Câˆ—-ALGEBRAS AND SELF-SIMILAR GROUPS Contents 1

C∗-ALGEBRAS AND SELF-SIMILAR GROUPS

VOLODYMYR NEKRASHEVYCH

Contents

1. Introduction 22. Self-similar and iterated monodromy groups 52.1. Self-similar groups and permutational bimodules 52.2. Iterated monodromy groups 82.3. Contracting groups and limit spaces 92.4. Limit solenoid 113. C∗-algebras associated with self-similar groups 133.1. Matrix recursions 133.2. Universal Cuntz-Pimsner algebra of a self-similar group 143.3. Another definition of OIMG(f) 153.4. Self-similar representations 153.5. Cuntz-Pimsner semigroup 163.6. The simple quotient of OG 173.7. The gauge action and the gauge-invariant subalgebra of OG 183.8. Functoriality of OG 193.9. Defining relations of the gauge-invariant sub-algebra 203.10. MG as a direct limit 244. Contracting groups 254.1. A finite set of defining relations of OG 254.2. Reconstruction of the limit space from the gauge action 264.3. K-theory of algebras associated to rational functions 315. Groupoid approach 345.1. Groupoids 345.2. Cuntz-Pimsner groupoid of a self-similar group 355.3. The algebraMG as a convolution algebra 375.4. Reduced algebras 375.5. Amenability 386. Regular self-similar groups and Ruelle algebras 396.1. Regular self-similar groups 396.2. Equivalence of regular groups 406.3. The algebra Of 406.4. Hyperbolic rational functions 426.5. The limit solenoid as a hyperbolic dynamical system 456.6. Ruelle algebra of the unstable equivalence 486.7. Ruelle algebra of the stable equivalence 507. A Fock space representation 51

7.1. Extensions of the algebras Of and Of by compact operators 517.2. Hausdorff dimension of the Julia sets of hyperbolic rational functions 54

1

2 VOLODYMYR NEKRASHEVYCH

References 58

1. Introduction

Self-similarity is an important type of symmetry appearing in fractal geometryand dynamics. Informally, an object is said to be self-similar if its parts are similarto the whole, i.e., if the structure of the object repeats at all scales. Self-similarityis encoded then by a semigroup (inverse semigroup, or a pseudogroup) of self-similarities, which are isomorphisms between the parts of the self-similar object ondifferent scales. For example, if J is the Julia set of a rational function f , then J isself-similar, with the self-similarities equal to the restrictions of the iterations of fonto pieces of J (since f will be typically expanding at the points of J and hencemap small pieces of J onto bigger pieces homeomorphically).

A classical idea in non-commutative geometry is to encode an invertible topologi-cal dynamical system (X, f) by the cross-product C∗-algebra C(X)of Z. Similarly,a natural algebra encoding a self-similar space would be some kind of cross-productof the algebra of continuous functions on the space with the self-similarity, i.e., withthe semigroup of self-similarities. The corresponding “cross-product” algebra is de-fined either as the convolution algebra of the groupoid of germs of the semigroup ofself-similarities (see [Dea95, Ren00]), or as the Cuntz-Pimsner algebra of a Hilbertbimodule naturally encoding the self-similarity (see [KW05]). Another class of al-gebras related to self-similar spaces are the cross-product algebras of groups actingon the boundary of a hyperbolic space (see [LS96, Ana97]). Here the self-similarityof the ideal boundary (of the limit set) is naturally encoded by the group action.

The notion of self-similarity has recently entered the field of geometric grouptheory. Self-similar groups provide easy-to-study examples of “exotic” groups, verymuch like in fractal geometry the self-similar fractals are the most well studied“exotic” geometric objects. The first examples of groups, where self-similarity wasan essential tool are the Grigorchuk [Gri80] and Gupta-Sidki [GS83] groups. TheGrigorchuk group is a particularly simple example of an infinite finitely generatedtorsion group and is the first example of a group of intermediate growth [Gri85].For general theory of self-similar groups see [Nek05] and [BGS03].

Let us give a definition of self-similarity for groups. Let X be a finite alphabetand denote by X

∗ the set of finite words over the alphabet X. The set X∗ has a

natural structure of a roote tree in which every word v ∈ X∗ is connected by an

edge with all the words of the form vx, where x ∈ X. The root is the empty word.A faithful action of a group G on the rooted tree X

∗ is said to be self-similar if forevery g ∈ G and x ∈ X there exist h ∈ G and y ∈ X such that

g(xw) = yh(w)

for all w ∈ X∗.

Here the term “self-similarity” reflects the fact that in the conditions of thedefinition the action of the element g ∈ G on the sub-tree xX∗ (on the tails of thewords starting with x) can be identified with the action of another element h ∈ Gon the whole tree X

∗. So, here also the structure of a smaller part (the action onthe subtree) is isomorphic to the structure of the whole (the action on the wholetree).

C∗-ALGEBRAS AND SELF-SIMILAR GROUPS 3

Self-similar groups are not only exotic (counter-) examples, but they also appearnaturally as the iterated monodromy groups of self-coverings of topological spaces.The iterated monodromy group of a finite covering f : C1 −→ C of a path-connectedtopological space C by a subspace C1 is the quotient of the fundamental group π1(C)by the intersection of the kernels of the actions of π1(C) on the fibers of the iterationsof f . There is a natural representation of the iterated monodromy group as a groupacting self-similarly on a rooted tree (see [Nek05] and Subsection 2.2 of this article).

This construction relates self-similarity of groups with more classical self-similarityof dynamical systems. Iterated monodromy groups encode the combinatorics of self-coverings, which can be effectively used in their study (see, for instance [BN06b]).

The converse construction, called the “limit space”, produces a dynamical systemfrom a self-similar group. Let G be a self-similar group acting on the tree X

∗.Consider the space X

−ω of the left-infinite sequences of the form . . . x2x1 of theelements of X. We introduce the product topology on X

−ω. Two sequences . . . x2x1

and . . . y2y1 ∈ X−ω are said to be equivalent if there is a finite set N ⊂ G and a

sequence gk ∈ N such that gk(xk . . . x1) = yk . . . y1 for all k ≥ 1. The quotient JG ofthe space X

−ω by this equivalence relation is called the limit space of the self-similargroup G. It is easy to see that the equivalence relation is invariant under the shift. . . x2x1 7→ . . . x3x2, hence the shift induces a continuous self-map σ : JG −→ JG.We get in this way the limit dynamical system (JG, σ) of the self-similar group.

The two described constructions are inverse to each other: if the self-covering f :C1 −→ C is expanding, then the limit dynamical system of the iterated monodromygroup of f is topologically conjugate to the restriction of f onto the Julia set of f .The Julia set of f is defined as the set of accumulation points of the inverse orbit⋃

n≥1 f−n(t) of a point t ∈ C.

We get in this way two related self-similar structures: an expanding self-coveringσ : JG −→ JG and a contracting (see Subsection 2.3) self-similar (iterated mon-odromy) group G. Relation between these self-similar structure has a flair of du-ality, which is not clarified yet. A superficial aspect of this duality is the fact thatone has to write the words . . . x2x1 in the definition of the limit space in the di-rection opposite to the direction of the words on which the group acts. One getsanother example of “opposite directions” if one compares the computation of spec-tra of Laplacians of Schreier graphs of self-similar groups [BG00, GS06] with thecomputation of spectra of Laplacians on self-similar fractals [FS92].

One of the aims of this paper is to try to understand this duality in the spiritof non-commutative geometry. We define Cuntz-Pimsner-type C∗-algebras OG en-coding self-similarity of a group (G,X) and study their properties. It seems thatthe Cuntz-Pimsner algebras OG have more “algebraic” or “symbolic” nature thanthe Cuntz-Pimsner algebras associated to self-coverings of topological spaces. Inparticular, if the self-similar group G is contracting (e.g., if it is the iterated mon-odromy group of an expanding self-covering), then OG can be defined by a finite setof generators and relations with a simple definition of the gauge action on the gen-erators. These presentations are closely related with the symbolic dynamics of theassociated limit dynamical system. For instance, the presentation for the Cuntz-Pimsner algebra of the iterated monodromy group of a quadratic polynomial iseasily written in terms of the “kneading sequence” of the polynomial.

We show that under some natural conditions on the groupG (satisfied in the casewhen G is the iterated monodromy group of an expanding self-covering) the algebra


OG is nuclear, purely-infinite and simple. Similar results are also true in the dualsituation of the Cuntz-Pimsner algebras of expanding self-coverings (see [KW05]).We also show that the limit dynamical system (JG, σ) of a self-similar group can bereconstructed from the algebra OG and the associated gauge action of the circle.This construction uses asymptotic commutation, which probably can be used tounderstand the duality better. As a corollary we get that two expanding self-coverings are topologically conjugate if and only if the associated gauge actions onthe Cuntz-Pimsner algebras of the iterated monodromy groups are conjugate.

The results about the structure of the algebras OG imply that these algebras areclassified by their K-theory (due to results of E. Kirchberg and N.C. Phillips [KP00,Phi00]). We compute the K-theory of the Cuntz-Pimsner algebras OG associatedwith the iterated monodromy groups of complex hyperbolic rational functions andof the respective guage-invariant sub-algebras. We show that the K-theory dependson a few topological invariants (degree of the map, number of cycles, the g.c.d. andsum of their lengths).

In the last part of the paper, we relate the algebras OG associated with iter-ated monodromy groups of expanding self-coverings f : J −→ J with the Ruellealgebras of the natural extension of f . The natural extension of f is the inverse

limit of the iterations of f together with the homeomorphism f induced by f onthe inverse limit. If f is expanding, then the natural extension is an example ofa Smale space: it has a local direct product structure with expanding and con-

tracting directions of f . The natural extension was studied in the case of complexrational functions acting on the Riemann sphere by M. Lyubich, Y. Minsky andV. Kaimanovich [LM97, KL05]. The Ruelle algebras associated to Smale spaceswere studied by J. Kaminker, I. Putnam and J. Spielberg [KPS97, Put96, PS99].They proved, in particular, a KK-theoretical Spanier-Whitehead duality betweenthe Ruelle algebras of the stable and unstable foliation of a Smale space.

We show that in the case of the natural extension of an expanding self-coveringf , the Ruelle algebra of the unstable foliation is strongly Morita equivalent to theCuntz-Pimsner algebra OIMG(f) of the iterated monodromy group of f , while theRuelle algebra of the stable foliation is strongly Morita equivalent to the Cuntz-Pimsner algebra of the self-covering f (as defined in [Dea95, Ren00, KW05]). Hencethe results of J. Kaminker, I. Putnam and J. Spielberg imply a Spanier-Whiteheadduality between the Cuntz-Pimsner algebras of self-similar groups and their limitdynamical systems.

The Cuntz-Pimsner algebras of self-similar groups and their limit dynamicalsystems are have similar properties (in the spirit of Sullivan’s dictionary [Sul85])with the cross-product algebras of the boundary actions of groups. For exampleC. Anantharaman-Delaroche has shown in [AD97] that the K-theory of the cross-product algebra of the action of a Fuchsian group on its limit set depends onlyon the signature of the group. H. Emerson showed non-commutative Poincareduality for the cross-product algebras of the action of a Gromov hyperbolic groupon its boundary in [Eme03]. It would be interesting to understand the “Sullivan’sdictionary” better in the spirit of non-commutative geometry.


2. Self-similar and iterated monodromy groups

We remind here the basic facts and definitions of the theory of self-similar groups,iterated monodromy groups and their limit spaces. A more detailed account andproofs can be found in [Nek05].

2.1. Self-similar groups and permutational bimodules.

Definition 2.1. A self-similar group (G,X) is a faithful action of a group G onthe set of finite words X

∗ over a finite alphabet X such that for every g ∈ G andevery x ∈ X there exist h ∈ G and y ∈ X such that

g(xw) = yh(w)

for all w ∈ X∗.

We write the above equality formally as

(1) g · x = y · h.A convenient way to interpret this equality is to identify the elements G with therespective transformations of X

∗ and identify the letters x of X with the “creationoperators”

w 7→ xw.

Then (1) becomes a true equality of compositions of transformations of X∗.

It follows from the definition of a self-similar group that for every g ∈ G andv ∈ X

∗ there exists a unique element h ∈ G such that g(vw) = g(v)h(w) for allw ∈ X

∗. We call h the restriction (or section) of g at v and denote it g|v. We havethe following properties of restrictions:

g|v1v2 = (g|v1) |v2 , (g1g2)|v = g1|g2(v)g2|v, g|−1v = g−1|g(v).

We also write

g · v = u · hif u = g(v) and h = g|v, i.e., if g(vw) = uh(w) for all w ∈ X

∗. This is also a trueequality of compositions of elements of G and creation operators.

The set X∗ has a natural structure of a rooted tree: the root is the empty word

and two words are connected by an edge if and only if they are of the form v and vxfor v ∈ X

∗ and x ∈ X. A self-similar action is always an action by automorphismsof the rooted tree X

∗.The boundary of the tree X

∗ is naturally identified with the space Xω of infinite

sequences of the form x1x2 . . ., for xi ∈ X. The space Xω has a natural topology

of a direct product of discrete sets X and a natural measure equal to the directproduct of the uniform distributions of X (called the uniform Bernoulli measure).

If (G,X) is a self-similar group, then the set X · G = {x · g : x ∈ X, g ∈ G} oftransformations

w 7→ xg(w)

is closed under pre- and post-compositions with elements of G. We get hence a setwith commuting left and write actions of G on it. These actions are computed bythe rules

h · (x · g) = h(x) · (h|xg), (x · g) · h = x · (gh).The set X ·G together with these G-actions is called the permutational bimodule

associated with the self-similar group.


More generally, a covering permutational bimodule over a group G is a set Mwith commuting left and right actions of G on it such that the right action is freeand has a finite number of orbits. Here an action is said to be free if x · g = ximplies g = 1.

A covering bimodule is said to be d-fold if it has d right orbits.

Definition 2.2. Two self-similar groups (G1,X1) and (G2,X2) are said to be equiv-alent if the associated bimodules are isomorphic, i.e., if there exists a group iso-morphism φ : G1 −→ G2 and a bijection F : X1 ·G1 −→ X2 ·G2 such that

F (g ·m · h) = φ(g) · F (m) · φ(h)

for all g, h ∈ G1 and m ∈ X1 ·G1.

It is proved in [Nek05] (Proposition 2.3.4) that equivalent self-similar groupshave conjugate actions on the respective trees of words.

Suppose that M is a covering d-fold bimodule over a group G. A basis of M isa set X ⊂ M , which is a right orbit transversal of M , i.e., such that every orbit ofthe right action contains a unique element of X. It follows that |X| = d and everyelement of M is written uniquely in the form x · g for x ∈ X and g ∈ G. Then forevery g ∈ G and x ∈ X the element g · x is written uniquely in the form y · h forsome y ∈ X and h ∈ G. We use this to define recursively an action of G on the treeX∗ by the rules

g(xw) = yh(w)

for all w ∈ X∗, whenever g · x = y · h in M .

The obtained action of G on X∗ is called the associated self-similar action and

is denoted (G,M,X), or just (G,X).We will often need the following technical definition.

Definition 2.3. A self-similar action (G,X) is self-replicating (or recurrent) if theleft action of G on the associated bimodule M = X ·G is transitive, i.e., if for anyx, y ∈ X there exists g ∈ G such that g · x = y · 1.

Example 2.1. Let G = Zn be a free abelian group. Let A be an n×n matrix withintegral entries such that detA = d. Then A(Zn) is a subgroup of index d in G.

Let M be the set of all affine transformations of Rn of the form

~x 7→ A−1(~x + ~g),

where g ∈ Zn is arbitrary. We use right notation for the elements of M : if S~v isthe transformation A−1(~x+ ~v), then we write

~x · S~v = A−1(~x+ ~v).

The group G acts on Rn in a natural way by translations. We also consider thisaction to be a right one:

~x · T~v = ~x+ ~v

for ~v ∈ Zn.

Then the set M is a covering d-fold bimodule over the group of translations Gwith respect to the compositions of transformations of Rn:

~x · S~u · T~v = A−1(~x+ ~u) + ~v = A−1(~x+ ~u+A(~v)),

hence

S~u · T~v = S~u+A(~v)


and~x · T~v · S~u = A−1(~x + ~v + ~u),

henceT~v · S~u = S~v+~u.

We see that S~u1and S~u2

belong to one right orbit if and only if ~u1−~u2 ∈ A(Zn).Consequently, a set {S~ui

} is a basis of the bimodule M if and only if the set {~ui}is a coset transversal of Z

n modulo the subgroup A(Zn) of index d.If X = {xi = S~ui

}i=1,...,d is a basis, then the associated action is given by therecurrent formula

T~v · xi = xj · TA−1(~ui+~v−~uj),

where j is uniquely determined by the condition ~ui + ~v − ~uj ∈ A(Zn).This action is associated to a “numeration system” on Z

n given by the digit set{~ui} and the base A. Every element of Zn is uniquely represented as a formal sum

~ui0 +A(~ui1) +A2(~ui2) + · · · .The constructed self-similar action describes addition of the elements of Zn to suchformal series. For more on such numeration systems, see [Nek05] Section 2.2.9and [BJ99].

Example 2.2. Let us take in the previous example G = Z (i.e., n = 1), A = 2,and the basis given by the coset transversal {0, 1} of Z modulo 2Z. We get then thebinary adding machine action. It is generated by a single transformation a actingon the binary words accordingly to the recurrent rule

a(0w) = 1w, a(1w) = 0a(w).

It describes the process of adding 1 to a diadic integer.

Definition 2.4. LetM1 andM2 be permutational bimodules over a groupG. Thentheir tensor product M1⊗M2 is the quotient of the direct product M1×M2 by theidentification

m1 ⊗ g ·m2 = m1 · g ⊗m2

for m1,m2 ∈M and g ∈ G.

It is easy to see that M1 ⊗M2 is a well defined G-bimodule with respect to theactions

g1 · (m1 ⊗m2) · g2 = (g1 ·m1)⊗ (m2 · g2).Hence for a given permutational bimodule M the bimodules M⊗n are defined.

If M is a covering d-fold bimodule, then M⊗n is a covering dn-fold bimodule.We define M⊗0 to be the group G with the natural left and right actions.If X is a basis of M , then the set X

n = {x1 ⊗ x2 ⊗ · · · ⊗ xn : xi ∈ X} is a basisof M⊗n. We will usually just write x1x2 . . . xn = x1 ⊗ x2 ⊗ · · · ⊗ xn.

Then every element of M⊗n is uniquely written in the form v · g, where v ∈ Xn

and g ∈ G. Consequently, for every v ∈ Xn and g ∈ G there exist unique u ∈ X

n

and h ∈ G such thatg · v = u · h

in M⊗n. It is easy to see that the map v 7→ u on Xn coincides with the associated

self-similar action (G,X,M) of g. Accordingly to this we write u = g(v). We denote

h = g|v,which also agrees with the corresponding notation for self-similar groups.


Note that the action (G,M,X) on Xn is conjugate with the natural left action of

G on the set of right orbits of G on M⊗n. Here the conjugator maps a right orbitR ∈M⊗n/G to the unique element v ∈ X

n belonging to R.

2.2. Iterated monodromy groups. Let C be a path connected and locally pathconnected topological space and let C1 ⊆ C be an open subset. Suppose thatf : C1 −→ C is a d-fold covering map, i.e., a continuous map such that for everyz ∈ C there exists a neighborhood U of z such that f−1(U) is a disjoint union ofd sets Ui for which f : Ui −→ U is a homeomorphism. We call then f a partialself-covering of C.

There is a natural permutational bimodule Mf over the fundamental groupπ1(C, t) of C, associated with a partial self-covering f : C1 −→ C. It consists ofthe homotopy classes of paths starting at the base-point t and ending in an inverseimage t′ ∈ f−1(t) of t under f . The right action of π1(C, t) on M is given just byappending the loops to such paths:

` · γ = `γ

for every ` ∈Mf and γ ∈ π1(C, t). Here we compose paths as functions, i.e., in theproduct `γ the loop γ is passed before the path `.

The left action on Mf is defined using inverse images of the loops under f :

γ · ` = γ``,

where γ` is the inverse image of γ under f uniquely defined by the condition thatthe beginning of γ` is the end of `.

It is easy to see that the right action of the fundamental group on the bimoduleMf is free and has d orbits. The orbits are in a one-to-one correspondence withthe end-points of their elements.

If we identify the right orbits with the corresponding end-points, then the leftaction of G on the set of right orbits of Mf will be identified with the monodromyaction of π1(C, t) on f−1(t): the image of a point z ∈ f−1(t) under the action ofγ ∈ π1(C, t) is the end of the inverse image of γ starting at z.

The following proposition is proved in [Nek05] (Proposition 5.2.3).

Proposition 2.1. Let f1 : C1 −→ C and f2 : C2 −→ C be partial self-coverings.Then the tensor product of bimodules Mf1 ⊗ Mf2 is isomorphic to Mf1◦f2 . Theisomorphism maps `1 ⊗ `2 to the path

f−12 (`1)`2`2 ∈Mf1◦f2 ,

where f−12 (`1)`2 is the unique inverse image of `1 under f2 starting at the end of

`2.

Let f : C1 −→ C be a partial self-covering. Then the backward orbit

Tf =⊔

n≥0

f−n(t)

of the basepoint has a natural structure of a vertex set of a rooted tree with theroot t ∈ f−0(t) in which a vertex z ∈ f−n(t) for n ≥ 1 is connected to the vertexf(z) ∈ f−(n−1)(t). The group π1(C, t) acts on each of the levels of this tree by themonodromy action, described above. These actions obviously agree with the treestructure and hence we get an action of π1(C, t) on the tree Tf .


