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Cuntz-Krieger algebras and wavelets on fractals

We consider representations of Cuntz--Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron-Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.


Introduction
A class of representations of the Cuntz algebra O N called permutative representations were studied and classified in [3,4,10]. Besides interest in their own right within the field of operator algebras, Cuntz algebras representations have very interesting applications to wavelets, fractals, and dynamical systems, see [3,4]. Some of these results have been extended to the more general class of Cuntz-Krieger algebras (see [20][21][22]), where representations of these algebras are related to Perron-Frobenius operators of certain measure space transformations. Similar representations of Cuntz-Krieger algebras were considered in the context of limit sets of Schottky groups and actions on trees in [5][6][7] for arithmetic applications to Arakelov geometry and p-adic Mumford curves.
In this paper we look at representations of the Cuntz-Krieger algebra having a underlying self-similarity structure. The concept of self-similarity has proved to be fundamental in mathematics as well as in diverse applications, related to the renormalization of structures on nested families of scales. In the theory of wavelets, the scales may be represented in resolutions taking the form of nested systems of linear spaces, while in C * -algebra theory it gives rise to representations of algebras on generators and relations such as those that define the Cuntz and Cuntz-Krieger algebras.
Cuntz-Krieger algebras arise naturally from semibranching function systems on measure spaces, where the partial inverses σ i of the coding map σ are not defined everywhere. The resulting algebra is generated by partial isometries S i associated to the maps in the semibranching function system, and the relations between these generators involve a matrix A with entries equal to zero or one, which describes the decomposition of the domains of the σ i as a union of ranges of other σ j in the same family.
Conversely, a Cuntz-Krieger algebra O A defined by generators and relations in terms of an N × N -matrix A as above determines a semibranching function system on the limit set of infinite sequences in an alphabet on N letters with the admissibility condition that consecutive letters i j can appear in a word if and only if the corresponding entry in the matrix is A i j = 1. One can identify this limit set A as a Cantor set inside the interval [0, 1] by considering points whose N -adic digital expansion satisfies the admissibility condition. On this Cantor set the action of the maps σ i become simple shifts in the N -adic expansion and the representation of O A on the Hilbert space L 2 ( A , µ), with respect to the Hausdorff measure of the appropriate dimension, has an especially simple form, and so does also the Perron-Frobenius operator for the shift map σ , which is expressed in terms of the generators of the algebra. The Hausdorff dimension of the limit set A is computed using the Perron-Frobenius theorem for the non-negative matrix A, which also shows that the components of the Perron-Frobenius eigenvector of A give the measures of the ranges of the maps σ i in the normalized Hausdorff measure of dimension the Hausdorff dimension of A , which is the unique probability measure satisfying the self-similarity condition for the fractal set A .
The Perron-Frobenius eigenvector of the matrix A t determines a fixed point for the Perron-Frobenius operator for the shift map σ on the limit set A , which in turn gives a KMS state for an associated time evolution on the algebra O A at inverse temperature equal to the Hausdorff dimension of A .
One can construct as in [17] further measures on A , using operator valued measures and square-integrable functions of unit norm. As in the case of the Cuntz algebras, by analyzing the Fourier transforms of these measures, one sees that one can approximate them with Dirac measures supported at truncations of the N -adic expansions.
Besides the Cantor set A ⊂ [0, 1], there is another fractal set that one can associate to the same matrix A, namely a Sierpinski fractal S A inside the unit cube, given by points (x, y) whose digits in the N -adic expansion satisfy the condition that A x i y i = 1. The Hausdorff dimension of these sets is simply computed in terms of the number of non-zero entries in A. The shifts in the N -adic expansion determine a semibranching function systems on S A , where, unlike in the case of A , the maps are everywhere defined, hence they give rise to an action of a Cuntz algebra or rank depending on the number of non-zero entries in A. There is a natural embedding of A into S A induced by the shift map on A . The action of the Cuntz algebra determines via this embedding of A into S A an action of a Cuntz-Krieger algebra.
We show how to use the representation of the algebra O A to construct an orthonormal system of wavelets on L 2 ( A , dµ).
We then consider the Ruelle transfer operator for the shift σ on A , with nonnegative valued potential W satisfying the Keane condition that the sum of the values over preimages under σ adds up to one. We show that one can construct from these measures on A t , for the transpose matrix A t , in terms of random walks where the probabilities assigned to words of a given length in the alphabet depends upon the values of the potential W . A simple example of a potential satisfying the Keane condition is given in terms of trigonometric functions.
The example of the continued fraction expansion on the Hensley Cantor sets considered in [24,25] is described as an example where the general results of this paper can be applied.
As an application we also show how the technique we described to construct wavelets on the Cantor sets A can be adapted to construct families of graph wavelets, using Cuntz-Krieger algebras associated to finite graphs with no sinks. Graph wavelets are considered a useful tool for spatial network traffic analysis [8].

