performed in Khrennikov [17]. In Paper IVthe structure of fuzzy cycles is investigated by analytic methods. We alsopresent an algorithm for calculating the number of fuzzy cycles.

In this thesis we also consider perturbed monomial systems over the fieldof p-adic numbers. This is done in Paper I. This systems are generated bypolynomials

hq(x) = xn + q(x), (4.6)

1This concept has nothing to do with the fuzzy set theory.

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References

where the perturbation q(x) is a polynomial whose coefficients have a smallp-adic absolute value. We investigate the connection between monomial andperturbated monomial systems by use of Hensels lemma. As in the mono-mial case the interesting dynamics of perturbated systems are essentially

located on the unit sphere in Qp. Sufficient conditions on the perturba-tion for the two systems to have similar properties are derived. By similarproperties we mean that there is a one to one correspondence between fixedpoints and cycles of the two kinds of systems.

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