It follows from Proposition 2.1 that the action of the fundamental group on Tf isconjugate with the associated action (π1(C, t),Mf ,X) on X

∗, where X is an arbitrarybasis of the bimodule Mf , i.e., a collection of d paths starting in t and ending ineach of the inverse images of t under f .

Definition 2.5. The iterated monodromy group of a partial self-covering f : C1 −→C is the quotient of π1(C, t) by the kernel of the monodromy action on the tree Tf .

As a straightforward corollary of the above definitions we get the following re-current formula for computation of the iterated monodromy group.

Theorem 2.2. Let f : C1 −→ C be a partial d-fold self-covering. Choose a basisX = {`x} of the bimodule Mf , i.e., a collection of d paths starting in the basepointt and ending in each inverse image of t under f . Then the associated action ofIMG (f) on X

∗ is given by the recurrent formula

γ(`xw) = `y(`−1y γx`x)(w),

where w ∈ X∗ is arbitrary, γ ∈ π1(C, t) is a loop defining an element of IMG (f),

the path γx is the unique inverse image of γ starting at the end of `x and `y ∈ X issuch that γx and `y have a common end.

Example 2.3. A rational function f ∈ C(z) acting on the Riemann sphere C issaid to be post-critically finite if the orbit of every critical point of f is finite.

Let f be a post-critically finite rational function and let Pf be the union of theforward images of the critical points under iterations of f . Then f is a partial

self-covering of the punctured sphere C \ Pf . The iterated monodromy group ofthis partial self-covering is the iterated monodromy group of f .

Direct calculations show that the iterated monodromy group of the polynomial z2

coincides with the binary adding machine action of Z (as described in Example 2.2).The iterated monodromy group of the polynomial z2 − 1 (for which the post-

critical set Pf is equal to {0,−1}) is generated by two elements a, b, which are givenby the recurrent

a(0w) = 1w, a(1w) = 0b(w), b(0w) = 0w, b(1w) = 1a(w).

For more on the iterated monodromy groups of rational functions and polyno-mials see [Nek05, BN06a, BGN03].

Example 2.4. The action of Zn associated to a matrix A, described in Example 2.1coincides with the iterated monodromy group of the self-covering of the torus Rn/Zn

induced by the matrix A.

2.3. Contracting groups and limit spaces.

Definition 2.6. A self-similar group (G,X) is said to be contracting if there is afinite set N ⊂ G such that for every g ∈ G there exists n such that g|v ∈ N forall v ∈ X

∗ of length |v| ≥ n. If the group is contracting, then the smallest set Nsatisfying this condition is called the nucleus of the group.

More generally, we define contracting permutational bimodules and their nucleiby the same definition (using the corresponding definition of g|v).

It is proved in [Nek04] that the property of a covering bimodule to be contractingdoes not depend on the choice of the basis. In particular, if a self-similar group iscontracting, then all equivalent self-similar groups are also contracting.


Example 2.5. The self-similar action of Zn defined by a matrix A (see Exam-ple 2.1) is contracting if and only if the matrix A is expanding (i.e., has spectrumoutside of the unit circle).

Definition 2.7. We say that a self-covering f : C1 −→ C of a complete Riemannianmanifold C is expanding if the derivative of f at every point of C1 is an expandinglinear operator and if for some z ∈ C the set Jf of accumulation points of theinverse orbit

⋃n≥1 f

−n(z) is compact. Then the set Jf does not depend on z, andis called the Julia set of f .

This is not the classical definition of a Julia set (see, for instance [Mil99]), butit will coincide with the classical one for the case of a hyperbolic rational function.

If f : C1 −→ C is an expanding partial self-covering, then the iterated monodromygroup (and the bimodule Mf) are contracting by Theorem 5.5.3 of [Nek05].

Example 2.6. If f is a post-critically finite rational function, then IMG (f) iscontracting. This follows from the uniform expansion of f on a neighborhood ofthe Julia set, if f is hyperbolic (if the orbit of every critical point converges to anattracting cycle). For the rest of post-critically finite rational functions (which willbe sub-hyperbolic) one has to consider C as an orbifold and show that f is expandingwith respect to an orbifold metric. More on orbifolds associated to post-criticallyfinite rational functions, see [Mil99]. Theory of iterated monodromy groups ofself-coverings of orbspaces is given in [Nek05].

Let M be a contracting covering G-bimodule. Choose a basis X of M . Let X−ω

be the space of left-infinite sequences . . . x2x1 over the alphabet X. We introducethe direct product topology on X

−ω (where X is discrete).Consider also the space X

−ω ×G where G is also discrete.

Definition 2.8. Two sequences . . . x2x1 · g and . . . y2y1 · h ∈ X−ω ×G are said to

be asymptotically equivalent if there is a sequence gn ∈ G assuming a finite set ofvalues and such that

gn · xnxn−1 . . . x1 · g = ynyn−1 . . . y1 · hin M⊗n for all n.

The sequences . . . x2x1 and . . . y2y1 ∈ X−ω are asymptotically equivalent if there

is a sequence gn ∈ G assuming a finite set of values such that

gn(xnxn−1 . . . x1) = ynyn−1 . . . y1

for all n.

So, if . . . x2x1 · g and . . . y2y1 · h ∈ X−ω ×G are asymptotically equivalent, then

. . . x2x1 and . . . y2y1 ∈ X−ω are asymptotically equivalent.

It is easy to see that asymptotic equivalence is an equivalence relation. It isproved in [Nek05] that the sequences . . . x2x1 · g and . . . y2y1 · h are equivalent ifand only if there exists a sequence gn of elements of the nucleus such that

gn · xn = yn · gn−1, and g0g = h.

The same statement is also true for the asymptotic equivalence on X−ω , except that

condition g0g = h is void.

Definition 2.9. The quotient of the space X−ω×G by the asymptotic equivalence

relation is called the limit G-space and is denoted XG.


The quotient of X−ω by the asymptotic equivalence relation is called the limit

space of (G,X) and is denoted JG.

We have a right action of G on the limit G-space XG coming from the naturalright action of G on X

−ω ·G. This action is proper and co-compact.The asymptotic equivalence relation is invariant under the shift

. . . x2x1 7→ . . . x3x2

on X−ω, hence the shift induces a continuous map σ : JG −→ JG on the limit space.

We obtain the limit dynamical system (JG, σ).It is proved in [Nek05] that the limit dynamical system and the action of G on

XG do not depend (up to topological conjugacies) on the choice of the basis X ofthe bimodule M . In particular the limit G-space XG and the limit space JG do notdepend on the basis.

It is easy to see that the limit space JG is the space of orbits of the action of Gon XG. The action is proper and this gives us a structure of an orbispace on JG.

The next theorem shows that the constructions of the iterated monodromy groupand the limit dynamical system are converse to each other. The theory becomescomplete only when we define iterated monodromy groups of partial self-coveringsof orbispaces and show how the limit dynamical system is a partial self-covering.For more details see Chapters 4 and 5 of [Nek05].

Theorem 2.3. Let f : C1 −→ C be an expanding partial self-covering. Then theiterated monodromy group IMG (f) is contracting and the limit dynamical system(JIMG(f), σ) is topologically conjugate to the dynamical system (Jf , f) of the actionof f on its Julia set.

Theorem 2.3 gives a symbolic finite-to-one encoding of the points of the Juliaset Jf by the sequences . . . x2x1 ∈ X

−ω. Note that this encoding depends on thechoice of a basis of the associated bimodule Mf .

A proof (of a more general version) of Theorem 2.3 is given in [Nek05], Theo-rem 5.5.3.

Example 2.7. If f : Rn/Zn −→ Rn/Zn is an expanding self-covering of the torusgiven by an expanding matrix A, then the Julia set of f is the whole torus Rn/Zn.Consequently, the limit dynamical system of the self-similar action of Zn describedin Example 2.1 is topologically conjugate to f acting on the torus. The obtainedencoding of Rn/Zn comes from the natural extension of the “A-adic” numerationsystem on Zn to Rn (see Section 6.2 of [Nek05]).

2.4. Limit solenoid. Let XZ be the space of bi-infinite sequences (xn)n∈Z over the

alphabet X. We write sometimes the sequence (xn)n∈Z as

. . . x−1x0 . x1x2 . . . ,

where dot marks the place between the coordinates number 0 and 1.The shift map σ : X

Z −→ XZ is given by

σ(. . . x−1x0 . x1x2 . . .) = . . . x−2x−1 . x0x1 . . . ,

i.e., if (yn)n∈Z = σ((xn)n∈Z), then yn = xn−1 for all n ∈ Z.

Definition 2.10. Let (G,X) be a contracting self-similar group. Two sequences(xn)n∈Z and (yn)n∈Z are asymptotically equivalent with respect to the action of G


if there exists a sequence gn ∈ G, for n ∈ Z assuming a finite set of values, suchthat

gn(xnxn+1 . . .) = ynyn+1 . . .

for all n ∈ Z.The quotient of the space X

Z by the asymptotic equivalence relation is called thelimit solenoid of the self-similar group and is denoted SG.

The shift σ on XZ preserves the asymptotic equivalence relation, hence it induces

a homeomorphism of SG, which we will also denote σ.The following description of the asymptotic equivalence relation and properties

of the solenoid are proved in [Nek05] Section 5.7.

Proposition 2.4. Sequences (xn)n∈Z and (yn)n∈Z are asymptotically equivalent ifand only if there exists a sequence (gn)n∈Z of the elements of the nucleus such thatgn · xn = yn · gn+1 for all n ∈ Z.

The limit solenoid SG is a compact metrizable space of finite topological dimen-sion.

If the action of the group G is transitive on the levels Xn of the tree X

∗, then thelimit solenoid is connected.

We will also need the following description of the topology on SG.

Proposition 2.5. Let Um be the set of pairs ((xn)n∈Z, (yn)n∈Z) such that

g(x−m . . . xm) = y−m . . . ym

for some g ∈ N . Let Um be the image of Um in SG ×SG.Then Um is a neighborhood of the diagonal and for every neighborhood of the

diagonal U ⊂ SG ×SG there exists m such that Um ⊂ U .

Proof. The projection map π : XZ × X

Z −→ SG × SG is a continuous surjectionbetween compact spaces, hence it is closed. The complement of Um is closed, henceits image under π is closed in SG ×SG. Consequently, the set

Um \ π(XZ × XZ \ Um)

is open. It follows from Proposition 2.4 that this set does not contain the diagonal,hence Um is a neighborhood of the diagonal.

Let U ⊂ SG × SG be a neighborhood of the diagonal. Then π−1(U) is aneighborhood of the diagonal of X

Z × XZ saturated with respect to the asymptotic

equivalence. It follows from Proposition 2.4 and the definition of the direct producttopology, that π−1(U) contains Um for some m, since it is the saturation of the setof pairs ((xn)n∈Z, (yn)n∈Z) such that x−m . . . xm = y−m . . . ym. �

The following description of the dynamical system (SG, σ) is proved in [Nek05].

Proposition 2.6. The dynamical system (SG, σ) is the natural extension of thelimit dynamical system (JG, σ), i.e., the solenoid is the inverse limit of the sequence

JGσ←− JG

σ←− JGσ←− · · ·

and the homeomorphism σ : SG −→ SG is induced by the shift σ : JG −→ JG onthe inverse limit.


We get hence a natural surjective continuous map (projection) SG −→ JG givenby the projection onto the first term of the inverse sequence. This projection isinduced by the map

. . . x−1x0 . x1x2 . . . 7→ . . . x−1x0.

3. C∗-algebras associated with self-similar groups

3.1. Matrix recursions. Let (G,X) be a self-similar group and let M = X ·G bethe respective permutational G-bimodule.

The linear C-span Φ of M becomes then a bimodule over the group ring CG,in the usual sense. The corresponding right CG-module ΦCG is free with the basisX = {x · 1 : x ∈ X}, hence is isomorphic to CGX.

We introduce on Φ the CG-valued inner product setting for α, β ∈ C, g, h ∈ Gand x, y ∈ X

〈αx · g|βy · h〉 ={αβg−1h if x = y

0 otherwise,

hence making the basis X of the right module orthonormal. Note that the definedinner product satisfies the conditions

(2) 〈v1| v2〉∗ = 〈v2| v1〉 , 〈v1| a · v2〉 = 〈a∗ · v1| v2〉 , 〈v1| v2 · a〉 = 〈v1| v2〉 · afor all v1, v2 ∈ Φ and a ∈ CG. Hence Φ is pre-Hilbert bimodule over the ∗-algebraCG.

The structure of the left module on Φ is given by the (structural) homomorphism

φ : CG −→ EndCG ΦCG∼= Md

(CGX

),

which we will call matrix recursion of the self-similar group.It is given on the elements g ∈ G by φ(g) = (axy)x,y∈X, where

axy =

{g|y if g(y) = x0 otherwise

and extended onto CG by linearity.

Example 3.1. The matrix recursion for the binary adding machine (see Exam-ple 2.2) is

φ(a) =

(0 a1 0

),

since a · 0 = 1 · 1 and a · 1 = 0 · a in the associated permutational bimodule.The matrix recursion on IMG

(z2 − 1

)(see Example 2.3) is given by

φ(a) =

(0 b1 0

), φ(b) =

(1 00 a

).

If Φ1 and Φ2 are two bimodules over the algebra CG, then their tensor productΦ1⊗Φ2 = Φ1⊗CGΦ2 is the quotient of the tensor product of vector spaces Φ1⊗CΦ2

by the linear span of the elements

v1 ⊗ a · v2 − v1 · a⊗ v2,for all v1 ∈ Φ1, v2 ∈ Φ2 and a ∈ CG. It is a CG-bimodule with respect to theoperations

a · (v1 ⊗ v2) = (a · v1)⊗ v2, (v1 ⊗ v2) · a = v1 ⊗ (v2 · a).


The inner product on Φ1 ⊗ Φ2 is given by

〈v1 ⊗ v2|u1 ⊗ u2〉 = 〈v2| 〈v1|u1〉 · u2〉 .Every endomorphism α of the right CG-module (Φ1)CG induces an endomor-

phism of the right CG-module (Φ1⊗Φ2)CG just by its action on the first coordinate:

v1 ⊗ v2 7→ α(v1)⊗ v2.We get in this way a homomorphism

EndCG(Φ1)CG −→ EndCG(Φ1 ⊗ Φ2)CG.

It is easy to see that composition of this homomorphism with the structural homo-morphism

CG −→ EndCG(Φ1)CG

of Φ1 is the structural homomorphism

CG −→ EndCG(Φ1 ⊗ Φ2)CG

of Φ1 ⊗ Φ2.In particular, for one bimodule Φ we get a sequence of homomorphisms

CG −→ Md(CG) −→Md2(CG) −→ Md3(CG) −→ . . .

of the matrix algebras Mdn(CG) ∼= EndCG(Φ⊗n)CG. We call these homomorphismand their compositions

φn,m : Mdn(CG) −→Mdm(CG)

for m > n, matrix recursions associated with the bimodule Φ (or the self-similargroup G, if Φ = C〈X ·G〉 is the self-similarity bimodule).

3.2. Universal Cuntz-Pimsner algebra of a self-similar group. Let Φ bea pre-Hilbert bimodule over a ∗-algebra A. For any pair x, y ∈ Φ we have theassociated “rank one” operator θx,y on Φ given by the equality

θx,y(z) = x 〈y| z〉 .It is easy to see that θx,y is an adjoinable endomorphism of Φ.

Let F (Φ) be the linear span of the operators θx,y. Suppose that the homomor-phism φ : A −→ EndA(Φ) defining the left action of A on Φ assumes values inF (Φ).

The Cuntz-Pimsner algebra OΦ is defined then (compare with Theorem 3.12of [Pim97]) as the universal C∗-algebra generated by A and operators Sx for everyx ∈ Φ satisfying the relations:

(1) αSx + βSy = Sαx+βy for all x, y ∈ Φ and α, β ∈ C.(2) Sx · a = Sx·a and a · Sx = Sa·x for all x ∈ Φ and a ∈ A.(3) S∗

x · Sy = 〈x| y〉 for all x, y ∈ Φ.(4) If φ(a) =

∑i αiθxi,yi , then a =

∑i αiSxiS

∗yi

in OΦ.

Let (G,X) be a self-similar group and let Φ be the linear span of the associatedpermutationalG-bimodule. Then the defining relations of the corresponding Cuntz-Pimsner algebra can be rewritten as in the following definition.

Definition 3.1. A (universal) Cuntz-Pimsner algebra OG = O(G,X) of a self-similargroup (G,X) is the universal C∗-algebra generated by the sets G and Sx (for x ∈ X)satisfying the following relations


(1) all relations of G (including the condition that the elements of G are uni-tary);

(2) relations

S∗xSx = 1,

∑

x∈X

SxS∗x = 1

(called Cuntz algebra relations, see [Cun77]);(3) for all g ∈ G, x ∈ X:

g · Sx = Sg(x) · g|x.Note that the last relation is equivalent to the relation

g =∑

x∈X

Sg(x)g|xS∗x.

The algebra OG may be seen as a cross-product of the group G with the self-similarity.

3.3. Another definition of OIMG(f). Let f : C1 −→ C be a partial self-coveringand let Mf be the associated permutational bimodule over π1(C). Hence we candefine the associated Cuntz-Pimsner algebra OMf

.There is an elegant definition of the algebra OMf

which does not use groupsexplicitly.

Choose a basepoint t ∈ C and consider the tree of preimages

Tf =⊔

n≥0

f−n(t).

The algebra OMfis then the universal C∗-algebra generated by operators Sγ

labeled by paths γ in C connecting the points of the set Tf and satisfying thefollowing defining relations.

(1) If γ1 and γ2 are homotopic, then Sγ1 = Sγ2 .(2) S1t = 1, where 1t is the trivial path at the basepoint t.(3) Sγ1γ2 = Sγ1Sγ2 , where γ1γ2 is the concatenation of the paths γ1 and γ2

(which is defined if the end of the path γ2 coincides with the beginning ofthe path γ1).

(4) Sγ−1 = S∗γ .

(5) Sγ =∑

δ∈f−1(γ) Sδ, where f−1(γ) is the set of deg f paths which are lifts

of γ by the covering f .

If we fix a collection {`x}x∈X of paths starting in t and ending in the points of thefirst level f−1(t) of the tree Tf (i.e., a basis of the bimodule Mf), then it is easy tosee that every generator Sγ can be written in the form S`x1

· · ·S`xnSαS

∗`ym· · ·S∗

`y1

for some loop α ∈ π1(C, t). It follows (by Theorem 2.2) that the defining relationsof OMf

are equivalent to the above relations (1)–(5).If f is an expanding self-covering of a complete Riemann manifold, then OMf

is

isomorphic to OIMG(f). We will denote OIMG(f) = Of .

3.4. Self-similar representations. If ρ is a unitary representation of a self-similargroup G on a Hilbert space H , then Φ⊗H is defined in the natural way (since H isa left G-module with respect to the representation ρ) as the quotient of the tensorproduct Φ⊗C H by the subspace spanned by the elements of the form

v ⊗ (ρ(g)(u))− (v · g)⊗ u,


for all v ∈ Φ, g ∈ G and u ∈ H . The C-valued inner product on Φ⊗H is given by

〈v1 ⊗ u1| v2 ⊗ u2〉 = 〈u1| ρ(〈v1| v2〉)(u2)〉 .If X is a basis of Φ, then Φ⊗H as a Hilbert space is isomorphic to

⊕x∈X

x⊗H ∼=H |X|, where x⊗H is an isomorphic copy of H . The representation Φ⊗ ρ acts on avector x⊗ v ∈ x⊗H by the rule

ρ(g)(x ⊗ v) = g(x)⊗ ρ(g|x)(v),

i.e., the representation Φ⊗ ρ is equal to the composition of the matrix recursion φwith ρ.

Definition 3.2. A representation ρ : G −→ U(H) is said to be self-similar if it isunitarily equivalent to Φ⊗ ρ.

The following is straightforward.

Proposition 3.1. A unitary representation of G is self-similar if and only if it canbe extended to a representation of the associated Cuntz-Pimsner algebra OG.

Example 3.2. The action of G on Xω preserves the Bernoulli measure µ. We get

consequently a unitary representation ρ of G on L2(Xω , µ). This representation canbe extended to a representation of OG by putting

ρ(Sx)(f)(w) =

{ √d · f(w1) if w = xw1,

0 otherwise.

The obtained representation is the natural representation of OG on L2(Xω , µ).

Example 3.3. Another class of self-similar representations of G are the permuta-tional ones. We say that a non-empty set W ⊂ X

ω is self-similar if W =⋃

x∈XxW .

If W is a self-similar G-invariant subset of Xω , then the permutational representa-

tion of G on `2(W ) is self-similar, since it can be extended to a representation ofOG by

ρ(Sx)(f)(w) =

{f(w1) if w = xw1,0 otherwise.

Such representations of OG are also called permutational. See for instance [BJ99],where permutational representations of the Cuntz algebra are studied.

3.5. Cuntz-Pimsner semigroup. Let us denote by 〈G,X〉 the inverse semigroupgenerated by the elements Sx, S

∗x and G in OG.

We will use a multi-index notation:

Sx1x2...xn = Sx1Sx2 . . . Sxn

andS∗

x1x2...xn= S∗

xn. . . S∗

x2S∗

x1.

We also agree that S∅ is the identity.The following is a direct corollary of the defining relations in OG.