Representations of Cuntz-Krieger Algebras
Let A be an N × N matrix with entries in {0, 1}. For consistency with the notation we adopt later in the paper, it is convenient to index the entries Recall that the Cuntz-Krieger algebra O A associated to such a matrix A is the C * -algebra generated by N (non-zero) partial isometries S 0 , . . . , S N −1 satisfying the relations The algebra O A is uniquely determined by the relations (2.1) and (2.2) and it is linearly spanned by the set of S α S β with words α and β in {0, . . . , N − 1} with possibly different lengths |α| and |β|, see [9]. We are especially interested here in representations of O A as bounded operators on Hilbert spaces of the form H = L 2 (X, µ), for (X, µ) a measure space. The representations we are interested in, which include the cases of the arithmetic applications mentioned above, are all constructed in terms of what we refer to as a semibranching function system, which will be concretely realized in terms of a shift map on a Cantor-like fractal set and its partial inverses. Definition 2.1 Consider a measure space (X, µ) and a finite family {σ i } i∈I , #I = N , of measurable maps σ i : D i → X , defined on measurable subsets D i ⊂ X . The family {σ i } is a semibranching function system if the following holds.
(1) There exists a corresponding family where we denote by R i the range R i = σ i (D i ). (2) There is a Radon-Nikodym derivative Thus, the maps of the semibranching function system are partial inverses of the coding map σ . Notice that the reverse composition σ i • σ is only defined when the image of x under σ lands in the domain D i of σ i .
Given a semibranching function system {σ i } N −1 i=0 with coding map σ , one can construct an associated family of linear operators {T i } N −1 i=0 acting on the Hilbert space L 2 (X, µ) by setting with ψ ∈ L 2 (X, µ), where χ R i is the characteristic function of R i ⊂ X .

Lemma 2.2 The adjoint of the operator T i of (2.4) is of the form
where we used the fact that the σ i are positive real valued. This gives (2.5).
We then see easily that the operators T i and T * i satisfy the following relation.

Proposition 2.3
The operators T i of (2.4) and their adjoints (2.5) satisfy the relations T i T * i = P i , where P i is the projection given by multiplication by where Q i is the projection given by multiplication by the characteristic function χ D i .
Proof We write explicitly the action of the operator T i T * i on elements ξ ∈ L 2 (X, dµ). We have Equivalently, we can write Notice then that one has so that we obtain which gives T i T * i = P i , the range projection on L 2 (X, µ) realized by the multiplication operator by the characteristic function of the set R i . By the assumptions (2.3) on the semibranching function system we know that the projections P i are orthogonal and that i P i = 1.
We then consider the product T * i T i . We have where we used again (2.6). This gives T * i T i = Q i , where Q i is the domain projection given by multiplication by the characteristic function χ D i . Unlike the range projections P i , the domain projections Q i are, in general, not orthogonal.
When the maps σ i are defined everywhere on X , one obtains from the operators T i and T * i a representation of the Cuntz algebra O N in the following way. Proposition 2.4 Let {σ i } be a semibranching function system on X , where the σ i are defined on all of X , that is, D i = X for all i = 0, . . . , N − 1. Then the operators T i define a representation of the Cuntz algebra O N on the Hilbert space H = L 2 (X, µ). Namely, they satisfy the relations Proof Under the assumption that the semibranching function system has D i = X for all i ∈ I , we obtain from Proposition 2.3 above that the operators T i and T * i of (2.4) and (2.5) satisfy T * i T i = 1. Moreover, we know from Proposition 2.3 that T i T * i = P i , the range projections given by multiplication by the characteristic functions χ R i . Since these range projections are orthogonal and the union of the R i exhausts X up to sets of measure zero, we obtain that i T i T * i = 1. In the case where the maps σ i are not defined everywhere on X , but only on smaller domains D i ⊂ X , one can then use the operators T i and T * i of (2.4) and (2.5) to construct representations of Cuntz-Krieger algebras, when the domains D i have the property that The examples considered in [5][6][7] are particular cases of this general procedure.
We then obtain since we have from (2.8) that Since the projections P i are orthogonal, we then obtain This gives (2.2) with S i = T i . Similarly, we have where Q i is the projection given by multiplication by χ D i . Using again (2.8) this then gives We describe below an important special case of semibranching function system, which gives rise to representations of Cuntz-Krieger algebras of the type described in Proposition 2.5.