Proposition 3.2. Every element of the semigroup 〈G,X〉 can be uniquely writtenin the form SvgS

∗u for u, v ∈ X

∗ and g ∈ G, and

S∗vSu =

Su′ if u = vu′

S∗v′ if v = uv′

0 in all the other cases (i.e., if v and u are incomparable)


The inverse semigroup 〈G,X〉 acts on the sets X∗ and X

ω by the rule

Sx(v) = xv

and the original action of G.The action on X

ω is an action by partial homeomorphisms (by homeomorphismsbetween cylindrical subsets). The action on X

∗ can be also interpreted as an actionby partial automorphisms (sometimes called hyerarchomorphisms, see [Ner03]).

3.6. The simple quotient of OG. We assume starting from here that the groupG is countable.

Definition 3.3. A point w ∈ Xω is G-generic if for any element h ∈ 〈G,X〉 one of

the following cases takes place:

(1) w does not belong to the domain of h,(2) h(w) 6= w,(3) w is fixed by h together with every point of a neighborhood of w.

It is easy to see that the set of points w not satisfying conditions (1)–(3) ofthe definition for a given h is a closed nowhere dense set. Consequently, the setof G-generic points is residual (co-meager). In particular, there exists a countable〈G,X〉-invariant set W (which will be automatically G-invariant and self-similar).

Theorem 3.3. Let ρ be a representation of OG on a Hilbert space H and let πbe the permutational representation of OG on `2(W ) for a 〈G,X〉-invariant set ofG-generic points W . Denote by ‖ · ‖ρ and ‖ · ‖π the respective C∗-norms on OG.

Then for every a ∈ OG we have

‖a‖π ≤ ‖a‖ρ.Proof. It is sufficient to prove the inequality for a ∈ C〈G,X〉. It is also sufficient toconsider the case when W is the 〈G,X〉-orbit of one point w0 ∈ X

ω .Let a =

∑mk=1 γkgk 6= 0, where γk ∈ C, gk ∈ 〈G,X〉. We may assume that g1 = 1.

For every ε > 0 there exists a non-zero element f of `2(W ) with a finite supportsuch that ‖a(f)‖ ≥ (1− ε)‖a‖π · ‖f‖.

Let f =∑

u∈S αuδu, where αu ∈ C, and S ⊂ W is a finite set. Here δu ∈ W isthe characteristic function of u.

Then

(3) a(f) =

m∑

k=1

∑

u∈S

αuγk · gk(u) =∑

v∈S′

∑

gk(δu)=v

αuγk

δv,

where S′ =⋃m

i=1 gk(S).Every point u ∈ S belongs to the orbit 〈G,X〉(w0), thus there exists hu ∈ 〈G,X〉

such that hu(w0) = u.Let W be the set of all the points w ∈ X

ω for which the equality gkhu(w) =glhv(w) is equivalent to the equality gk(u) = gl(v) for all u, v ∈ S and all 1 ≤ k, l ≤m (in particular, w has to belong to the domain of gkhu, if w0 does). The set Wcontains the point w0, and since w0 is G-generic, it contains a neighborhood of w0.

Thus there exists a finite beginning r ∈ X∗ of the word w0 ∈ X

ω , such that everyevery word w ∈ X

ω which begins with r belongs to W .Let us take r sufficiently long, so that r belongs to the domain of gkhu whenever

w0 does, andgkhu(w0) = glhv(w0)⇔ gkhu(r) = glhv(r),


and

gkhu(w0) 6= glhv(w0)⇔ gkhu(r) and glhv(r) are incomparable.

Fix a vector e ∈ H of norm 1, and define now a vector

f =∑

u∈S

αuρ(huSr)(e) ∈ Hdn

.

Every vector ρ(huSr) (e) belongs to the subspace ρ(huSr)(H) = ρ(Shu(r))(H).Since for different u ∈ S the words hu(r) are incomparable, ρ(huSr) (e) are orthog-

onal and ‖f‖ =∑

u∈S |αu|2 = ‖f‖.We also have that the vectors ρ(gkhuSr) (e) and ρ(glhvSr) (e) for u, v ∈ S are

orthogonal if gk(u) 6= gl(v).If gk(u) = gl(v), then gkhu(r) = glhv(r) and moreover, gkhu(rw) = glhv(rw) for

all w ∈ Xω . We have gkhu(rw) = gkhu(r)q1(w) and glhv(rw) = glhv(r)q2(w) for

q1, q2 ∈ G such that gkhuSr = Sgkhu(r)q1 and glhv ·Sr = Sglhv(r) ·q2. Hence q1 = q2and ρ(gkhuSr) (e) = ρ(glhvSr) (e)

Then∥∥∥ρ(a)

(f)∥∥∥ = ‖a(f)‖ and,

‖a‖ρ‖f‖ ≥ ‖a(f)‖ = ‖a(f)‖ ≥ (1 − ε)‖a‖π‖f‖ = (1− ε)‖a‖π‖f‖,hence

‖a‖ρ ≥ (1 − ε)‖a‖πfor every ε > 0, thus ‖a‖ρ ≥ ‖a‖π, which finishes the proof. �

If π : OG −→ B(H) is a representation, then we denote by OGπ the image of πin B(H).

We have the following corollary of the last theorem.

Corollary 3.4. Let π be the permutational representation of OG on a set of G-generic points of X

ω. Then the C∗-algebra OGπ does not depend on the choice ofW .

Let us denote the algebra OGπ from the corollary by OGmin . The next result isalso a direct corollary of Theorem 3.3.

Corollary 3.5. The algebra OGmin is the unique simple unital quotient of OG.

A similar result is obtained in [Nek04], where it is also shown that the algebraOGmin is purely infinite.

3.7. The gauge action and the gauge-invariant subalgebra of OG. Denoteby T = {z ∈ C : |z| = 1} the complex unit circle. Define for z ∈ T

Γz(g) = g

for all g ∈ G and

Γz(Sx) = zSx

for all x ∈ X.It is easy to see that the Γz(g) and Γz(Sx) satisfy the defining relations of OG

from Definition 3.1. Since Γz−1 ◦ Γz is identical, this implies that Γz is extendedto an automorphism of OG. The obtained action of T on OG is called the gaugeaction. It is easy to see that this action is continuous.


Since Γz(SvgS∗u) = z|v|−|u|SvgS

∗u for all v, u ∈ X

∗ and g ∈ G, the map

a 7→∫

Γz(a) dz

is a conditional expectation onto the closed linear span of the elements SvgS∗u for

g ∈ G and v, u ∈ X∗ such that |v| = |u|. This linear span is the set of fixed points

of the gauge action.Let Mk be the closed linear span in OG of the elements SvgS

∗u for g ∈ G and

v, u ∈ Xk. In particular,M0 is the C∗-subalgebra of OG generated by G, which we

will denote A.If follows from Proposition 3.2 that for v1, v2, u1, u2 ∈ X

k and g1, g2 ∈ G

Sv1g1S∗u1Sv2g2S

∗u2

=

{Sv1g1g2S

∗u2

if u1 = v20 otherwise.

Consequently,Mk is isomorphic to the algebra Mdk(A) of dk × dk-matrices overA.

Every element g ∈ G can be written in OG as a sum

(4) g =∑

x∈X

Sg(x)g|xS∗x,

which shows thatMk+1 ⊃Mk.Consequently, the space of fixed points of the gauge action is the C∗-algebra

equal to the closure of the ascending union⋃

k≥0Mk.

The subalgebra equal to the span of the elements SvS∗u for |v| = |u| is the

Glimm’s uniformly hyperfinite algebra Md∞ . Its diagonal subalgebra, generated bythe projections SvS

∗v is isomorphic to the algebra C(Xω) of continuous functions on

Xω. The isomorphism identifies a projection SvS

∗v with the characteristic function

of the cylindrical set vXω . It is easy to see that the action of G ⊂ MG on thediagonal algebra by conjugation coincides with the action by conjugation of G onC(Xω).

It is proved in [Nek04] that if the group (G,X) is self-replicating (see Defini-tion 2.3), then the gauge invariant sub-algebra of OG is generated by G and thesubalgebra C(Xω).

3.8. Functoriality of OG. Let (G1,X1) and (G2,X2) be self-similar groups andlet Mi = Xi ·Gi be the associated covering bimodules.

A morphism of self-similar groups is, by definition, a morphism of the associatedbimodules, i.e., a pair (FG, FM ), where FG : G1 −→ G2 is a homomorphism ofgroups and FM : M1 −→M2 is a map such that

FM (g · x · h) = FG(g) · FM (x) · FG(h)

for all g, h ∈ G1 and x ∈M1.

Example 3.4. Let f : C1 −→ C and f ′ : C′1 −→ C′ be partial self-coverings andF : C −→ C′ a continuous map such that F (C1) ⊂ C′1 and F ◦ f = f ′ ◦ F , i.e.,F semi-conjugates the coverings f and f ′. If we choose the basepoints t ∈ C andF (t) ∈ C′, then the map F induces a morphism F∗ : IMG (f) −→ IMG (f ′) ofthe self-similar iterated monodromy groups. It acts just by taking images of loopsγ ∈ IMG (f) and paths ` ∈Mf under the action of F .


Example 3.5. In the other direction, if F : (G1,X1) −→ (G2,X2) is a morphismof self-similar contracting groups, then it naturally induces a semi-conjugation ofthe limit dynamical systems, which follows directly from the definition of the limitG-space (see [Nek05] Definition 3.2.1).

The following property of Cuntz-Pimsner algebras follows directly from the def-inition (due to their universality).

Proposition 3.6. Every morphism of self-similar groups F : (G1,X1) −→ (G2,X2)induces a homomorphims of the Cuntz-Pimsner algebras F∗ : OG1 −→ OG2 , whichagrees with the gauge actions.

Example 3.6. There is a natural morphism (embedding) of the binary addingmachine action into the iterated monodromy group of any post-critically finitequadratic polynomial f , since quadratic polynomials act near infinity as z2, andIMG

(z2)

is the binary adding machine. This embedding induces a surjective semi-

conjugacy of the double self-covering of the circle (of the action of z2 on the unitcircle) onto the Julia set of f , coinciding with the Carateodori loop. This embeddinginduces also a homomorphism OZ −→ OIMG(f), which is injective, since, as we willsee later, the Cuntz-Pimsner algebra of the adding machine is simple.

3.9. Defining relations of the gauge-invariant sub-algebra.

Definition 3.4. Let R be the set of relations in G (including the condition thatthe elements of G are unitaries) together with the relations

Sv1g1S∗u1Sv2g2S

∗u2

=

{Sv1g1g2S

∗u2

if u1 = v20 otherwise

and

SvgS∗u =

∑

x∈X

Svg(x)g|xS∗ux,

where u, v, ui, vi ∈ X∗, g, gi ∈ G and |u| = |v|, |u1| = |v1| = |u2| = |v2|.

Denote byMG the universal C∗-algebra generated by partial isometries denotedSvgS

∗u, where v, u ∈ X

∗, |v| = |u| and g ∈ G, satisfying the relations R.

We know that the relations R are satisfied in OG, hence the gauge-invariantsub-algebra of OG is a quotient of MG. We will see later that actually MG isisomorphic to the gauge-invariant sub-algebra of OG, i.e., that every representationofMG is contained in a representation of OG (see Theorem 3.7).

It follows from the definition ofMG that

MG∼= Md ⊗MG,

where Md is the algebra of |X|× |X|-matrices over C and the isomorphism identifiesan element SxvgS

∗yu for x, y ∈ X, g ∈ G and v, u ∈ X

∗, |v| = |u|, with SxS∗y⊗SvgS

∗u ∈

Md⊗MG after the usual identification of the elements SxS∗y with the matrix units

in Md. Similarly, we get the isomorphisms

MG∼= Mdn ⊗MG,

when we take x, y ∈ Xn.

In particular for every w ∈ X∗ the map

SvgS∗u 7→ SwvgS

∗wu


extends to an injective (non-unital) homomorphism Ew : MG −→ MG. We canwrite the maps Ew in terms of the above isomorphisms ofMG with Mdn ⊗MG ashomomorphisms

Mdn ⊗MG −→Mdn+|w| ⊗MG

acting trivially onMG and embedding Mdn = 〈SvS∗u : v, u ∈ X

n〉 into Mdn+|w| =〈SvS

∗u : v, u ∈ X

n+|w|〉 via the mapping SvS∗u → SwvS

∗wu.

Fix some x0 ∈ X and consider the direct limit of the forward iterations of Ex0

(5) MG

Ex0−→MG

Ex0−→MG

Ex0−→ · · · .It follows directly from the above arguments that the direct limit is isomorphic

to the direct limit of

MGEx0−→Md ⊗MG

Ex0−→Md2 ⊗MGEx0−→ · · · ,

i.e., to the stabilization K⊗MG, where K is the algebra of compact operators.The map Ex0 induces an automorphism of K⊗MG, which we will denote S.

Let us denote byMG(n) the nth term of the direct sequence (5) starting numer-

ation from 0, i.e., the image of Mdn ⊗MG in K ⊗MG. We have S(MG(n+1)) ⊂

MG(n). The automorphism S acts onMG

(0) just as the map Ex0 :

S(a) = Sx0aS∗x0∈MG

(0).

Consider the cross-product (K ⊗MG) oS Z and let US be the generator of Z.The cross-product is generated by the elements of the form aUn

S for n ∈ Z anda ∈ K⊗MG. We have aUn

S bUmS = aSn(b)Um+n

S .

Theorem 3.7. (1) The sub-algebra of the cross-product (K⊗MG) oS Z gen-

erated by MG(0) and the elements

Tx = SxS∗x0US ∈ MG

(0) · US

is isomorphic to OG.(2) Every representation ρ of MG can be extended to a representation of the

cross-product (K⊗MG) oS Z and a representation of OG.(3) The gauge-invariant sub-algebra of OG is isomorphic to MG.(4) The cross-product (K⊗MG) oS Z is isomorphic to K⊗OG.

Proof. We have

T ∗xTx = U∗

SSx0S∗xSxS

∗x0US = U∗

SSx0S∗x0US = S−1(Sx0S

∗x0

) = 1

and

TxT∗x = SxS

∗x0USU

∗SSx0S

∗x = SxS

∗x ∈MG

(0),

hence∑

x∈XTxT

∗x is equal to the unit ofMG

(0).Let us show by induction that for every v ∈ X

n we have

Tv = SvS∗xn0Un

S .

It is true for n = 1. Suppose that it is true for v, let us prove it for xv. We have

TxTv = SxS∗x0USSvS

∗xn0Un

S =

SxS∗x0· USSvS

∗xn0U∗

S · Un+1S = SxS

∗x0· Sx0SvS

∗xn+10

· Un+1S =

SxvS∗xn+10

Un+1S


It follows that for every SvgS∗u ∈ MG

(0), where v, u ∈ Xn and g ∈ G we have

TvgT∗u = SvS

∗xn0Un

S g(UnS )∗Sxn

0S∗

u =

SvS∗xn0Sn(g)Sxn

0S∗

u =

SvS∗xn0Sxn

0· g · S∗

xn0Sxn

0S∗

u = SvgS∗u.

It follows that for g ∈MG(0) we have

g =∑

x∈X

Tg(x)g|xT ∗x ,

hence the generators g ∈ G and Tx, x ∈ X satisfy the defining relations of the

algebra OG, hence the sub-algebra of (K⊗MG) oS Z generated byMG(0) and Tx,

for x ∈ X, is a quotient of OG, i.e., every representation of this sub-algebra is arepresentation of OG. Therefore, (2) will imply (1). We have seen that the definingrelations of MG are satisfied in the gauge-invariant sub-algebra of OG, hence (2)implies (3).

Let us prove (2). Suppose that ρ is a representation of MG on a Hilbert spaceH . For every n we have the representation Φ⊗n⊗ ρ of Mdn ⊗MG. Recall that thisis the representation on the direct sum Φ⊗n ⊗H =

⊕v∈Xn v ⊗H of |X|n copies of

H given by the formula

(Φ⊗n ⊗ ρ)(Svv1gS∗uu1

)(u ⊗ ξ) = v ⊗ ρ(Sv1gS∗u1

)(ξ),

for v, u ∈ Xn, v1, u1 ∈ X

∗, |v1| = |u1|, ξ ∈ H and w ⊗ ξ ∈ w ⊗H for w ∈ Xn and

ξ ∈ H denotes the vector ξ in the respective copy of H in the direct sum Φ⊗n⊗H .Denote by Hn for n ≥ 1 the space Φ⊗n ⊗ H , by H−n for n ≥ 1 the space

ρ(Sxn0S∗

xn0)(H) ⊂ H and by H0 the space H itself.

Similarly, we denote by ρn the representation Φ⊗n ⊗ ρ of MG on Hn for n ≥ 0and by ρ−n the representation given by

ρ−n(SvgS∗u)(ξ) = ρ(S∗

xn0 vgSxn

0 u)(ξ)

for every ξ ∈ H−n = ρ(S∗xn0Sxn

0). It is easy to check that ρ−n is a representation of

MG.Let H =

⊕n∈Z

Hn and let ρ =⊕

n∈Zρn be the respective representation ofMG

on the direct sum H.We have isometric embeddings Hn −→ Hn+1 equal to the identical embeddings

for n < 0 and mapping v⊗ ξ ∈ Hn = Φ⊗n ⊗H to x0v⊗ ξ ∈ Φ⊗(n+1)H = Hn+1 forn ≥ 0. The direct sum of these embeddings is an isometry E : H −→ H.

Let us show that E agrees with the embedding Ex0 :MG −→MG and with therepresentation ρ, i.e., that

(6) E (ρ(a)(ξ)) = ρ(Ex0(a))(E(ξ))

for all a ∈MG and ξ ∈ H.It is sufficient to check this equality for a = SvgS

∗u and ξ ∈ Hn. Suppose that

n ≥ 0. We may assume that SvgS∗u is of the form Sv1v2gS

∗u1u2

for some v1, u1 ∈ Xn

and ξ = u1⊗ζ for ζ ∈ H . (If u1 is not a beginning of u, then ρ(SvgS∗u)(u1⊗ζ) = 0.)

Then

ρ(Sv1v2gS∗u1u2

)(u1 ⊗ ζ) = v1 ⊗ ρ(Sv2gS∗u2

)(ζ),

where u1, v1 ∈ Xn, ζ ∈ H .


Similarly

x0v1 ⊗ ρ(Sv2gS∗u2

)(ζ) = ρ(Sx0v1v2gS∗x0u1u2

)(x0u1 ⊗ ζ),hence

E (ρ(SvgS∗u)(ξ)) = ρ(Ex0(SvgS

∗u))(E(ξ)).

For n < 0 we have for every a = SvgS∗u ∈MG and ξ ∈ Hn

ρ(a)(ξ) = ρ(Sx−n0 vgS

∗x−n0 u

)(ξ).

The image of this vector in Hn+1 is equal to

E(ρ(Sx−n

0 vgS∗x−n0 u

)(ξ))

= ρ(Sx0vgS∗x0u)(E(ξ)),

since E is identical on Hn ⊂ Hn+1 and

ρ(Sx0vgS∗x0u)(E(ξ)) = ρ(Sx−n−1

0 x0vgS∗x−n−10 x0u

)(ξ)

by the definition of ρn+1.

Let H be the direct limit of H under iterations of the isometry E. We get

then a representation π of the algebra K ⊗MG on H as the direct limit of therepresentations ρ on H with respect to the embeddings E and Ex0 . Note that theoriginal representation ρ of MG is a sub-representation of the restriction of the

representation π ontoMG(0) ∼=MG.

The embeddings E : H −→ H induces a unitary operator U on the direct limit

H. It follows directly from (6) that Uπ(a)U∗ = π(S(a)), where S, as before, is theautomorphism of K ⊗MG induced by Ex0 . Consequently, setting π(S) = U wedefine a representation π of (K⊗MG) oS Z.

We have proved that every representation of MG can be extended to a repre-sentation of the cross-product.

The initial sub-space H of the direct limit H is π(MG(0)) = ρ(MG)-invariant

and π(SxS∗x0S) = ρ(SxS

∗x0

)U -invariant for every x ∈ X.The restriction of π(SvgS

∗u) onto H coincides, by definition, with ρ(SvgS

∗u) for

every SvgS∗u ∈ MG and hence together with the restrictions of π(SxS

∗x0S) onto H

we get a representation of OG. We have proved (2), which implies (1) and (3).It remains to prove that (K⊗MG) oS Z is isomorphic to K⊗OG, where OG is

now identified with the sub-algebra generated byMG(0) and Tx for x ∈ X.

Define for n ≥ 0

T (n)x = SxS

∗x0US ,

where SxS∗x0

is seen as an element of MG(n). Consider the sub-algebra Cn of

(K⊗MG) oS Z generated byMG(n) and T

(n)x for all x ∈ X.

Note that

T (n−1)x = Sx0SxS

∗x0S∗

x0US = Sx0xS

∗x0x0

Sx0S∗x0US = Sx0xS

∗x0x0

T (n)x0,

where all products of Sy are elements of MG(n). Consequently, Cn−1 ⊂ Cn.

The elements of the form SvgS∗u · Um

S for SvgS∗u ∈ MG

(n) and m ≥ 0 generatethe cross-product algebra (K ⊗MG) oS Z. The image of the element SvgS

∗u · Um

S

inMG(n+m) is equal to

Sxm0SvgS

∗uS

∗xm0· Um

S = Sxm0SvgS

∗uS

∗xm0Sxm

0S∗

xm0· Um

S = Sxm0 vgSxm

0 uT(n+m)xm0

∈ Cn+m,

hence the union of the sub-algebras Cn is dense.


Recall that the map Cn −→ OG given by

SvgS∗u 7→ SvgS

∗u and T (n)

x 7→ Sx,

for SvgS∗u ∈MG

(n) ⊂ Cn on the left-hand side and SvgS∗u ∈ OG on the right-hand

side of the first equality, is an isomorphism.The identical embedding Cn ↪→ Cn+1, after identifications of OG with Cn and

Cn+1, is given by

SvgS∗u 7→ Sx0SvgS

∗uS

∗x0

and

Sx 7→ Sx0xS∗x0x0

Sx0 = Sx0SxS∗x0.

Thus, it is the embedding OG ↪→ OG∼= Md ⊗OG given by

a 7→ Sx0aS∗x0

for all a ∈ OG. We conclude (in the same way as for MG) that the direct limit ofthese embeddings is isomorphic to K⊗OG. Since

⋃Cn is dense in (K⊗MG)oS Z,

we get the isomorphism of K⊗OG with (K⊗MG) oS Z. �

3.10. MG as a direct limit. Let C∗(G) be the universal C∗-algebra of the groupG. For every pair n < m the matrix recursion

φn,m : Mdn(CG) −→Mdm(CG)

is extended to a homomorphism of C∗-algebras

φn,m : Mdn(C∗(G)) −→Mdm(C∗(G)).

Proposition 3.8. The algebra MG is isomorphic to the direct limit of the C∗-algebras Mdn(C∗(G)) with respect to the homomorphisms φn,m : Mdn(C∗(G)) −→Mdm(C∗(G)) induced by the matrix recursion.

Proof. The direct limit is generated by the images of the elements

SvgS∗u ∈Mdn(C∗(G)),

for v, u ∈ Xn and g ∈ G. These images obviously satisfy the defining relations of

MG, hence the direct limit is a homomorphic image ofMG.On the other hand, if we have any collection of partial isometries SvgS

∗u satisfying

the defining relations ofMG, then we get representations of Mdn(C∗(G)) forming acommutative diagram with the homomorphisms φn,m, hence these representationsdefine a representation of the direct limit. �

In particular, the algebra A ⊂ MG generated by G (i.e., the closed linear spanof G inMG) is isomorphic to the completion of the algebra C∗(G) with respect tothe norm

‖a‖0 = limn→∞

‖φ0,n(a)‖ = infn≥0‖φ0,n(a)‖.

If k is any field, then we also can define an analog of the algebra A puttingAk to be the quotient of kG by the union of the kernels of the matrix recursionsφ0,n : kG −→ Mdn(kG). This algebra was defined and studied for the first time byS. Sidki in [Sid97]. See also [Bar06] for other examples of such algebras.

Example 3.7. If (G,X) is the binary adding machine action of G = Z, then MZ

is the Bunce-Deddence algebra [BD75]. This follows from the matrix recursion forthe adding machine (see Example 3.1).


4. Contracting groups

Definitions of contracting groups and their properties are given in Subsection 2.3.

4.1. A finite set of defining relations of OG. Let (G,X) be a contracting actionand let N be its nucleus.

It follows that the gauge-invariant sub-algebra MG of OG is the closure of theascending union of the finite-dimensional subspaces Mn(N ) equal to the linearspan of the sets

{SvgS∗u : v, u ∈ X

n, g ∈ N}.The spaceMn(N ) has dimension |X|2n · |N | and we have

Mn(N ) ⊂Mn+1(N ).

There exists (by the definition of a contracting action) a number n0 is such thatg1g2|v ∈ N for all gi ∈ N and v ∈ X

n0 . Then

M0(N ) · M0(N ) ⊂Mn0(N )

and, more generally,Mn(N ) · Mn(N ) ⊂Mn+n0(N ).

Proposition 4.1. Let G be a group together with a contracting covering G-bimodule

M . Let En be the subgroup of the elements g ∈ G such that g · v = v · 1 for allv ∈ M⊗n. Suppose that

⋃n≥1 En coincides with the kernel of the associated self-

similar action (G,X). Let G be the quotient of G by the kernel of the action.

Then the canonical epimorphism G −→ G induces an isomorphism OG −→ OG

of C∗-algebras.

Proof. If an element of G is trivial in G, then it follows from the conditions of

proposition, that its image in Mdn(G) under the nth iterate of the matrix recursionwill be trivial for all sufficiently big n, hence this element will be equal to 1 in OG.

Consequently, all relations of G will be present in OG, which implies that OG∼=

OG. �

Theorem 4.2. If the group (G,X) is contracting and N is its nucleus, then thealgebra OG is generated by {Sx}x∈X ∪ N and is defined by the following finite setof relations

(1) Cuntz relationsS∗

xSx = 1

(2) decompositions

g =∑

x∈X

Sg(x)g|xS∗x

for every g ∈ N ,(3) all relations g1g2g3 = 1 of length at most three which are true for the

elements of the nucleus N in the group G and relations gg∗ = g∗g = 1 forg ∈ N .

Proof. Note that the remaining Cuntz algebra relation∑

x∈XSxS

∗x is contained in

(2) for g = 1. Since every element of C〈G,X〉 can be written as a linear combinationof the elements of the form SvgS

∗u for u, v ∈ X

∗ and g ∈ N , the set {Sx}x∈X ∪N generates OG. The rest follows from Proposition 4.1 and Propositions 2.13.2and 2.13.4 of [Nek05]. �


Example 4.1. For the binary adding machine action of Z (see Example 2.2) the

algebra OZ, which is isomorphic to Oz2 , is given by the presentation

C∗〈S0, S1, a : S∗xSx = 1, S0S

∗0 + S1S

∗1 = 1, aa∗ = 1, a∗a = 1, a = S0aS

∗1 + S1S

∗0 〉,

We can rewrite this as the presentation

C∗〈S, a | S∗S = 1,

a∗a = aa∗ = 1,

SS∗ + aSS∗a∗ = 1,

a2S = Sa〉,for S = S0 and aS = S1.

Example 4.2. If G = IMG(z2 − 1

)is the Basilica group, then OG = Oz2−1 is

generated by S0, S1 satisfying the Cuntz algebra relations and unitaries a and bsuch that

a = S1S∗0 + S0bS

∗1 , b = S0S

∗0 + S1aS

∗1 ,

which is equivalent to the presentation

C∗〈S, a | S∗S = 1,

a∗a = aa∗ = 1,

SS∗ + aSS∗a∗ = 1,

a2S = S2S∗ + (Sa)2S∗a∗〉.

Example 4.3. More generally, the algebras Oz2+c for a post-critically finite qua-dratic polynomial z2 + c is generated by the Cuntz algebra O2 = 〈S0, S1〉 and oneunitary element a satisfying the relation

a = S1S∗0 + S0(1− SvS

∗v + SvaS

∗v )S∗

1 ,

if the critical point of f is periodic under iterations of f and

a = 1− S0uS∗0u − S1vS

∗1v + S0u(S1S

∗0 + S0S

∗1 )S∗

0u + S1vaS∗1v,

if it is pre-periodic. Here u and v are some finite (possibly empty) binary wordsrelated to the “kneading sequence” of the polynomial. These presentations followdirectly from the description of the iterated monodromy groups of post-criticallyfinite quadratic polynomials given in [BN06a].

The next theorem is proved in the same way as Proposition 4.1, (see also Propo-sition 3.8).

Theorem 4.3. Let G and G be as in Proposition 4.1. Then the algebra MG is

isomorphic to the direct limit of the matrix algebras Mdn(C∗(G)) with respect tothe homomorphisms induced by the matrix recursions.

4.2. Reconstruction of the limit space from the gauge action. Recall thatthe gauge action of the unit circle T = {z ∈ C : |z| = 1} on OG is given byΓz(g) = g for g ∈ G and Γz(Sx) = zSx. We will show here how to reconstruct thelimit dynamical system (JG, σ) of a contracting group from the C∗-algebra OG andthe gauge action on it.

Throughout this subsection (G,X) is a contracting self-similar group and N isits nucleus.


The subalgebraMG is uniquely determined as the algebra of the fixed points ofthe gauge action.

Let Tx for x ∈ X by any collection of d elements of OG such that Γz(Tx) = zTx

for all z ∈ T, x ∈ X, and Tx generate a Cuntz algebra; that is, T ∗xTx = 1 for all

x ∈ X and∑

x∈XTxT

∗x = 1.

It is easy to see that the map ∆ :MG −→MG defined as

∆(a) =∑

x∈X

TxaT∗x

is an injective unital endomorphism ofMG.Consider the direct limit C∗-algebra of the iteration of this embedding, i.e., of

the sequence

MG∆−→MG

∆−→MG∆−→ · · ·

and denote it BG.The nth term of this direct sequence (starting numeration from 0) is, as usual,

identified with the tensor product Mdn ⊗MG. Then the endomorphism ∆ acts as

the usual diagonal embedding on Mdn . Consequently, the algebra BG is isomorphicto the tensor product ofMG with the Glimm’s UHF algebra Md∞ .

The embedding ∆ induces an automorphism of BG, denoted ∆. Let us denote

by MG(n) = Mdn ⊗MG for n ≥ 0 the image in BG = Md∞ ⊗MG of the nth

term of the direct sequence (starting numeration from 0). It is a subalgebra of BG

isomorphic toMG, the union of such sub-algebras is dense and

∆(MG(n)) =MG

(n−1).

Definition 4.1. Denote by BG the asymptotical center of BG, i.e., the set of all

elements a ∈ BG such that

limn→∞

‖[∆n(a), b]‖ = 0

for all b ∈ BG.

Lemma 4.4. The asymptotic center BG is a C∗-sub-algebra of BG.

Proof. If ‖a1 − a2‖ < ε, then

‖[∆n(a1), b]− [∆n(a2), b]‖ =

‖∆n(a1)b − b∆n(a1)−∆n(a2)b+ b∆n(a2)‖ ≤2 ‖∆n(a1)−∆n(a2)‖ · ‖b‖ = 2‖a1 − a2‖ · ‖b‖ ≤ 2ε‖b‖,

which shows that BG is closed.It is clearly a ∗-algebra, since , and

[∆n(a∗), b] = [∆n(a), b∗]∗,

[∆n(a1 + a2), b] = [∆n(a1), b] + [∆n(a2), b],

and

[∆n(a1a2), b] =

∆n(a1)∆n(a2)b −∆n(a1)b∆

n(a2) + ∆n(a1)b∆n(a2)− b∆n(a1)∆

n(a2) =

∆n(a1)[∆n(a2), b] + [∆n(a1), b]∆

n(a2),

hence ‖[∆n(a1a2), b]‖ ≤ ‖a1‖ · ‖[∆n(a2), b]‖+ ‖a2‖ · ‖[∆n(a1), b]‖ . �


The automorphism ∆ preserves the algebra BG (since a ∈ BG if and only if∆(a) ∈ BG).

Proposition 4.5. Let Dn = Cdn ⊂MG

(n) = Mdn ⊗MG be the linear span of the

projections TvT∗v for v ∈ X

n. Then Dn ⊂ Dn+1, and

D =⋃

n≥0

Dn

is a commutative subalgebra of Md∞ ⊗MG isomorphic to the algebra C(X−ω) of

continuous functions on X−ω. The isomorphism maps TvT

∗v ∈ Dn to the charac-

teristic function of the cylindrical set X−ωv.

Proof. The element TvT∗v ∈ Dn is equal to

∑x∈X

TxvT∗xv ∈ Dn+1, which agrees with

the identification of TvT∗v with the characteristic function of X

−ωv. This impliesthat

⋃n≥0Dn is a ∗-algebra isomorphic to the algebra of continuous finite range

functions on X−ω. It is easy to see that restriction of the BG-norm onto

⋃n≥0Dn

coincides with the sup-norm on the algebra of functions, hence we get the necessaryisomorphism of the completions. �

Denote by DG the C∗-algebra C(X−ω) ∩ BG, called the diagonal sub-algebra ofBG.

Theorem 4.6. The C∗-algebras BG and DG and their endomorphism ∆ do notdepend on the choice of the isometries Tx (i.e., they depend only on the algebra OG

and the gauge action).The diagonal algebra DG ⊂ BG is isomorphic to the algebra C(JG) of continuous

functions on the limit space of G. The endomorphism ∆ of DG is induced by thelimit dynamical system σ : JG −→ JG.

Proof. Let us prove at first that the algebras do not depend on the choice of Tx.Let Rx be another set of isometries such that Γz(Rx) = zRx,

∑x∈X

RxR∗x = 1 and

R∗xRx = 1 for every x ∈ X.

Denote by ∆R the corresponding diagonal embedding ∆R(a) =∑

x∈XRxaR

∗x.

Consider the element

U =∑

x∈X

RxT∗x

of OG. We have

UU∗ =∑

x∈X

RxT∗x

∑

y∈X

TyR∗y =

∑

x∈X

RxR∗x = 1

and similarly UU∗ = 1, so that U is unitary.We also have Γz(U) =

∑x∈X

zRxz−1T ∗

x = U , hence U ∈ MG. The diagonal

embedding ∆ defined with respect to the set {Tx} is given then by

∆(a) =∑

x∈X

TxaT∗x =

∑

x∈X

TxR∗x ·∑

y∈X

RyaR∗y ·∑

x∈X

RxT∗x = U∗∆R(a)U.

Consequently,

∆n(a) = U∗∆R(U∗) . . . ∆n−1R (U∗)∆n

R(a)∆n−1R (U) . . . ∆R(U)U.


We get hence the following commutative diagram of C∗-algebras

MG∆R−→ MG

∆R−→ MG∆R−→ · · ·yU0

yU1

yU2

MG∆−→ MG

∆−→ MG∆−→ · · ·

where Uk is the inner automorphism

x 7→ U∗∆R(U∗) . . . ∆n−1R (U∗)x∆n−1

R (U) . . . ∆R(U)U

(and U0(x) = x). As a direct limit of this diagram we get an isomorphism U∞

between the direct limit algebras, conjugating the actions of ∆R and ∆. Conse-quently, U∞ is an isomorphism between the asymptotical centers of the direct limitalgebras.

Let us show that this isomorphism preserves the diagonal sub-algebras.

Note that ∆nR(U) =

∑vx∈Xn+1 RvxT

∗xR

∗v. Let us prove by induction that for

every v ∈ Xn we have

R∗v∆n−1

R (U) . . . ∆R(U)U = T ∗v

Suppose that the statement is true for v, consider vx instead, for x ∈ X. Then

R∗vx∆n

R(U) = R∗vx

∑

wy∈Xn+1

RwyT∗yR

∗w = T ∗

xR∗v,

hence

R∗vx∆n

R(U) . . . ∆R(U)U = T ∗xR

∗v∆n−1

R (U) . . . ∆R(U)U = T ∗xT

∗v = T ∗

vx.

Consequently for every n ≥ 0 and v ∈ Xn we have

Un(RvR∗v) = TvT

∗v ,

which implies that U∞ is an isomorphism between the direct limits of the diagonalalgebras.

We have proved that BG, DG and the action of ∆ on them do not depend on thechoice of {Tx}.

Therefore, it is sufficient to prove our theorem for the standard generating set

{Sx}x∈X of the Cuntz algebra OX. From now on ∆(a) =∑

x∈XSxaS

∗x.

Lemma 4.7. An element a ∈ BG belongs to the asymptotic center if and only if

limn→∞

‖[∆n(a), g]‖ = 0

for every g ∈ N ⊂ G ⊂MG(0).

Proof. By the same argument as in Lemma 4.4, if ‖b1 − b2‖ < ε, then

‖[∆n(a), b1]− [∆n(a), b2]‖ ≤ 2ε ‖∆n(a)‖ = 2ε‖a‖,which shows that if a asymptotically commutes with every element of a dense subset

of BG, then it belongs to BG.We also have equalities

[∆n(a), b1 + b2] = [∆n(a), b1] + [∆n(a), b2],

[∆n(a), λb] = λ[∆n(a), b],

and[∆n(a), b1b2] = [∆n(a), b1]b2 + b1[∆

n(a), b2]


which imply that every element a asymptotically commuting with every element of

a generating set of BG belongs to BG.Suppose that A is a generating set of MG. Let A(n) be the image of this

generating set in MG(n). Then

⋃n≥1A

(n) is a generating set of BG.

Let a ∈ BG and b ∈ A(k). Then for all n

‖[∆n(a), b]‖ =∥∥[∆n+k(a),∆k(b)]

∥∥ ,

since ∆ is an automorphism. But ∆k(b) belongs to A(0). Consequently it is suf-

ficient to consider only a generating set of MG(0). The set of elements SvgS

∗u for

g ∈ N and v, u ∈ X∗, |v| = |u|, is a generating set ofMG

(0).Suppose that a ∈MG and |v| = |u| = k. Then

∥∥∥[∆k(a), SvgS∗u]∥∥∥ =∥∥∥∥∥∥

∑

w∈Xk

SwaS∗wSvgS

∗u − SvgS

∗u

∑

w∈Xk

SwaS∗w

∥∥∥∥∥∥=

‖SvagS∗u − SvgaS

∗u‖ = ‖Sv[a, g]S

∗u‖ = ‖[a, g]‖.

For every a ∈ BG and for every ε > 0 there exists m and am ∈MG(m) such that

‖a− am‖ < ε. Then

lim supn→∞

‖[∆n(a), SvgS∗u]‖ ≤ 2ε ‖SvgS

∗u‖+ lim sup

n→∞‖[∆n(am), SvgS

∗u]‖ =

2ε+ lim supn→∞

∥∥[∆k(∆n−k(am)), SvgS∗u]∥∥ =

2ε+ lim supn→∞

∥∥[∆n−k(am), g]∥∥ ≤ 2ε+ 2ε‖g‖+ lim sup

n→∞

∥∥[∆n−k(a), g]∥∥ =

4ε+ lim supn→∞

‖[∆n(a), g]‖ ,

since for n ≥ m+ k the element ∆n−k(am) belongs to MG(0), where the action of

∆ coincides with ∆. This proves that in order to check that a belongs to BG, it issufficient to check asymptotic commutation with elements of N . �

We are ready now to prove that the algebra DG = D ∩BG is isomorphic to thealgebra C(JG) of continuous functions on the limit space of G.

Since the limit space JG is the quotient of X−ω by the asymptotic equivalence

relation, the algebra C(JG) is in a natural way a sub-algebra of C(X−ω) ∼= D. Afunction f ∈ C(X−ω) belongs to C(JG) if and only if it is constant on each of theasymptotic equivalence classes.

Let us show that C(JG) ⊂ D. Take any f ∈ C(JG) ⊂ C(X−ω). For every

ε > 0 there exists m and a linear combination fm =∑

v∈Xm λvSvS∗v ∈ Dm of

characteristic functions of cylindrical sets, such that ‖f − fm‖ < ε. Note that

∆n(fm) ∈ MG(0) for all n ≥ m.

The images of the cylindrical sets X−ωv and X

−ωu in JG, for v, u ∈ Xm intersect if

and only if there exists g ∈ N such that g(v) = u (see Proposition 3.6.8 of [Nek05]).If they intersect, then |λv−λu| < 2ε, since f will have a common value in two pointsof the cylindrical sets.


Consider ∆n(fm) =∑

v∈Xm+n λvSvS∗v ∈MG

(0) for some n ≥ m. Then

‖[∆n(f), g]‖ ≤

2ε+ ‖[∆n(fm), g]‖ = 2ε+

∥∥∥∥∥∑

v∈Xn+m

λvSvS∗vg − λvgSvS

∗v

∥∥∥∥∥ =

2ε+

∥∥∥∥∥∑

v∈Xn+m

(λvSvg|g−1(v)S∗g−1(v) − λvSg(v)g|vS∗

v)

∥∥∥∥∥ =

2ε+

∥∥∥∥∥∑

v∈Xn+m

(λg(v) − λv)Sg(v)g|vS∗v

∥∥∥∥∥ =

2ε+ maxv∈Xn+m

|λg(v) − λv| < 4ε,

hence f ∈ D. We have proved that C(JG) ⊂ D.Take now a function f ∈ C(X−ω) \ C(JG). Then there exist two sequences

. . . x2x1 and . . . y2y1 ∈ X−ω such that f(. . . x2x1) 6= f(. . . y2y1), but the sequences

are asymptotically equivalent, i.e., there exists a sequence gn ∈ N such thatgn(xn . . . x1) = yn . . . y1. Denote |f(. . . x2x1) − f(. . . y2y1)| = δ. For every ε > 0

there exists m and fm ∈ Dm such that ‖f − fm‖ < ε. If n ≥ m and ∆n(fm) =∑v∈Xn+m λvSvS

∗v , then |λxn+m...x1−f(. . . x2x1)| < ε and |λyn+m...y1−f(. . . y2y1)| <

ε. Consequently |λxn+m...x1 − λyn+m...y1 | > δ − 2ε.We can choose an increasing sequence nk ≥ m of indices such that gnk+m = g is

constant. Then for every k we have

‖[∆nk(f), g]‖ ≥ ‖[∆nk(fm), g]‖ − 2ε = max|v|=nk+m

|λg(v) − λv| − 2ε > δ − 4ε,

which proves that f /∈ D.

It is easy to see that the action of ∆ on C(X−ω) = D is induced by the shift. . . x2x1 7→ . . . x3x2 on X

−ω (it is sufficient to check this on the characteristicfunctions of the cylindrical sets). This implies that the action of ∆ on C(JG) = Dis induced by the limit dynamical system σ : JG −→ JG. �

4.3. K-theory of algebras associated to rational functions. Theorem 4.3makes it possible to compute the K-theory of MG and OG for many contractinggroups. Let us consider some examples.

Theorem 4.8. Let f be a post-critically finite hyperbolic rational function of degreed. Denote by c the number of attracting cycles of f , by k the sum and by l thegreatest common divisor of their lengths.

Then

K0(MIMG(f)) = Z[1/d], K1(MIMG(f)) = Zk−1

and

K0(OIMG(f)) = Z/(d− 1)Z⊕ Zc−1, K1(OIMG(f)) = Z/lZ⊕ Z

c−1.

Proof. It follows from Theorem 5.5.3 of [Nek05] and the fact that f is expandingon the complement of the post-critical set Pf of f with respect to the Poincare

metric, that the associated bimodule Mf over the fundamental group π1(C \ Pf )is contracting. The fundamental group is the free group F|Pf |−1 of rank |Pf | − 1generated by loops around each post-critical point except one. It will be more


convenient for us to present the fundamental group as 〈γz1 , . . . , γzp : γz1 · · · γzp =1〉, where γzi are closed disjoint simple loops around the post-critical points zi.

Let us show that the fundamental group G = π1(C \ Pf ) satisfies the conditions

of Proposition 4.1. We have to prove that every element g ∈ G which is trivial inIMG (f) belongs to the group E =

⋃n≥1 En. Suppose that g does not belong to E .

This means that for every n we have g ·v = v ·g|v for all v ∈ Xn and at least one g|v

does not belong to E . The set of possible values of g|v is finite (since the bimoduleis contracting), hence we can find two words v and u ∈ X

∗ such that g|v = g|vu andg|v /∈ E . It will follow that g|v = g|vun for all n ≥ 1. If γ is a loop representingg|v, then g|vun is of the form `nγn`

−1n for some path `n and an fn-preimage γn of

γ, which is a loop. Moreover, the path `n+1 is a continuation of the path `n. Thelength of the path γn tends to zero as n goes to infinity and the path `n convergesto a path from the basepoint to a point of the Julia set. But this clearly contradictsto the condition that g|v = g|vun is a non-trivial element of the fundamental group.

Consequently, the fundamental group G of C \ Pf satisfies the conditions ofTheorem 4.3 for G = IMG (f).

It is known that K0(C∗(F|Pf |−1)) is Z, generated by the class of 1 and the group

K1(C∗(F|Pf |−1)) is Z|Pf |−1 generated by the classes of the free generators of F|Pf |−1.

Hence, K1(C∗(π1(C \ Pf ))) is the abelian group given by the presentation

〈uz1 ,uz2 , . . . ,uzp : uz1 + uz2 + · · ·+ uzp = 0〉,

where uzi is the class of the generator γzi of the fundamental group of C \ Pf .It is easy to see that the matrix recursions on the fundamental group induces

the homomorphisms of the K0 groups Z −→ Z : m 7→ dm. Consequently, the groupK0(MG) is isomorphic, by Theorem 4.3 and continuity of K-theory, to the directlimit of the sequence

Zd·−→ Z

d·−→ · · · ,

i.e., to Z[1/d].It follows from the recurrent formula in Theorem 2.2 that the endomorphism of

K1 induced by the matrix recursion is the map

T (uzi) =∑

zj∈Pf∩f−1(zi)

uzj .

The group K1(MIMG(f)) is isomorphic hence to the direct limit of iterations of T .Let us denote by Cf the set of points belonging to the attracting cycles of f .

Then Cf ⊂ Pf , and if z ∈ Pf \ Cf , then Tm(uz) = 0 for all m big enough. Notethat for every m ≥ 1 each ux, for x ∈ Pf , appears exactly once in a sum Tm(uy) =∑

z∈f−m(y)∩Pfuz for one uy. Form big enough we have Tm(

∑z∈Cf

uz) =∑

z∈Pfuz.

One has to take m such that for every z ∈ Pf the point fm(z) belongs to a cycle.This implies that K1(MIMG(f)) is the abelian group given by the presentation

〈vz : z ∈ Cf ,∑

z∈Af

vz = 0〉,

i.e., is isomorphic to Zk−1.


Pimsner six-term exact sequence [Pim97] yields

Z[

1d

] 1−d−→ Z[

1d

]−→ K0(OG)x

yK1(OG) ←− K1(MIMG(f))

id−T←− K1(MIMG(f))

We get K0(OG) ∼= Z/(d− 1)Z⊕ ker(id− T ), and

K1(OG) ∼= K1(MIMG(f))/Ran(id− T ).

Since T is the cyclic permutation vf(z) 7→ vz of the generators vz ofK1(MIMG(f)),the kernel of id−T is generated by the elements of the form

∑z∈C vz , where C ⊂ Cf

is a cycle of the action of f on Pf . Hence ker(id−T ) is the abelian group generatedby c elements of sum equal to zero, i.e., ker(id− T ) ∼= Zc−1.

The quotient of K1(MIMG(f)) by the range of id− T is the abelian group givenby the presentation

〈vz : z ∈ Cf ,∑

z∈Af

vz = 0,vf(z) = vz〉,

that is, the group generated by c generators w1, . . . ,wc modulo the relation

k1w1 + k2w2 + · · ·+ kcwc = 0,

where ki are the lengths of the cycles of f on Cf . This implies that K1(OIMG(f)) ∼=Z/lZ⊕ Zc−1. �

Corollary 4.9. Let f be a post-critically finite hyperbolic polynomial of degree d.Let k be the sum of the lengths of its finite attracting cycles, and let c be theirnumber. Then

K0(MIMG(f)) = Z[1/d], K1(MIMG(f)) = Zk

and

K0(OIMG(f)) = Z/(d− 1)Z⊕ Zc, K1(OIMG(f)) = Z

c.

Corollary 4.9 implies, in particular, that the K-theory of OIMG(f) for hyperbolicquadratic polynomials f is (K0, [1]0,K1) = (Z, 0,Z) and does not depend on f . Wewill see later that these algebras (at least in the case of a periodic critical point)are classifiable by their K-theory, hence the algebra OIMG(f) does not depend on f .It would be interesting to describe isomorphism between these algebras explicitly.We will see later that this algebra is also isomorphic to the cross-product algebraof the action of a hyperbolic quadratic polynomial on its Julia set.

Note also that these algebras have the same K-groups, by results of M. Laca andJ. Spielberg [LS96] and C. Anantharaman-Delaroche [AD97], as the cross-productC∗-algebra associated with the natural action of the modular group PSL(n,Z) onthe boundary of the upper half-plane (or on the unit circle, for the disc model).However, the image of 1 in K0 is different. It is equal to zero in our algebras andto 1 ∈ Z for the boundary action of PSL(n,Z) (see Theorem 2.1 of [AD97]).


5. Groupoid approach

5.1. Groupoids. The standard reference for groupoids and their C∗-algebras isthe monograph [Ren80]. See also [Pat99, AR00], and [KS02] for the treatment ofthe non-Hausdorff case. We only fix here notations.

A groupoid is a small category of isomorphisms. We usually identify a groupoidG with the set of morphisms and identify an object of the category with the identityautomorphism (called a unit) of the object. Then the source of γ ∈ G is definedby s(γ) = γ−1γ and the range by r(γ) = γγ−1. We denote by G(0) the set of unitsof the groupoid.

A topological groupoid is a groupoid G with a topology on it such that thegroupoid operations (inversion, composition, source and range maps) are continuouson their domains and the source and range maps are open. We also always requirethat every point of G has a compact Hausdorff neighborhood and that the space ofunits G(0) and the spaces s−1(x) and r−1(x) are Hausdorff for every x ∈ G(0).

A topological groupoid is r-discrete (or etale) if the source and range maps arelocal homeomorphisms, i.e., if the space of units is open.

The main examples of r-discrete groupoids are groupoids of germs of inversesemigroups.

LetX be a locally compact Hausdorff space and let G be an inverse semigroup (ora pseudogroup) of local homeomorphisms of X (i.e., of homeomorphisms betweenopen subsets of X). Then a G-germ is a pair (g, x) ∈ G×X up to the equivalenceidentifying two germs (g1, x1) and (g2, x2) if x1 = x2 and g1 and g2 coincide on aneighborhood of x1.

The groupoid operations are given by

(g, x)−1 = (g−1, g(x)), (g1, x1)(g2, x2) = (g1g2, x2)

where the product is defined if and only if g2(x2) = x1.The topology on the groupoid of G-germs is given by the basis of open sets of

the form

UU,g = {(g, x) : x ∈ U},where g ∈ G and U ⊂ Dom g is an open set.

It is not hard to check that the groupoid of germs of an inverse semigroup oflocal homeomorphisms is an r-discrete groupoid.

Suppose that G is a Hausdorff r-discrete groupoid. Let Cc(G) be the space ofcontinuous complex functions with compact support on G. It is a normed involutivealgebra with respect to the operations

f∗(γ) = f(γ−1), (f · g)(γ) =∑

γ1γ2=γ

f(γ1)g(γ2)

and the norm

‖f‖I = max (‖f‖s, ‖f‖r) ,where

‖f‖s = supx∈G(0)

∑

s(γ)=x

|f(γ)|

and

‖f‖r = supx∈G(0)

∑

r(γ)=x

|f(γ)|.


The universal C∗-algebra C(G) of the groupoid G is the universal envelopingC∗-algebra of the ‖ · ‖I-completion of Cc(G).

Consider the representation λx of Cc(G) on `2(Gx) given by

λx(f)(ξ(γ)) =∑

γ1γ2=γ

f(γ1)ξ(γ2).

One can show that ‖λx(f)‖ ≤ ‖f‖I, hence the norm

‖f‖reg = supx∈G(0)

‖λx(f)‖

is a C∗-norm on Cc(G). The completion Cr(G) of Cc(G) by this norm is the reducedC∗-algebra of the groupoid.

If G is not Hausdorff, then instead of Cc(G) one has to take the linear span ofthe functions G −→ C vanishing outside a compact Hausdorff set K and continuouson an open neighborhood of K. Then the universal and the reduced algebra of G isdefined in the same way as for Hausdorff groupoids. This approach was suggestedby A. Connes in [Con82].

If the groupoid is not r-discrete, then the sums in all the definitions are replacedby the integrals over an appropriately defined Haar system (see [Ren80, Pat99]).

5.2. Cuntz-Pimsner groupoid of a self-similar group. Let (G,X) be a self-similar group. Recall that the inverse semigroup 〈G,X〉 = {SvgS

∗u : v, u ∈ X

∗, g ∈G} acts faithfully on the space X

ω by the local homeomorphisms

SvgS∗u(uw) = vg(w)

with domain uXω and range vXω .Let OG be the groupoid of germs of this inverse semigroup of local homeomor-

phisms. We call OG the Cuntz-Pimsner groupoid of the self-similar group (G,X).

Theorem 5.1. The C∗-algebra OG is isomorphic to the universal convolution C∗-algebra of the Cuntz-Pimsner groupoid OG.

Proof. For every SvgS∗u ∈ 〈G,X〉 the set of germs of SvgS

∗u is compact, open and

Hausdorff and such sets cover the groupoid OG. This implies that every element ofCc(OG) is a finite sum

∑fi, where each fi is a continuous complex valued function

on the set of germs of an element of 〈G,X〉 (see Lemma 1.3 of [KS02]).Let SvgS

∗u be an arbitrary element of 〈G,X〉 and let C(SvgS

∗u) be the space of

continuous functions on the set of germs of SvgS∗u. We have shown that

Cc(OG) =∑

SvgS∗u∈〈G,X〉

C(SvgS∗u).

The restriction of the norm ‖ ·‖I onto C(SvgS∗u) is the usual sup-norm on the space

of continuous functions on the compact space U of germs of SvgS∗u.

The map

(SvgS∗u, uw) 7→ w

is a homeomorphism of U with Xω . For every n the preimage of the partition⊔

t∈Xn tXω = Xω by this homeomorphism is the partition of U into the sets of germs

of the elements Sug(t)g|tS∗ut of 〈G,X〉. It is well known that the linear span of the

characteristic functions of cylindrical sets of Xω is dense in C(Xω) (with respect to

the sup-norm). Consequently, the linear span of the characteristic functions of thesets of germs of Sug(t)g|tS∗

ut is dense in C(SvgS∗u) with respect to the norm ‖ · ‖I .


Let us construct the isomorphism F : OG −→ C(OG). Set F (SvgS∗u) to be equal

to the characteristic function of the set of germs of SvgS∗u. Extend F by linearity

onto the linear span C〈G,X〉 of 〈G,X〉 ⊂ OG. We have proved that F (C〈G,X〉) isdense in Cc(OG). A straightforward argument shows that F respects the algebraicoperations (involution and multiplication).

It remains to show that for every a ∈ C〈G,X〉 the norm ‖F (a)‖I is an upperbound of ‖a‖ for any C∗-norm ‖ · ‖ on the ∗-algebra C〈G,X〉 (since OG is theuniversal enveloping algebra of C〈G,X〉).

Let ρ be a representation of C〈G,X〉 on a Hilbert space H . Let a =∑αiSvigiS

∗ui

be an arbitrary element of C〈G,X〉. Let n be an integer greater that length of everyvi and ui. Then every vector ~x ∈ H is decomposed into the sum

∑v∈Xn ρ(SvS

∗v )~x,

where the components ~xv = ρ(SvS∗v )~x of the decomposition are pairwise orthogonal.

Denote, for u, v ∈ Xn

αu,v =∑

ui ≺ u and vi ≺ v

|αi|.

Here and below we write ui ≺ u if u begins with ui. If ui ≺ u ∈ Xn, then for

every w ∈ Xω the word uw belongs to the domain of SvigiS

∗ui

, otherwise no wordof the form uw ∈ X

ω belongs to the domain of SvigiS∗ui

. Consequently,

∑

v∈Xn

αu,v =∑

ui≺u

|αi| =∑

γ∈OG, s(γ)=uw

|F (a)(γ)| ≤ ‖F (a)‖s,

and similarly∑

u∈Xn αu,v ≤ ‖F (a)‖r.Let ~x, ~y ∈ H be arbitrary and let ~xu = ρ(SuS

∗u)~x and ~yv = ρ(SvS

∗v) be their

components in the decomposition H =⊕

v∈Xn ρ(Sv)(H). Note that if u does notbegin with ui, then ρ(S∗

ui)~xu = ρ(S∗

uiSuS

∗u)~x = 0. Then, using Cauchy-Schwarz

inequality and the fact that ρ(gi) are unitary and S∗vi

and S∗ui

are partial isometries,


we get

| 〈ρ(a)~x| ~y〉 | =

∣∣∣∣∣∣

∑

i,u,v∈Xn

⟨αiρ(SvigiS

∗ui

)~xu

∣∣ ~yu

⟩∣∣∣∣∣∣=

∣∣∣∣∣∣

∑

i,u,v∈Xn

⟨αiρ(giS

∗ui

)~xu

∣∣ ρ(S∗vi

)~yu

⟩∣∣∣∣∣∣≤

∑

u,v∈Xn

∑

ui≺u,vi≺v

|αi|∣∣⟨ρ(giS

∗ui

)~xu

∣∣ ρ(S∗vi

)~yv

⟩∣∣ ≤

∑

u,v∈Xn

∑

ui≺u,vi≺v

|αi| · ‖~xu‖ · ‖~yv‖ =∑

u,v∈Xn

αu,v‖~xu‖ · ‖~yv‖ =

∑

u,v∈Xn

α1/2u,v ‖~xu‖ · α1/2

u,v ‖~yv‖ ≤

∑

u,v∈Xn

αu,v‖~xu‖2

1/2 ∑

u,v∈Xn

αu,v‖~yv‖2

1/2

≤

(‖F (a)‖s

∑

u∈Xn

‖~xu‖2)1/2(

‖F (a)‖r∑

v∈Xn

‖~yv‖2)1/2

=

‖F (a)‖1/2s ‖F (a)‖1/2

r ‖~x‖ · ‖~y‖ ≤ ‖F (a)‖I · ‖~x‖ · ‖~y‖,which implies ‖ρ(a)‖ ≤ ‖F (a)‖I . �

5.3. The algebra MG as a convolution algebra. Let us denote by MG thegroupoid of germs on X

ω of the gauge-invariant sub-semigroup of 〈G,X〉{SvgS

∗u : |v| = |u|, g ∈ G}.

Proposition 5.2. If the action (G,X) is self-replicating (see Definition 2.3), thenthe groupoid MG coincides with the groupoid of germs of the action of G on X

ω.

Proof. If (G,X) is self-replicating, then for every u, v ∈ Xn and g ∈ G there exists

h ∈ G such that g · u = v · h. But then the germs of g on uXω coincide with thegerms of SvhS

∗u. �

The next theorem is proved in the same way as Theorem 5.1, using Theorem 3.7.

Theorem 5.3. The algebraMG is the universal convolution algebra of the groupoidMG.

5.4. Reduced algebras. Let us denote by OGred andMGred the reduced algebrasof the groupoids OG and MG, respectively. These algebras are quotients of thealgebras OG andMG.

Lemma 5.4. The following conditions are equivalent.

(1) The groupoid OG is Hausdorff.(2) The groupoid MG is Hausdorff.(3) For any g ∈ G the interior of the set of fixed points of the action of g on

Xω is closed.


Proof. If two germs γ1, γ2 do not have disjoint neighborhoods, then s(γ1) = s(γ2),r(γ1) = r(γ2) and the germ γ−1

1 γ2 does not have disjoint neighborhoods with theunit at s(γ2). Consequently, if we want to check that a groupoid is Hausdorff, it issufficient to check that every unit has disjoint neighborhoods with every non-unitelement.

If |v| 6= |u|, then no germs of the local homeomorphism

vw 7→ ug(w)

is trivial (for instance, because then the local homeomorphism multiplies the uni-form Bernoulli measure by a constant different from 1). Consequently, OG is Haus-dorff if and only if MG is Hausdorff.

Suppose that the germ of the transformation

vw 7→ ug(w)

at vw0 = vx1x2 . . . is not a unit, but can not be separated from the trivial germ atvw0, where g ∈ G and v, u ∈ X

∗ are words of equal length. This is equivalent to thecondition that v = u and that the germ of g at w0 has no disjoint neighborhoodswith the germ of identity at w0, but itself is non-trivial. This in turn is equivalentto the condition that w0 is a fixed point of g and that for every neighborhood U ofw0 there exists a non-empty open subset of U on which g acts trivially, but g doesnot act trivially on U , i.e., that w0 belongs to the boundary of the interior of theset of fixed points of f . �

Proposition 5.5. If one of the equivalent conditions of Lemma 5.4 holds, then thealgebras OGred andMGred are simple, and thus isomorphic to OGmin andMGmin,respectively. The algebra OGred is then purely infinite.

Proof. The orbits of the groupoids OG and MG are dense in Xω , hence by Propo-

sition 4.6 of [Ren80], their reduced algebras are simple. It is proved in [Nek04] thatthe algebra OGmin is purely infinite. Another way to prove pure infiniteness is touse a criterion from [Ana97]. �

5.5. Amenability. Throughout this subsection (G,X) is a contracting self-replicatinggroup and the groupoid MG of germs of the action of G on X

ω is Hausdorff.For the definition of an amenable groupoid see [Ren80, AR00].

Theorem 5.6. The groupoids OG and MG are amenable.

Proof. The groupoid MG is the groupoid of germs of the action of G on Xω. It is

proved in [Nek05] that this groupoid is of polynomial growth, hence it is amenableby [AR00], Proposition 3.2.32.

Amenability of OG follows then from the amenability of MG in the same way asamenability of the Cuntz groupoid follows from the amenability of the uniformlyhyperfinite groupoid (see [Ren80]). One constructs the direct limit of the groupoidMG whose convolution algebra is K ⊗MG (using Theorem 3.7, see also the proofof Theorem 6.11 below). The direct limit groupoid will be amenable by Proposi-tion 5.3.37 of [AR00].

Then the groupoid OG is obtained as a restriction of the cross-product of thedirect limit with Z, hence it is also amenable by Corollary 5.3.23 of [AR00]. �

Corollary 5.7. The algebras MG and OG are isomorphic to the reduced algebrasof the groupoids MG and OG, respectively, and are nuclear C∗-algebras.


Proposition 5.8. The algebrasMG and OG are simple. The algebra OG is purelyinfinite and satisfies the Universal Coefficient Theorem.

Proof. Follows from Proposition 5.5 of our paper and Lemma 3.5 and Proposi-tion 10.7 of [Tu99]. �

Another proof of this proposition (but just for the case of regular self-similargroups) follows from the results of [PS99] and Corollary 6.12 below.

6. Regular self-similar groups and Ruelle algebras

6.1. Regular self-similar groups. The following regularity condition will elimi-nate the case when the groupoid of germsMG is non-Hausdorff and will allow us toavoid the necessity of introduction of an orbispace structure on the limit solenoid.

Definition 6.1. We say that a self-similar group (G,X) is regular if for every g ∈ Gand every w ∈ X

ω either g(w) 6= w or w is fixed under the action of g together willall points of some its neighborhood (compare with Definition 3.3).

It is sufficient to check the regularity condition for the elements g of the nucleusof the group, if the group is contracting. Since the regularity condition impliesthat the set of fixed points of an element g ∈ G is open, every regular group has aHausdorff groupoid of germs (see Lemma 5.4).

Proposition 6.1. Let (G,X) be a contracting self-similar groups. Then the follow-ing conditions are equivalent:

(1) The group (G,X) is regular.(2) The limit orbispace JG has no singular points, i.e., the action of G on the

limit G-space is free.(3) The map σ : JG −→ JG is a covering.

Proof. We use here known results on limit orbispaces of contracting groups, whichcan be found in [Nek05]. If JG has no singular points, then σ : JG −→ JG is acovering, since it is a partial self-covering of the orbispace JG. On the other hand,if JG has singular points, then the isotropy groups of JG are faithfully representedin the iterated monodromy group (G,X) of σ, hence the map σ is not everywhere|X|-to-one and can not be a covering. Consequently, (2) and (3) are equivalent.

Suppose that there exists a singular point of JG. It is easy to see that then thereexists a singular point represented by a periodic sequence v−ω. Moreover, it followsfrom the description of the asymptotic equivalence on the limit G-space that wemay assume that for some non-trivial element g of the nucleus we have g · v = v · g.Then the sequences vω ∈ X

ω is a fixed point of g, but since g|vn = g 6= 1, the elementg does not act trivially on any neighborhood of vω . Hence the group (G,X) is notregular.

In the other direction, if (G,X) is not regular, then there exists an element ofthe nucleus g and an infinite word w ∈ X

ω such that g(w) = w but g|v 6= 1 for anyfinite beginning v of w. Since the nucleus N is finite, this implies that there existsa non-trivial element h ∈ N and a finite word u ∈ X

∗ such that h · u = u · h. Butthen h is a non-trivial element of the isotropy group of the point u−ω ·1 of the limitG-space. �


Though the regularity condition is rather restrictive, it includes sufficiently manyexamples, for instance iterated monodromy groups of hyperbolic rational functions,as the following proposition implies.

Proposition 6.2. Let f : C1 −→ C be an expanding partial self-covering of a com-plete Riemannian manifold and suppose that IMG (f) is finitely generated. ThenIMG (f) is contracting and regular.

Proof. It is proved in [Nek05] that in the conditions of the proposition the iteratedmonodromy group is contracting and its limit space is homeomorphic as an orbis-pace to the Julia set J of f . Since J is just a topological space, the limit orbispacehas no non-trivial isotropy groups. This implies, by Proposition 6.1, that the groupIMG (f) is regular. �

Lemma 6.3. If the action (G,X) is regular and contracting, then there exists k suchthat for every v ∈ X

k and any two elements g, h of the nucleus either g(v) 6= h(v)or g|v = h|v.Proof. By definition of a regular action, for every w = x1x2 . . . ∈ X

ω there exists ksuch that for every g, h ∈ N either h−1g(x1 . . . xk) 6= x1 . . . xk or (h−1g)|x1...xk

= 1.It follows from compactness of X

ω that we may choose k common for all w ∈ Xω.

Then for every x1 . . . xk ∈ Xk either g(x1 . . . xk) 6= h(x1 . . . xk) or g(x1 . . . xk) =

h(x1 . . . xk) and 1 = (h−1g)|x1...xk= h|−1

x1...xkg|x1...xk

. �

6.2. Equivalence of regular groups.

Theorem 6.4. Let (Gi,X) for i = 1, 2 be contracting regular self-similar groups.Then the following conditions are equivalent.

(1) The self-similar groups (G1,X) and (G2,X) are equivalent.(2) The limit dynamical systems (JG1 , σ) and (JG2 , σ) are topologically conju-

gate.(3) There exists an isomorphism between the Cuntz-Pimsner algebras OG1 andOG2 respecting the gauge actions.

Proof. Equivalence of (1) and (2) follows from the theory of iterated monodromygroups and limit spaces, see [Nek05]. The condition (1) obviously implies (3). Theimplication (3)⇒(2) follows from Theorem 4.6 of our paper. �

6.3. The algebra Of . Let f : C1 −→ C be an expanding d-fold self-covering,as in Definition 2.7. If Jf is the Julia set of f , then f : Jf −→ Jf is a d-fold self-covering, which is topologically conjugate to the limit dynamical systemσ : JIMG(f) −→ JIMG(f) of the iterated monodromy group. Hence, the dynamicalsystem f : Jf −→ Jf is uniquely determined by the iterated monodromy groupof f (as a self-similar groups), and hence it is uniquely determined by the gauge

action on OIMG(f). We will denote OIMG(f) = Of .There exists another natural Cuntz-Pimsner algebra associated to self-coverings

of a topological space. This algebra was studied by V. Deaconu [Dea95], V. Deaconuand P. S. Muhly [DM01], T. Kajiwara and Y. Watatani [KW05] and also in a moregeneral form by J. Renault [Ren00].

It can be seen as the cross-product of C(Jf ) by the action of f and is definedeither as the convolution algebra of the groupoid generated by the germs of f , or asa Cuntz-Pimsner algebra of a bimodule over the algebra of functions on JG, whichis naturally defined by the self-covering (see [KW05]).


These two approaches are equivalent (see [DM01]), so we will describe themwithout proving isomorphism of the defined algebras.

Denote by Φ the algebra C(Jf ), which can be seen as a right Hilbert moduleover itself, where the right action is defined by the rule

(ζ · g)(x) = ζ(x)g(f(x))

for ζ ∈ Φ, g ∈ C(Jf ), x ∈ Jf and the C(Jf )-valued inner product is given by

〈ζ1| ζ2〉 (x) =∑

y∈f−1(x)

ζ1(y) · ζ2(y),

for ζ1, ζ2 ∈ Φ and x ∈ Jf . It is easy to prove that Φ is a projective HilbertC(Jf )-module (already complete).

The rank one operators θζ1,ζ2(ξ) = ζ1 · 〈ζ2| ξ〉 act then by

θζ1,ζ2(ξ)(x) = ζ1(x) 〈ζ2| ξ〉 (f(x)) = ζ1(x)∑

y∈f−1(f(x))

ζ2(y)ξ(y).

The left action of φ : C(Jf ) −→ End Φ is given by usual multiplication inC(Jf ) = Φ. Let us show that φ(C(Jf )) belongs to the span of θζ1,ζ2 .

Let {Ui}i∈I be a finite covering of Jf by open subsets such that f−1(Ui) is adisjoint union of opens sets Ui,j , for j = 1, . . . , d, such that f : Ui,j −→ Ui is ahomeomorphism.

Find a partition of unity ϕi such that supp(ϕi) ⊂ Ui. Denote by ϕi,j ∈ Φ thecomposition of f : Ui,j −→ Ui with ϕi extended as 0 outside of Ui,j . Then for everyx ∈ supp(ϕi,j) and y ∈ f−1(f(x)) \ {x} we have ϕi,j(y) = 0. Consequently,

θ(ϕi,j)1/2,(ϕi,j)1/2(ψ)(x) = (ϕi,j(x))1/2

∑

y∈f−1(f(x))

(ϕi,j(y))1/2ψ(y) = ϕi,j(x)ψ(x)

for every x ∈ Jf . It follows that for every g ∈ C(Jf ) we have∑

i∈I,j=1,...,d

θg(ϕi,j)1/2,(ϕi,j)1/2(ψ)(x) =∑

i∈I,j=1,...,d

g(x)ϕi,j(x)ψ(x) = g(x)ψ(x),

hence

φ(g) =∑

i∈I,j=1,...,d

θg(ϕi,j)1/2,(ϕi,j)1/2 .

The Cuntz-Pimsner algebra Of is defined then, for instance, using Theorem 3.12of [Pim97] as the universal C∗-algebra generated by operators Sζ for ζ ∈ Φ andπ(g) for g ∈ C(Jf ), modulo the relations

(1) αSζ1 + βSζ2 = Sαζ1+αζ2 for all ζ1, ζ2 ∈ Φ and α, β ∈ C;(2) Sζ · π(g) = Sζ·π(g) for all ζ ∈ F and g ∈ C(Jf ), where ζ · π(g) is multipli-

cation in the right module Φ;(3) S∗

ζ1Sζ2 = π(〈ζ1| ζ2〉) for all ζ1, ζ2 ∈ F ;

(4) π(g) =∑SζiS

∗ξi

, if φ(g) =∑θζi,ξi .

Another way to construct Of is to define it as a universal convolution algebra ofthe groupoid

Of = {(x, n, y) ∈ Jf × Z× Jf : ∃k, l ≥ 0, n = k − l, fk(x) = f l(y)}with multiplication given by

(x,m, y)(y, n, z) = (x,m+ n, z).


The topology on Of is given by the basis of open sets of the form

UU,V,k,l = {(x, k − l, y) : x ∈ U, y ∈ V },

where U, V and k, l are such that f−l◦fk : U −→ V is a univalent homeomorphism.The isomorphism Ψ between the algebra Of and the universal convolution alge-

bra of the groupoid Of is given by

Ψ(Sζ)(γ) =

{ζ(x) if γ = (x, 1, y) for y = f(x),0 otherwise;

and

Ψ(π(g))(γ) =

{g(x) if γ = (x, 0, x),0 otherwise.

It is an easy exercise to check the defining relations (1)–(4) in this case. Acomplete proof of the fact that Ψ is an isomorphism can be found in [DM01].

The elements of the form (x, 0, y) form a closed sub-groupoid of Of , which wewill denote Mf . The convolution algebra of Mf will be denotedMf . The algebraMf is also invariant under a natural gauge action on Of .

Similarly to the case of self-similar groups, the groupoid Mf is an increasingunion of the groupoids

Mn = {(x, 0, y) : fn(x) = fn(y)}.The groupoid Mn can be seen as the groupoid of “deck transformations” for thecovering fn : Jf −→ Jf .

Note that the groupoid Mn is compact, hence the convolution algebra C∗(Mn)coincides with the convolution algebra of continuous functions C(Mn). It is alsonot hard to see that the groupoid Mn is principal and hence it can be identifiedwith the corresponding subspace of Jf × Jf .

We have the following properties of the algebra Of . For a proof see [Ana97]Proposition 4.2 and [Dea95].

Theorem 6.5. The algebra Of is nuclear, purely infinite, simple and satisfies theUCT.

6.4. Hyperbolic rational functions. Let f ∈ C(z) be a post-critically finitehyperbolic rational function, i.e., a rational function of degree more than one suchthat the orbit of every critical point of f eventually belongs to a superattractingcycle (i.e., a cycle containing a critical point).

Then f is uniformly expanding on a neighborhood of its Julia set, the groupIMG (f) is contracting, finitely generated, self-replicating and regular, the limitdynamical system σ : JIMG(f) −→ JIMG(f) is conjugate with the action of f on itsJulia set Jf (see Theorem 2.3).

Denote Of = OIMG(f) and Mf =MIMG(f).

Theorem 6.6. If f is a post-critically finite hyperbolic rational function the

Ki(Mf ) ∼= Ki(Mf )

and

Ki(Of ) ∼= Ki(Of ).


Proof. The idea of computation is very similar to the one used in Theorem 4.8.In that case we did not know the structure of the groups IMG (f) well enough, sowe have approximated MIMG(f) by the matrix algebras over the free group (thefundamental group of the punctured sphere). Here also, the Julia set of f is toocomplicated, so we approximate it by the sphere with holes.

Namely, let C0 be the Riemann sphere C minus open disjoint discs around thepost-critical points of f such that Jf ⊂ C0 and f−1(C0) ⊂ C0. One of the waysto construct such a set is the following. It is known (see, for instance [Mil99])

that the Fatou components of f (i.e., the connected components of C \ Jf ) areconformally equivalent to the open unit disc and moreover, that there exist bi-conformal isomorphisms ΨU : U −→ D of every component with the open unit discsuch that if f(U1) = U2, then

ΨU2(f(z)) = (ΨU1(z))m

for every z ∈ U1, where m = 1 if there is no critical point in U1 and m is equalto the local degree of f at teh (necessarily unique) critical point in U1. If z0 is apost-critical point of f , then ΨU (z0) = 0, where U is the Fatou component to whichz0 belongs.

Then we can take C0, for instance, to be the sphere minus the sets

Ψ−1U ({z : |z| < 1/2})

for all Fatou components U containing post-critical points.It is known that K0(C(C0)) is isomorphic to Z and is generated by the class of

the unit and that K1(C(C0)) is isomorphic to the cohomotopy group π1(C0). Inother words, every element of K1(C0) is the class of a unitary u ∈ C(C0), i.e., of afunction from C0 into the unit circle T.

The class of a projection p ∈ Mn(C(C0)) in K0 coincides with the class of itstrace (which will be a constant function n ∈ N on C0) and the class of a unitary u ∈Mn(C(C0)) in K1 coincides with the class of its determinant (which is a continuousfunction on C0 with values on T).

The class of a function u : C0 −→ T is uniquely determined by the windingnumbers wz of composition of u with the curves going in the positive directionalong the boundaries of the holes around post-critical points z. A collection ofnumbers wz is realized by a unitary u if and only if their sum is equal to zero.Hence, K1(C(C0)) is isomorphic to the sub-group of ZPf consisting of the vectorswith zero sum of coordinates. We will write an element of K1 by

∑z∈Pf

wzez,

where wz is the winding number corresponding to the boundary of the disc aroundz ∈ Pf . The only condition on the coefficients wz is

∑z∈Pf

wz = 0.

Denote Cn = f−n(C0) (where f−n denotes inverse image). Then Cn ⊂ Cn+1,

(7)⋂

n≥0

Cn = Jf ,

and f : Cn+1 −→ Cn is a d-fold covering map.In particular, the nth iteration fn of f is a dn-fold covering

fn : Cn −→ C0.Denote byMn the convolution algebra of the groupoid of the “deck transforma-

tions” of fn, i.e., the groupoid of germs (f−n, z1) · (fn, z2) on Cn, where z1, z2 ∈ Cnare such that fn(z1) = fn(z2). It follows that theMn is Morita equivalent to C(C0)


and that the isomorphism of K-groups is given by trace (for K0) and determinant(for K1).

Note also that for m > n restriction of every element ofMn onto Cm ⊂ Cn is anelement of Mm (since fn(z1) = fn(z2) implies fm(z1) = fm(z2)). We get hencenatural homomorphisms φn,m :Mn −→ Mn+1 such that φn2,n3 ◦ φn1,n2 = φn1,n3

for all n1 < n2 < n3. In particular, φ0,n : C(C0) −→Mn is restriction of continuousfunctions C0 −→ C onto Cn. This implies that completion of C(C0) with respect tothe norm limn→∞ ‖φ0,n(a)‖ is isomorphic to C(Jf ).

It follows from (7) that the direct limit of the homomorphisms φn,m is the algebraMf defined for the self-covering f : Jf −→ Jf . Consequently, the K-groups ofMf

are the direct limits of the K-groups ofMn (which are isomorphic to the K-groupsof C(C0)) under the maps induced by the homomorphisms φn,m.

Computing traces we see that φn,n+1 induces multiplication by d onK0(C(C0)) =K0(C

∗(Mn)) = K0(C∗(Mn+1)). Computation of determinants (a particular, but

a typical case of which is the “r-times-around embedding”, see [Bla98], Exer-cise 10.11.4) shows that φn,n+1 induces onK1(C(C0)) the endomorphism F mapping

the element∑

z∈Pfwzez to

∑z∈Pf

wzef(z) =∑

z∈Pf

(∑t∈Pf∩f−1(z) wt

)ez.

The K-groups ofMf are the direct limits of these homomorphisms. We concludethat K0(Mf ) ∼= Z[1/d].

There exists m such that fm(z) belongs to a cycle for every z ∈ Pf . Let Cf

be the union of cycles of f on Pf . Then Fm(K1(C(C0))) is equal to the group ofelements of the form

∑z∈Cf

wfez for which∑

z∈Cfwf = 0. The homomorphism F

induces an automorphism of this group. Consequently, K1(Mf ) ∼= Z|Cf |−1 = Zk−1.The algebra Of is also a Cuntz-Pimsner algebra (see [KW05]) and the Pimsner

six-term exact sequence in this case is

Z[

1d

] 1−d−→ Z[

1d

]−→ K0(Of )x

yK1(Of ) ←− K1(Mf )

id−F←− K1(Mf )

Note that in our case the groups K1(Mf ) and the endomorphism F are in somesense transposed versions of the groups K1(MIMG(f)) and T from Theorem 4.8.

Here K1(Mf ) is the subgroup of Z|Cf | consisting of elements∑

z∈Cfwzez such

that∑

z∈Cfwz = 0, while K1(MIMG(f)) is the abelian group generated by vz for

z ∈ Cf modulo the relation∑

z∈Cfvz = 0. The automorphism T is also in some

sense transposed to F :

T (vf(z)) = vz, F

∑

z∈Cf

wzez

=

∑

z∈Cf

wzef(z).

In our case we also have K0(Of ) ∼= Z/(d− 1)Z⊕ ker(id− F ) and

K1(Of ) ∼= K1(Mf )/(id− F )(K1(Mf )).

Kernel of id− F consists of the sums∑

z∈Cfwzez for which wz = wf(z), i.e., of

the elements whose winding numbers corresponding to the holes of one cycle areequal. The sum of the winding numbers has to be equal to zero, so that ker(id−F )is isomorphic to Z

c−1.


Choose an element z0 ∈ Cf . Then the group K1(Mf ) is freely generated by theelements hz = ez0 − ez for z ∈ Cf \ {z0}. The image of id− F is generated by

ez0 − ez − ef(z0) + ef(z) = hz + hf(z0) − hf(z),

where hz0 is 0, by definition. Hence, K1(Of ) is the abelian group generated by aset hz for z ∈ Cf modulo the relations

hf(z) = hz + hf(z0).

Note that hz0 = 0 is a corollary of the defining relations. Let us choose one elementyi, i = 0, . . . , c− 1 in each cycle of Cf , so that y0 = z0. Then the group K1(Of ) isgenerated by hyi for i = 1, . . . , c− 1 and hf(z0). If yi for i = 0, . . . , c − 1, belongsto a cycle of length li, then hyi = lihf(z0) + hyi , i.e., lihf(z0) = 0. It is easy tosee that these equalities are defining relations of K1(Of ) for the generating set{hyi}i=1,...,c−1 ∪ {hf(z0)}. Consequently, K1(Of ) is isomorphic to Z/lZ ⊕ Zc−1,where l is the greatest �

We see that if f is a hyperbolic quadratic rational function with two attracting

cycles of co-prime lengths, then K0(Of ) = K0(Of ) = K1(Of ) = K1(Of ) = Z,

hence the algebras Of and Of are isomorphic and do not depend on f . However,

Theorem 4.6 implies that together with the gauge action the algebras Of uniquelydetermines the action of f on its Julia set.

An interesting question would be to clarify the relation between the algebras

Mf andMf .

6.5. The limit solenoid as a hyperbolic dynamical system. In the followingwe assume that (G,X) is a contracting, self-replicating and regular group.

Proposition 6.7. The asymptotic equivalence relation is a shift of finite type. Inparticular, the dynamical system (SG, σ) is finitely presented.

Recall that a subset F of AZ for a finite alphabet A is called a shift of finitetype if there exits a finite set of words P ⊂ A∗ such that a sequence belongs to F ifand only if it has no subwords belonging to P . In our proposition we consider theequivalence relation to be a subset of (X× X)Z = X

Z × XZ.

Proof. Let k be as in Lemma 6.3 and let R be the set of pairs of sequences(. . . x−1x0x1 . . . , . . . y−1y0y1 . . .) such that for every n ∈ Z there exists gn ∈ N suchthat gn(xnxn+1 . . . xn+k) = ynyn+1 . . . yn+k. It is obviously a shift of finite typeand the asymptotic equivalence relation is a subset of R. In the other direction, theelement gn ∈ N is defined uniquely by the pair (xnxn+1 . . . xn+k, ynyn+1 . . . yn+k).We also have gn+1 = gn|xn , but this implies that . . . x−1x0x1 . . . and . . . y−1y0y1 . . .are asymptotically equivalent. �

We say that two points ξ, ζ ∈ SG are stably (unstably) equivalent if for everyneighborhood of the diagonal U ⊂ SG ×SG there exists nU ∈ Z such that

(σn(ξ), σn(ζ)) ∈ Ufor all n ≥ nU (n ≤ nU , respectively).

Proposition 6.8. Let the points ξ, ζ ∈ SG be represented by sequences (xk)k∈Z

and (yk)k∈Z, respectively.


The points ξ, ζ are stably equivalent if and only if there exists n ∈ Z such thatthe sequences . . . xn−1xn and . . . yn−1yn ∈ X

−ω are asymptotically equivalent, i.e.,represent the same point of JG.

The points ξ, ζ are unstably equivalent if and only if there exists n ∈ Z such that

g(xnxn+1 . . .) = ynyn+1 . . .

for some element g of the nucleus.

Proof. It is easy to see that stable and unstable equivalence follows from the con-ditions in the proposition and Proposition 2.5.

Let us show that the converse implications hold. Let Um ⊂ XZ × X

Z and Um ⊂SG ×SG be the neighborhoods of the diagonal defined in Proposition 2.5.

Let ξ and ζ be the points of SG represented by the sequences (xi)i∈Z and (yi)i∈Z.Suppose that ξ and ζ are stably equivalent. Then for any m there exists nm suchthat the pair (σn ((xi)i∈Z) , σn ((yi)i∈Z)) belongs to Um for all n ≥ nm. Conse-quently, for every n ≥ nm there exists gn−m ∈ N such that

g−n−m(x−n−mx−n−m+1 . . . x−n+m) = y−n−my−n−m+1 . . . y−n+m.

But if 2m > k, where k is as in Lemma 6.3, then this will imply that g−n|x−n =g−n+1 for all n > nm−m and hence that . . . x−n−1x−n is asymptotically equivalentto . . . y−n−1y−n.

The proof of the statement about the unstable equivalence is analogous. �

Let Um, as in the proof of the above proposition, be the set of all pairs (ξ, ζ) ∈ S2G

which are represented by sequences ((xn), (yn)) such that g(x−mx−m+1 . . . xm) =y−my−m+1 . . . ym for some g ∈ N . Let k be as in Lemma 6.3. We may assume thatk is big enough so that N 3|v ⊂ N for all words v ∈ X

∗ of length at least k.Suppose that (ξ, ζ) ∈ Uk. Represent them by a pair of sequences

ξ = . . . x−1x0 . x1 . . . ,

ζ = . . . y−1y0 . y1 . . . .

If g ∈ N is such that g(x−k . . . xk) = y−k . . . yk (which is uniquely defined by(x−k . . . x−1, y−k . . . y−1), then let r = g|x−k...x0 . Define the [ξ, ζ] to be the pointrepresented by the sequence

. . . x−1x0 . r−1(y1y2 . . .).

Lemma 6.9. The map [·, ·] : Uk −→ SG is well defined, continuous and satisfiesthe conditions

[ξ, ξ] = ξ, [ξ, [η, ζ]] = [ξ, ζ], [[ξ, η], ζ] = [ξ, ζ], σ([ξ, η]) = [σ(ξ), σ(η)]

for all ξ, η, ζ ∈ SG for which the respective expressions are defined.

Proof. Let us prove that the point [ξ, ζ] is well defined. Suppose that ξ is repre-sented by . . . x−1x0x1 . . . and by . . . x′−1x

′0x

′1 . . . and that the point ζ is represented

by . . . y−1y0y1 . . . and by . . . y′−1y′0y

′1 . . .. There exist sequences gn and hn in the

nucleus such that gn · xn = x′n · gn+1 and hn · yn = y′n · hn+1.It follows then that h−kgg

−1−k(x′−k . . . x

′k) = y′−k . . . y

′k, hence [ξ, ζ] defined with

respect to the new pair of sequences is represented by

. . . x′−1x′0(g1r

−1h−11 )(y′1y

′2 . . .) = . . . x′−1x

′0g1r

−1(y1y2 . . .),


which is asymptotically equivalent to . . . x−1x0r−1(y1y2 . . .). Consequently, the

map [·, ·] is well defined.The equality [ξ, ξ] = ξ is obvious. Let ξ, η, ζ be represented by the sequences (xn),

(yn) and (zn), respectively. If [ξ, [η, ζ]] is defined, then g(y−k . . . yk) = z−k . . . zk forsome g ∈ N and [η, ζ] = . . . y−1y0g|−1

y−k...y0(z1z2 . . .). There also exists h ∈ N such

that h(x−k . . . xk) = y−k . . . y0g|−1y−k...y0

(z1 . . . zk) and then

[ξ, [η, ζ]] = . . . x−1x0h|−1x−k...x0

g|−1y−k...y0

(z1z2 . . .).

We have

gh(x−k . . . xk) = g(y−k . . . y0g|−1y−k...y0

(z1 . . . zk)) = z−k . . . zk

and, by the choice of k

gh|x−k...x0 = g|y−k...y0h|x−k...x0 ∈ N .Consequently [ξ, ζ] is represented by the sequence

. . . x−1x0h|−1x−k...x0

g|−1y−k...y0

(z1z2 . . .) = [ξ, [η, ζ]].

If [ξ, η] is defined then for some g ∈ N we have g(x−k . . . xk) = y−k . . . yk and

[ξ, η] = . . . x−1x0g|−1x−k...x0

(y1y2 . . .).

If [[ξ, η], ζ] is defined, then there exists h ∈ N such that

h(x−k . . . x0g|−1x−k...x0

(y1 . . . yk)) = z−k . . . zk

and then[[ξ, η], ζ] = . . . x−1x0h

−1x−k...x0

(z1z2 . . .).

We have g|x−k...x0(x1 . . . xk) = y1 . . . yk, hence g|−1x−k...x0

(y1 . . . yk) = x1 . . . xk and

therefore h(x−k . . . xk) = z−k . . . zk, which implies that [[ξ, η], ζ] = [ξ, ζ].Let us prove the equality σ([ξ, η]) = [σ(ξ), σ(η)]. If both [ξ, η] and [σ(ξ), σ(η)]

are defined, where ξ and η are represented by (xn) and (yn) respectively, thenthere exist g, h ∈ N such that g(x−k . . . xk) = y−k . . . yk and h(x−k−1 . . . xk−1) =y−k−1 . . . yk−1. It follows from the uniqueness of h and g that h|−k−1 = g, whicheasily implies the necessary equality. �

It follows from Proposition 6.8 that [ξ, ζ] is stably equivalent to ξ and unstablyequivalent to ζ.

We will denote

Usk (ξ) = {ζ ∈ SG : [ξ, ζ] = ζ} Uu

k (ξ) = {ζ ∈ SG : [ζ, ξ] = ξ}.Then the elements of Us

k (ξ) are stably equivalent to ξ and the elements of Uuk (ξ)

are unstably equivalent to ξ.The following result is a direct corollary of Proposition 6.7, Lemma 6.9 and

Lemma 2 of [Fri87].

Proposition 6.10. The limit solenoid (SG, σ) is a Smale space, i.e., there existsa metric d on SG and a constant λ ∈ (0, 1) such that

d(σ(η), σ(ζ)) ≤ λd(η, ζ)for all η, ζ ∈ Us

k(ξ), and

d(σ−1(η), σ−1(ζ)) ≤ λd(η, ζ)for all η, ζ ∈ Uu

k (ξ).


Smale spaces were defined by Ruelle in [Rue78]. See also [Put96, KPS97, PS99,Rue78, Rue88] for the C∗-algebras associated with them.

6.6. Ruelle algebra of the unstable equivalence. We will often use the follow-ing definition.

Definition 6.2. Let X be a topological space represented as a union X =⋃

i∈I Ai

of subspaces Ai. Then the union topology on X defined by the covering {Ai}i∈I

is the direct limit of the identical embeddings⋃

i∈I1Ai ↪→

⋃i∈I2

Ai for all finitesubsets of indices I1 ⊂ I2 ⊂ I.

If every subset Ai ⊂ X is closed, then a subset A ⊂ X is closed in the uniontopology if and only if A ∩Ai is closed for every i ∈ I.

We have seen in Proposition 6.8 that the unstable equivalence relation U is aunion of the sets Uv,g,u of pairs of points (ξ, ζ) represented by sequences of the form

(. . . x−1x0 . vw ; . . . y−1y0 . ug(w))

where . . . x−1x0, . . . y−1y0 ∈ X−ω and w ∈ X

ω are arbitrary and g ∈ N , v, u ∈ X∗,

|v| = |u| are fixed.We introduce the union topology on U ⊂ SG × SG defined by the covering

U =⋃

g∈G,u,v∈X∗,|u|=|v| Uv,g,u.

This topology converts the unstable equivalence relation into a Hausdorff topo-logical groupoid, which we call the unstable groupoid. The natural Haar system

on the unstable groupoid is given by µξ0 = µξ

u × δξ, where µξu is the image of the

uniform Bernoulli measure on X−ω (see more about this Haar system in [KPS97]).

The shift σ : SG −→ SG defines an automorphisms of the stable and unstablegroupoids. We get hence the cross product U oσ Z defined by this automorphism.

The cross-product groupoid can be described as the set of triples (ζ1, n, ζ2) ∈SG×Z×SG such that σn(ζ1) is unstably equivalent to ζ2. Then Uoσ Z is a unionof the sets Uv,g,u,n of triples of the form

(. . . x−1x0 . vw, n , . . . y−1y0 . ug(w)) ,

where . . . x−1x0, . . . y−1y0 ∈ X−ω, n ∈ Z and w ∈ X

ω are arbitrary and g ∈ N andv, u ∈ X

∗ with |u| − |v| = n are fixed. Here we identify sequence with the points ofSG represented by them.

The topology of U oσ Z coincides with the union topology of the sub-spacesUv,g,u,n ⊂ SG × Z×SG.

Let us denote by C∗(U) the universal C∗-algebra of the unstable groupoid andby Ru the universal algebra of the cross-product groupoid U oσ Z.

It is proved in [PS99] that the groupoid U and the cross-product groupoid U oσ

Z are amenable, hence their universal algebras are isomorphic to their reducedalgebras.

Now we are going to described a generalized transversal of the groupoids U andU oσ Z in the sense of [PS99].

Choose a point ξ ∈ SG and let V s(ξ) be the stable equivalence class of ξ. If ξis represented by a sequence . . . x−1x0 . x1x2 . . ., then every point of V s(ξ) can berepresented by a sequence (yn) such that xn = yn for all n less than some n0.

Let us assume, for simplicity, that ξ is a fixed point of SG represented by thesequence . . . x0x0 . x0x0 . . . for some x ∈ X.


Following [PS99] let us introduce the union topology on V s(ξ) given by thedecomposition of V s(ξ) into the union of subsets

Vn = {(. . . x0x0v1 . w1, . . . x0x0v2 . w2) : vi ∈ Xn, wi ∈ X

ω} ,which are given with the relative topology of subsets of SG ×SG.

Consider the groupoid Uξ of the elements of U whose source and range belongto V s(ξ). This is an r-discrete groupoid with respect to the union topology givenby the decomposition into the union of the sets of the form

Un,v,u,g = {(. . . x0x0v1 . vw, . . . x0x0v2 . ug(w)) : w ∈ Xω , vi ∈ X

n}for n > 0, g ∈ N and v, u ∈ X

∗, |v| = |u|.The groupoid Uξ is σ-invariant, and we can construct the cross-product groupoid

Uξ oσ Z. The topology on it is the direct union topology of the sets

Um,n,v,u,g = {(. . . x0x0v1 . vw, m , . . . x0x0v2 . ug(w)) : w ∈ Xω, vi ∈ X

n}for n > 0, m ∈ Z, g ∈ N and v, u ∈ X

∗ such that |u| − |v| = m.

Theorem 6.11. The convolution algebra of the groupoids Uξ and Uξ oσ Z areisomorphic to K⊗MG and K⊗OG, respectively.

Proof. The groupoid Uξ is an ascending union of the groupoids

Un =⋃Un,v,u,g,

where the union is taken over all g ∈ N and v, u ∈ X∗ such that |v| = |u|.

Note that if u 6= v then the sequences . . . x0v . y1y2 . . . and . . . x0u . z1z2 . . . cannot represent the same point of SG. It follows that the groupoid Un is isomorphicto the groupoid MG under the isomorphism Un −→MG mapping the element

(. . . x0x0v1 . vw, . . . x0x0v2 . ug(w))

to the germ ofv1vX

ω −→ v2uXω : v1vw −→ v2ug(w)

at v1vw.The identical embedding Un ↪→ Un+1 is conjugated by these isomorphisms with

the embedding MG ↪→MG mapping a germ of

w1w −→ w2g(w)

at w1w to the germ ofx0w1w −→ x0w2g(w)

at x0w1w. Consequently, the embedding of the convolution algebras C∗(Un) ↪→C∗(Un+1) is conjugate with the embedding Ex0 :MG −→MG, see Subsection 3.9.It follows then from Theorem 3.7, that C∗(Uξ) ∼= K ⊗MG and C∗(Uξ oσ Z) ∼=K⊗OG. �

Corollary 6.12. The algebras C∗(U) and Ru are strongly Morita equivalent toMG and OG, respectively. Moreover, the algebra Ru is isomorphic to K⊗OG.

Proof. It is proved in [PS99] that U and Ru are strongly Morita equivalent to thealgebras C∗(Uξ) ∼= K⊗MG and C∗(Uξ oZ) ∼= K⊗OG, respectively. Consequently,C∗(U) and Ru are strongly Morita equivalent to MG and OG, respectively.

The algebras Ru and OG are separable, hence

Ru ∼= K⊗Ru ∼= K⊗ C∗(Uξ o Z) ∼= K⊗K⊗OG∼= K⊗OG,


since Ru is stable. �

This gives another proof of the facts that the algebra OG is nuclear, purelyinfinite and simple, since all these properties are known for Ru (see [PS99]).

6.7. Ruelle algebra of the stable equivalence. Let us perform the same re-duction, following [PS99] of the groupoid S of the stable equivalence and the crossproduct groupoid S oσ Z, as we did for the unstable equivalence, but using the al-gebra Oσ. Let ξ ∈ SG be any point represented by the sequence . . . x0x0 . x0x0 . . .and let V u(ξ) be the unstable equivalence class of ξ. Recall that a point repre-sented by (yk)k∈Z is unstably equivalent to ξ if and only if there exists n ∈ Z andan element g ∈ N of the nucleus of the group such that g(x0x0 . . .) = ynyn+1 . . ..Consequently, V u(ξ) is a union of the sets Vn of pairs of points of the form

(w1 . vg1(x0x0 . . .), w2 . ug2(x0x0 . . .))

where n ≥ 0 is fixed and v, u ∈ Xn, gi ∈ N , and wi ∈ X

−ω are arbitrary. Wetopologize V u(ξ) by the union topology, where Vn are given relative topology ofsubsets of SG ×SG.

Then the reduction Sξ is the set of pairs of stably equivalent points of V u(ξ) andhence it is a union of the sets Vn,v,u of pairs of points representable by sequencesof the form

(wv . v1g1(x0x0 . . .), wu . v2g2(x0x0 . . .))

for some w ∈ X−ω, vi ∈ X

n and gi ∈ N .The groupoid Sξ is σ-invariant, hence we can define Sξ oσ Z. We topologies it

by the union topology for the covering by the sets Vm,n,v,u of triples

(wv . v1g1(x0x0 . . .), m, wu . v2g2(x0x0 . . .))

for w ∈ X−ω, gi ∈ N and vi ∈ X

∗ such that |v| − |u| = m.

Proposition 6.13. The convolution algebras of the groupoids Sξ and Sξ o Z areisomorphic to K⊗Mσ and K⊗Oσ, respectively.

Proof. For every g ∈ N and v1 ∈ X∗ the set Av1,g,n of pairs of points of SG

representable by sequences of the form

(wv . v1g(x0x0 . . .), wu . v1g(x0x0 . . .))

for some w ∈ Xω and v, u ∈ X

n is a compact groupoid naturally isomorphic to Mn

(the isomorphisms is induced by the projection SG −→ JG).The identical embedding Av1,g,n ⊂ Av1,g,n+1 is conjugated with the embedding

Mn ↪→ Mn+1. It follows that the union Av1,g =⋃

n≥0Av1,g,n+1 is a groupoidisomorphic to Mσ.

Note that two sequences wv . v1g1(x0x0 . . .) and wu . v2g2(x0x0 . . .) for vi ∈ X∗

and gi ∈ G represent the same point of SG if and only if they coincide (by theregularity property). This implies that the groupoid Sξ is isomorphic to the directproduct of the groupoid Av1,g

∼= Mσ and the trivial groupoid on the countable setC = {vg(x0x0 . . .) : v ∈ X

∗, g ∈ N} (i.e., the groupoid of the trivial equivalencerelation C × C on C). Consequently, C(Sξ) is isomorphic toMσ ⊗K.

The proof of the statement about the cross-product is completely analogical. �

The next corollary is proved in the same way as the similar result for OG.


Corollary 6.14. The Ruelle algebra Rs is isomorphic to K⊗Oσ. The convolutionalgebra of the groupoid S is strongly Morita equivalent to the algebra Mσ.

It is known that the groupoid S o Z is amenable and that the Ruelle algebra Rs

is purely infinite and simple (see [PS99]). As a result we get.

Corollary 6.15. The algebra Oσ is purely infinite, simple, nuclear and satisfiesthe Universal Coefficients Theorem.

Note that the fact that Oσ (as well as OG) is purely infinite also easily followsfrom a general result of C. Anantharaman-Delaroche [Ana97] (using additionallythe fact that the corresponding groupoids are amenable and Hausdorff).

7. A Fock space representation

7.1. Extensions of the algebras Of and Of by compact operators. Letf : J −→ J be an expanding self-covering. Choose a basepoint t ∈ J and choosefor every z ∈ f−1(t) a path `x connecting t to z, where x ∈ X for some set oflabels X. These data define the self-similar iterated monodromy group (IMG (f) ,X)associated to f (see Theorem 2.2), so that the limit dynamical system (JG, σ) istopologically conjugate with the dynamical system (J, f) (by Theorem 2.3).

We will denote Of = OIMG(f).Consider the Fock space

`2(X∗) =

∞⊕

n=0

`2(X)⊗n

of `2(X).The Hilbert spaces `2(X)⊗n ∼= C|X|n is naturally isomorphic to the tensor prod-

uct Φ⊗n ⊗ ε, where Φ is the C-span of the permutational bimodule X · G seen asa CG-bimodule and ε is the trivial representation of G. Hence a get a representa-tion λ of G on `2(X)⊗n ∼= `2(Xn), which coincides with the natural permutationalrepresentation coming from the action of G on the nth level X

n of the tree X∗.

It follows from Proposition 2.1 that the action ofG on X∗ =

⊔n≥0 X

n is conjugate

with the action of G = IMG (f) on the tree Tf =⊔

n≥0 f−n(t) (see Definition 2.5).

The isomorphism Λ : X∗ −→ Tt conjugating the actions is given inductively by the

condition that the point Λ(xv) for x ∈ X and v ∈ Xn is the end of the preimage

of `x under fn−1 which starts at Λ(v). Equivalently, Λ(x1 . . . xn) is defined as theend of the path `x1 ⊗ · · · ⊗ `xn ∈Mfn , after identification of M⊗n

f with Mfn usingProposition 2.1.

For g ∈ G the operator λ(g) acts on the basic vectors δv of `2(X∗) as g acts onthe vertices of X

∗:

λ(g)(δv) = δg(v).

For x ∈ X let Lx be the creation operators

Lx(δv) = δxv.

Let L be the C∗-algebra generated by the operators λ(g) and Lx for g ∈ G andx ∈ X.

The algebra L can be also defined in purely topological terms in the spirit ofSubsection 3.3.


Namely, for every path γ in J starting in a point z1 ∈ f−n(t) and ending in apoint z2 ∈ f−m(t) define an operator Lγ on `2(Tf ) setting for z ∈ f−k(z1)

Lγ(δz) = δγ(z)

where γ(z) is the end of the lift of γ by fk which starts at z. We set Lγ(δz) = 0for all z ∈ Tt \

⋃k≥0 f

−k(z1).In particular, if γ is the trivial path at t, then Lγ is the identity operator. If γ

is the trivial path at z ∈ Tt, then Lγ is the projection onto the space of functionson Tt whose support is the rooted subtree Tz with the root z (the set of all inverseimages of z under iterations of f).

It is easy to see that if we identify Tf with X∗ by the isomorphism Λ, then Lγ

for a loop γ ∈ π1(J, t) will be identified with λ(g), where g is the image of γ inIMG (f). It is also not hard to see that the operator L`x will be identified withLx for every x. It follows that the algebra generated by the operators Lγ coincideswith the algebra L generated by the operators λ(g) and Lx.

A direct check shows that the operators λ(g) and Lx for g ∈ G and x ∈ X satisfy

almost all the defining relations of Of = OG. More precisely:

(1) All relations in G are obviously satisfied by λ(g), since these operatorsdescribe the action of G on X

∗.(2) For every x ∈ X we have L∗

xLx = 1.(3) If g(xw) = yh(w) for all w ∈ X

∗, then

λ(g)Lx = Lyλ(h)

.

However, ∑

x∈X

LxL∗x = 1− Pδ∅

,

where ∅ ∈ X∗ is the empty word and Pδ∅

= δ∅ · 〈δ∅| ·〉 is the orthogonal projectionon the space spanned by δ∅.

Since the algebra Of is simple, we have proved the following fact.

Proposition 7.1. The quotient of the C∗-algebra L by the ideal of compact opera-

tors is isomorphic to Of , where the isomorphism is induced by the map g 7→ λ(g),Lx 7→ Sx.

This proposition can be also proved using the definition via operators Lγ . The

operators Lγ also satisfy all defining relations of the algebra Of (as they are de-scribed in 3.3) except for relation (5) instead of which we have

Lγ −∑

α∈f−1(γ)

Lα = δz2 · 〈δz1 | ·〉 ,

where z1 and z2 are the beginning and the end of the path γ, respectively.Let us construct a similar extension by compact operators of the algebra Of .

Let ζ be a continuous function on J , seen as an element of the bimodule Φ overC(J), considered in 6.3. Define an operator Rζ on `2(Tt) by

Rζ(δz) = ζ(z)δf(z)

for z 6= t and Rζ(δt) = 0. Also define for every g ∈ C(f) an operator ρ(g) by

ρ(g)(δz) = g(z)δz.


Let us see how the operators Rζ and ρ(g) are related with the defining relations(1)–(4) of Of given in Subsection 6.3. The first relation αRζ1 + βRζ2 = Rαζ1+βζ2

is obviously true. For the second relation check that

ρ(g)Rζ(δz) = ρ(g)(ζ(z)δf(z) = g(f(z))ζ(z)δf(z) = Rζ·g(δz),

where ζ ·g is the right action of g on ζ ∈ Φ, which is given by (ζ ·g)(z) = ζ(z)g(f(z)).

Note that R∗ζ(δz) =

∑y∈f−1(z) ζ(y)δy. Consequently,

Rζ2R∗ζ1

(δz) = Rζ2

∑

y∈f−1(z)

ζ1(y)δy

=

∑

y∈f−1(z)

ζ1(y)ζ2(y)δz = ρ(〈ζ1| ζ2〉)(δz).

Finally, suppose that φ(g) =∑

i θζi,ξi in the bimodule Φ. It means that for everyψ ∈ Φ we have

g(z)ψ(z) =∑

i

ζi · 〈ξi|ψ〉 (z) =∑

i

ζi(z)∑

y∈f−1(f(z))

ξi(y)ψ(y),

which is equivalent to the conditions that ζi(z)ξi(y) = 0 for y ∈ f−1(f(z)) \ {z}and g(z) =

∑i ζi(z)ξi(z) (take ψ concentrated around z).

We have then, for z 6= t∑

i

R∗ξiRζi(δz) =

∑

i

R∗ξiζi(z)δf(z) =

∑

i

∑

y∈f−1(f(z))

ξi(y)ζi(z)δz =∑

i

ξi(z)ζi(z)δz = ρ(g)(δz).

It follows that the difference ρ(g)−∑i R∗ξiRζi is the rank one operator

δt 7→ g(t)δt.

We have proved the following proposition

Proposition 7.2. The quotient of the C∗-algebra R generated by the operators Rf

by the ideal of compact operators is isomorphic to the algebra Ofop opposite to the

algebra Of .

Theorem 7.3. For any L ∈ L and for any R ∈ R the operator [L,R] is compact.

Proof. It is sufficient to check this condition for L and R equal to the generatorsof the algebras L and R.

Let γ be a path from a point x ∈ f−n(t) to a point y ∈ f−m(t) and let ζ ∈ C(f).Consider then the commutators [Lγ , Rζ ] and [Lγ , ρ(ζ)].

We have for z ∈ f−k(x), k ≥ 1,

[Lγ , Rζ ](δz) = LγRζ(δz)−RζLγ(δz) =

Lγ(ζ(z)δf(z))−Rζ(δzγ ) = ζ(z)δf(zγ) − ζ(zγ)δf(zγ) = (ζ(z)− ζ(zγ))δf(zγ),

where zγ is the end of the lift of γ by fk starting at z. If z ∈ f−(n+k)(t) \ f−k(x),then [Lγ , Rζ ](δz) = 0.

The covering f is expanding, hence the distance between z and zγ goes to zerouniformly on z as k goes to infinity. Consequently |ζ(z) − ζ(zγ)| uniformly ap-proaches zero for k →∞, which implies that [Lγ , Rζ ] is compact. Compactness of[Lγ , ρ(ζ)] is proved analogously. �


Corollary 7.4. The C∗-algebra generated by R∪ L is an extension of the algebra

Of ⊗Ofop by the algebra of compact operators.

7.2. Hausdorff dimension of the Julia sets of hyperbolic rational func-

tions. Let f ∈ C(z) be a hyperbolic rational function. Then the iterated mon-odromy group of f is contracting and the limit dynamical system (JIMG(f), σ) istopologically conjugate to the action of f on its Julia set Jf .

Choose a basepoint t ∈ Jf and consider the tree Tt =⊔

n≥0 f−n(t). Let H =

`2(Tt).Let λ be the natural permutational representation of IMG (f) on H , as in the

previous subsection. It is faithful, by definition of the iterated monodromy group.Similarly, let ρ be the representation of the algebra C(Jf ) on H given by the

multiplication operators

ρ(f)(δz) = f(z)δz.

Again, this is the representation of C(Jf ) ⊂ R considered in the previous subsec-tion. It is faithful, since the set

⋃n≥0 f

−n(t) is dense in Jf .

By Theorem 7.3, the operators [λ(g), ρ(f)] are compact for all g ∈ G and f ∈C(Jf ).

Let us assume that∞ /∈ Jf (we can assume this, since Jf 6= C) and let Z ∈ C(Jf )be the restriction of the identical function z onto Jf .

Theorem 7.5. Let Bp = {A ∈ B(H) : Tr(|A|p) < ∞} be the Schatten ideal ofoperators on H = `2(Tt). Then for any non-trivial g ∈ IMG (f) the number

inf{p : [λ(g), ρ(Z)] ∈ Bp}

is equal to the Hausdorff dimension of Jf .

Proof. We will use the following results of C.T. McMullen (see Theorem 1.2 of [McM00]).The Poincare series of f is

Ps(f, x) =∑

y∈f−n(x)

|(fn)′(y)|−s.

The critical exponent δ(f) is the supremum of those s > 0 such that Ps(f, x) =∞for all x ∈ C.

If f is geometrically finite (in particular, if it is hyperbolic), then δ(f) is equalto the Hausdorff dimension of Jf and the Poincare series is divergent for s = δ(f)and for every x.

The Julia set Jf is compact and does not contain critical points. The functionf is uniformly expanding on a neighborhood of Jf (with respect to some conformalmetric).

It follows that there exist constants C > 0 and λ > 1 such that

|(fn)′(z)| > Cλn

for all z in an open neighborhood W of Jf . We can find W such that f−1(W ) ⊂W(see, for instance, the proof of Theorem 6.6).

We have for y → x

f(y)− f(x) = (y − x)f ′(x) + (y − x)2f ′′(x)/2 + o((y − x)2),


hence (by compactness of W ) there exist ε > 0 and C1 > 0 such that

∣∣∣∣f(x)− f(y)

(x − y)f ′(x)− 1

∣∣∣∣ < C1|x− y|

for all x, y ∈ W such that |x − y| < ε. Replacing ε by a smaller number, we mayassume that C1ε < 1.

We may also assume that ε is such that for every open subset U ⊂W of diameterless than ε and every n the set f−n(U) is a disjoint union of dn subsets Ui ⊂ Wsuch that fn : Ui −→ U is a homeomorphism.

Lemma 7.6. Let γ0, γ1, . . . , γn be a sequence of paths in W of length less than εsuch that f(γk) = γk−1. Denote by xk, yk the beginning and the end of the path γk

for k = 0, 1, . . . , n, respectively.Then

K1 <|x0 − y0|

|(fn)′(xn)| · |xn − yn|< K2

and∣∣∣∣(fn)′(xn)

(fn)′(yn)

∣∣∣∣ < K3

for some positive constants K1,K2 and K3 depending only on f , ε and W .

Note that we have necessarily K3 > 1.

Proof. The last inequality obviously follows from the first two.We have |xk − yk| < ε. If U0 is a disc of diameter less than ε containing the

path γ0, then the paths γk belong to open sets Uk such that fk : Uk −→ U0 is ahomeomorphism (by the choice of ε).

Let hk : U0 −→ Uk be the map inverse to fk : Uk −→ U0. Since |(fk)′(z)| > Cλk

for all z in W , we have |h′k(z)| < C−1λ−k for all z ∈ U . Consequently

|xk − yk| = |hk(x0)− hk(y0)| < C−1λ−k|x0 − y0| < C−1λ−kε.

Then

|(fn)′(xn)| = |f ′(xn)| · |f ′(xn−1)| · · · |f ′(x1)| =|x0 − y0||xn − yn|

·∣∣∣∣f ′(xn)(xn − yn)

xn−1 − yn−1

∣∣∣∣ ·∣∣∣∣f ′(xn−1)(xn−1 − yn−1)

xn−2 − yn−2

∣∣∣∣ · · ·∣∣∣∣f ′(x1)(x1 − y1)

x0 − y0

∣∣∣∣ <

|x0 − y0||xn − yn|

· (1 − C1|xn − yn|)−1 · · · (1 − C1|x1 − y1|)−1 <

|x0 − y0||xn − yn|

·∞∏

k=1

(1− C1C−1ελ−k)−1 = K−1

1 · |x0 − y0||xn − yn|


and

|(fn)′(xn)| = |f ′(xn)| · |f ′(xn−1)| · · · |f ′(x1)| =|x− y||xn − yn|

·∣∣∣∣f ′(xn)(xn − yn)

xn−1 − yn−1

∣∣∣∣ ·∣∣∣∣f ′(xn−1)(xn−1 − yn−1)

xn−2 − yn−2

∣∣∣∣ · · ·∣∣∣∣f ′(x1)(x1 − y1)

x0 − y0

∣∣∣∣ >

|x0 − y0||xn − yn|

· (1 + C1|xn − yn|)−1 · · · (1 + C1|x1 − y1|)−1 >

|x0 − y0||xn − yn|

·∞∏

k=1

(1 + C1C−1ελ−k)−1 = K−1

2 · |x0 − y0||xn − yn|

,

which finishes the proof of the lemma. �

Let Pf be the set of post-critical points of f , i.e., the closure of the union offorward orbits of critical points of f . If f is hyperbolic, then Pf has a finitenumber of accumulation points and contains the union of all attracting cycles off (see [Mil99]). If a point z does not belong to an attracting cycle, then startingfrom some n the set f−n(z) belongs to W (more precisely, f−n(z) converges to theJulia set in the Hausdorff metric).

It is easy to see that if x ∈ Pf , then the Poincare series Ps(f, x) is infinite forevery s > 0, hence we may discard the set Pf when finding the critical exponentδ(f).

On the other hand, the following lemma shows that all non-post-critical pointsare equivalent for our purpose.

Lemma 7.7. If x, y ∈ C\Pf then the Poincare series Ps(f, x) is convergent if andonly if the series Ps(f, y) is convergent.

Proof. If x ∈ C \ Pf , then there exists n such that f−n(x) ⊂ W . Consequently, itis sufficient to prove for x, y ∈W that Ps(f, x) is convergent if and only if Ps(f, y)is convergent.

Let γ be an arbitrary smooth path from x to y. We have a bijection φ :f−n(x) −→ f−n(y), for every n ≥ 0, mapping the beginning of a lift α ∈ f−n(γ)to its end.

There exists n0 such that all lifts of γ by fn0 are of length less than ε. Then byLemma 7.6 we have for every n > n0 and xn ∈ f−n(x)∣∣∣∣

(fn)′(xn)

(fn)′(F (xn))

∣∣∣∣ =

∣∣∣∣(fn−n0)′(xn)

(fn−n0)′(F (xn))

∣∣∣∣∣∣∣∣

(fn0)′(fn−n0(xn))

(fn0)′(fn−n0(F (xn)))

∣∣∣∣

< K3 ·∣∣∣∣


(fn0)′(fn−n0(F (xn)))

∣∣∣∣ = K3 ·∣∣∣∣


(fn0)′(F (fn−n0(xn)))

∣∣∣∣ ≤ K3 ·K4,

where K4 = maxz∈f−n0(x)

∣∣∣ (fn0)′(z)(fn0)′(F (z))

∣∣∣, since fn−n0(xn) ∈ f−n0(x).

Consequently,∑

z∈f−n(x)

|(fn)′(z)|−s ≤ (K3 ·K4)−s

∑

z∈f−n(y)

|(fn)′(z)|−s,

which finishes the proof of the lemma. �

We are ready now to prove the theorem. Let g be a non-trivial element ofIMG (f) and let γ be a loop representing it. Let z ∈ f−n(t) and let γz be the lift


of γ by fn starting at z. Denote by zγ the end of γz. Then

[ρ(Z), λ(g)](δz) = ρ(Z)λ(g)(δz)− λ(g)ρ(Z)(δz) =

zγδzγ − zδzγ = (zγ − z)δzγ = (zγ − z)λ(g)(δz),

hence the matrix of [ρ(Z), λ(g)]∗[ρ(Z), λ(g)] on `2(f−n(t)) is diagonal with theentries |zγ − z|2. It follows that

‖[ρ(Z), λ(g)]‖pp =∑

z∈f−n(t)

|zγ − z|p.

There exists n0 such that all lifts of γ by fn for n ≥ n0 are of length less than ε.We also assume that the action of g on the n0th level of the tree Tt is non-trivial.

Then, by Lemma 7.6 we have for all z ∈ f−n(t)

K1 <|fn−n0(z)− fn−n0(zγ)||(fn−n0)′(z)| · |z − zγ |

< K2,

hence

|fn−n0(z)− fn−n0(zγ)|K2|(fn−n0)′(z)| < |zγ − z| <

|fn−n0(z)− fn−n0(zγ)|K1|(fn−n0)′(z)|

Fixing the points z′ = fn−n0(z) and z′γ = fn−n0(zγ) = fn−n0(z)γ of f−n0(t)such that z′ 6= z′γ and changing n− n0 = k and z, we get estimates for p > 0:

K−p2 |z′γ − z′|p|(fk)′(z)|−p < |zγ − z|p < K−p

1 |z′γ − z′|p|(fk)′(z)|−p

for all z ∈ f−k(z′), hence

K−p2 |z′γ − z′|p

∑

z∈f−k(z′)

|(fk)′(z)|−p <

∑

z∈f−k(z′)

|zγ − z|p <

K−p1 |z′γ − z′|p

∑

z∈f−k(z′)

|(fk)′(z)|−p,

thus

K−p2 |z′γ − z′|pPp(f, z

′) <∑

k≥0

∑

z∈f−k(z′)

|zγ − z|p < K−p1 |z′γ − z′|pPp(f, z

′).

If [ρ(Z), λ(g)] ∈ Bp, then we can choose z′ ∈ f−n0(t) such that z′γ 6= z′ (since g isnon-trivial) and get an estimate

Pp(f, z′) < Kp

2 |z′γ − z′|−p∑

k≥0

∑

z∈f−k(z′)

|zγ − z|p ≤ Kp2 |z′γ − z′|−p ‖[ρ(Z), λ(g)]‖pp ,

implying p > δ(f).


On the other hand, if p > δ(f), then Pp(f, z′) is finite for every z′ ∈ f−n0(t) and

hence

‖[ρ(Z), λ(g)]‖pp =∑

n≥0

∑

z∈f−n(t)

|zγ − z|p =

n0−1∑

n=0

∑

z∈f−n(t)

|zγ − z|p +∑

z′∈f−n0(z)

∑

k≥0

∑

z∈f−k(z′)

|zγ − z|p ≤

n0−1∑

n=0

∑

z∈f−n(t)

|zγ − z|p +∑

z′∈f−n0(z)

K−p1 |z′γ − z′|pPp(f, z

′) <∞,

hence [ρ(Z), λ(g)] ∈ Bp. We have proved that p > δ(f) is equivalent to [ρ(Z), λ(g)] ∈Bp, which finishes the proof. �

References

[AD97] Claire Anantharaman-Delaroche, C∗-algebres de Cuntz-Krieger et groupes fuchsiens, Op-erator theory, operator algebras and related topics (Timisoara, 1996), Theta Found.,Bucharest, 1997, pp. 17–35.

[Ana97] Claire Anantharaman-Delaroche, Purely infinite C∗-algebras arising from dynamical sys-

tems, Bull. Soc. Math. France 125 (1997), no. 2, 199–225.[AR00] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de

l’Enseignement Mathematique, vol. 36, Geneve: L’Enseignement Mathematique; Uni-versite de Geneve, 2000.

[Bar06] Laurent Bartholdi, Branch rings, thinned rings, tree enveloping rings, Isr. J. Math. 154

(2006), 93–139.[BD75] John W. Bunce and James A. Deddens, A family of simple C∗-algebras related to weighted

shift operators, J. Functional Analysis 19 (1975), 13–24.[BG00] Laurent Bartholdi and Rostislav I. Grigorchuk, On the spectrum of Hecke type operators

related to some fractal groups, Proceedings of the Steklov Institute of Mathematics 231

(2000), 5–45.[BGN03] Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych, From fractal

groups to fractal sets, Fractals in Graz 2001. Analysis – Dynamics – Geometry – Stochas-tics (Peter Grabner and Wolfgang Woess, eds.), Birkhauser Verlag, Basel, Boston, Berlin,2003, pp. 25–118.

[BGS03] Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran Sunik, Branch groups, Handbookof Algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989–1112.

[BJ99] Ola Bratteli and Palle E. T. Jorgensen, Iterated function systems and permutation rep-

resentations of the Cuntz algebra, Mem. Amer. Math. Soc. 139 (1999), no. 663, x+89.[Bla98] Bruce Blackadar, K-theory for operator algebras, second ed., Mathematical Sciences Re-

search Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998.[BN06a] Laurent Bartholdi and Volodymyr V. Nekrashevych, Iterated monodromy groups of qua-

dratic polynomials I, (preprint), 2006.[BN06b] Laurent Bartholdi and Volodymyr V. Nekrashevych, Thurston equivalence of topological

polynomials, Acta Math. 197 (2006), no. 1, 1–51.[Con82] Alain Connes, A survey of foliations and operator algebras, Operator algebras and ap-

plications, Proc. Symp. Pure Math., vol. 38, Part 1, 1982, pp. 521–628.[Cun77] Joachim Cuntz, Simple C∗-algebras generated by isometries, Comm. Math. Phys. 57

(1977), 173–185.[Dea95] Valentin Deaconu, Groupoids associated with endomorphisms, Transactions of the A.M.S.

347 (1995), no. 5, 1779–1786.

[DM01] Valentin Deaconu and Paul S. Muhly, C∗-algebras associated with branched coverings,Proc. Amer. Math. Soc. 129 (2001), no. 4, 1077–1086 (electronic).

[Eme03] Heath Emerson, Noncommutative Poincare duality for boundary actions of hyperbolic

groups, J. Reine Angew. Math. 564 (2003), 1–33.


[Fri87] David L. Fried, Finitely presented dynamical systems, Ergod. Th. Dynam. Sys. 7 (1987),489–507.

[FS92] M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, PotentialAnal. 1 (1992), no. 1, 1–35.

[Gri80] Rostislav I. Grigorchuk, On Burnside’s problem on periodic groups, Functional Anal.Appl. 14 (1980), no. 1, 41–43.

[Gri85] , Degrees of growth of finitely generated groups and the theory of invariant means,Math. USSR Izv. 25 (1985), no. 2, 259–300.

[GS83] Narain D. Gupta and Said N. Sidki, On the Burnside problem for periodic groups,Math. Z. 182 (1983), 385–388.

[GS06] R. Grigorchuk and Z. Sunic, Asymptotic aspects of Schreier graphs and Hanoi Towers

groups, Comptes Rendus Mathematique, Academie des Sciences Paris 342 (2006), no. 8,545–550.

[KL05] Vadim A. Kaimanovich and Mikhail Lyubich, Conformal and harmonic measures on

laminations associated with rational maps, vol. 173, Memoirs of the A.M.S., no. 820,A.M.S., Providence, Rhode Island, 2005.

[KP00] Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact C∗-algebras in the

Cuntz algebra O2, J. Reine Angew. Math. 525 (2000), 17–53.[KPS97] J. Kaminker, I. F. Putnam, and J. Spielberg, Operator algebras and hyperbolic dynamics,

Operator Algebras and Quantum Field Theory (S. Doplicher, R. Longo, J.E.Roberts, andL. Zsido, eds.), International Press, 1997.

[KS02] M. Khoshkam and G. Skandalis, Regular representation of groupoid C∗-algebras and

applications to inverse semigroups, J. reine angew. Math. 546 (2002), 47–72.[KW05] Tsuyoshi Kajiwara and Yasuo Watatani, C∗-algebras associated with complex dynamical

systems, Indiana Univ. Math. J. 54 (2005), no. 3, 755–778.[LM97] Mikhail Lyubich and Yair Minsky, Laminations in holomorphic dynamics, J. Differ.

Geom. 47 (1997), no. 1, 17–94.[LS96] Marcelo Laca and Jack Spielberg, Purely infinite C∗-algebras from boundary actions of

discrete groups, J. Reine Angew. Math. 480 (1996), 125–139.[McM00] Curtis T. McMullen, Hausdorff dimension and conformal dynamics. II. Geometrically

finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, 535–593.[Mil99] John W. Milnor, Dynamics in one complex variable. Introductory lectures, Wiesbaden:

Vieweg, 1999.[Nek04] Volodymyr V. Nekrashevych, Cuntz-Pimsner algebras of group actions, Journal of Op-

erator Theory 52 (2004), no. 2, 223–249.[Nek05] Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs,

vol. 117, Amer. Math. Soc., Providence, RI, 2005.[Ner03] Yu. A. Neretin, Groups of hierarchomorphisms of trees and related Hilbert spaces, J.

Funct. Anal. 200 (2003), no. 2, 505–535.[Pat99] Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras,

Birkhauser Boston Inc., Boston, MA, 1999.[Phi00] N. Christopher Phillips, A classification theorem for nuclear purely infinite simple C∗-

algebras, Doc. Math. 5 (2000), 49–114 (electronic).[Pim97] Michael V. Pimsner, A class of C∗-algebras generalizing both Cuntz-Krieger algebras and

crossed products by Z, Free probability theory (Waterloo, ON, 1995), Amer. Math. Soc.,

Providence, RI, 1997, pp. 189–212.[PS99] I. F. Putnam and J. Spielberg, The structure of C∗-algebras associated with hyperbolic

dynamical systems, Journal of Functional Analysis 163 (1999), 279–299.[Put96] Ian F. Putnam, C∗-algebras from Smale spaces, Can. J. Math. 48 (1996), 175–195.[Ren80] Jean Renault, A groupoid approach to C∗-algebras, Lecture Notes in Mathematics, vol.

793, Springer-Verlag, Berlin, Heidelberg, New York, 1980.[Ren00] Jean Renault, Cuntz-like algebras, Operator theoretical methods (Timisoara, 1998),

Theta Found., Bucharest, 2000, pp. 371–386.[Rue78] D. Ruelle, Thermodynamic formalism, Addison Wesley, Reading, 1978.[Rue88] , Noncommutative algebras for hyperbolic diffeomorphisms, Invent. Math. 93

(1988), 1–13.[Sid97] Said N. Sidki, A primitive ring associated to a Burnside 3-group, J. London Math. Soc.

(2) 55 (1997), 55–64.


[Sul85] Dennis Sullivan, Quasi-conformal homeomorphisms and dynamics I, solution of the

Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401–418.[Tu99] Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, K-

Theory 17 (1999), 215–264.

Câˆ—-ALGEBRAS AND SELF-SIMILAR GROUPS Contents 1

Documents

Transcript of Câˆ—-ALGEBRAS AND SELF-SIMILAR GROUPS Contents 1