Cantor Sets and Subshifts of Finite Type
Let A be an alphabet in N letters, which we can identify with the set {0, . . . , N −1}. Let A be the set of all infinite admissible words in the alphabet A, where the admissibility condition is specified by an N × N matrix A with entries in {0, 1}. Namely, (2.10) We assume further that the matrix A has the property that A ii = 1 for each i = 0, . . . , N − 1, that is, that arbitrarily long strings made of the same letters are allowed in the words of A . The set A can be topologized as a Cantor set, for example by identifying it with the subset of the interval [0, 1] of numbers whose base N expansion satisfies the admissibility condition. However, notice that, when we choose to view A as a subset of the interval [0, 1], which is convenient in what follows, we identify the rational numbers with infinite periodic sequences rather than with a finite N -adic expansion, so as to be able to act with the shift map σ on all of A . More precisely, the map between the abstract set A and its image inside the interval [0, 1] is two-to-one on the periodic sequences. These are of measure zero in the interval, so for our measure theoretic argument we ignore the distinction and use the same notation for both sets.
Let δ A be the Hausdorff dimension of the set A , realized as a subset of the interval [0, 1] in this way. We can then consider the Hilbert space We consider on A the self-map given by the one-sided shift

Proposition 2.6
The shift σ is the coding map of the semibranching function system and (2.14) Proof We show that the maps of (2.12) form a semibranching function system. We have hence the condition (2.3) of a semibranching function system is satisfied. Moreover, the Radon-Nikodym derivative is well defined and positive, since the map σ i on A ⊂ [0, 1] is realized by contractions and translations. In fact, we can write the domain D i of the map σ i as On each R j the map σ i is the restriction of the map of the I j ⊂ [0, 1], where x 0 is the first digit in the N -adic expansion of w = 0.x 0 x 1 x 2 . . ., that maps it to the subset I j j of elements with first and second digit equal to j composed with a translation that maps isometrically I j j → I i j to the interval of all numbers with first digit i and second digit j. It is then clear that the shift map (2.11) is a coding map for this semibranching function system, since on each D i we have σ • σ i (w) = w.
One then sees easily that this gives a representation of the Cuntz-Krieger algebra O A of the type described in Proposition 2.5 above.

Proposition 2.7 The operators T i and T
Proof The result immediately follows from Proposition 2.5, upon noticing that the condition (2.15) is the needed relation (2.8). We are assuming A ii = 1 for all i, so the hypothesis of Proposition 2.5 are satisfied.
It is well known (see [9]) that the abelian C * -algebra C( A ) sits naturally inside the Cuntz-Krieger algebra O A as the C * -subalgebra generated by the range projections for arbitrary x i ∈ A and arbitrary n.

Perron-Frobenius Operator
Consider the operator T σ : L 2 (X, µ) → L 2 (X, µ) that composes with the coding map σ : X → X , It is well known in the theory of dynamical systems that one can associate to a self map σ : X → X of a measure space its Perron-Frobenius operator P σ . This is defined as the adjoint of the composition (2.16) by be a semibranching function system with coding map σ : X → X . Then the Perron-Frobenius operator P σ is of the form Notice the similarity of the Perron-Frobenius operator P σ to the operators T * i of (2.5) above. In fact, using (2.5) and Proposition 2.8, we easily get the following, which was observed already in [20].
be a semibranching function system with coding map σ : X → X . Then the Perron-Frobenius operator P σ is of the form Notice that, in some particular cases, the functions σ i may be constant, in which case (2.19) gives just a linear combination of the operators T * i . For example, in the cases considered in [6,7] the functions σ i are locally constant, while they are not in the case considered in [5].
In the case of representations as in Proposition 2.5, we can express the Perron-Frobenius operator in terms of the partial isometries S i in the following way.
In the case where the 1/2 σ i are constant over D i , the operator P σ belongs to the algebra O A .
Proof The hypothesis are the same as in Proposition 2.5, hence we know that the generators S i of the Cuntz-Krieger algebra O A in the representation on L 2 (X, µ) are given by the T i of (2.4). Then (2.19) gives (2.20). The case where the 1/2 σ i are constant over D i then follows immediately from (2.20), since P σ is then a linear combination of the S * i .
To avoid having to assume that the σ i are constant in the result above (although this will in fact be the case in the main example we will be considering later), one can more conveniently work with representations of the Cuntz-Krieger algebras on the Hilbert space of half-densities, analogous to the representations of the Cuntz algebra considered in [16].
Given a semibranching function system on X satisfying (2.8), we can construct representations of the Cuntz-Krieger algebra O A on the space of half-densities of X , in much the same way as we did in Proposition 2.7 on the space L 2 (X, dµ).

Proposition 2.11
Let {σ i } be a semibranching function system on X , where the σ i are defined on subsets D i ⊂ X satisfying (2.8), possibly up to sets of measure zero. LetH be the Hilbert space of half-densities on X . Consider the operators (2.22)

These operators define a representation of the Cuntz-Krieger algebra O A .
Proof To compute the adjoints S * i we check We then check that the operatorsS i andS * i satisfy the Cuntz-Krieger relations (2.1) and (2.2). We have which shows thatS iS * i =P i , the range projection given by multiplication by the characteristic function of R i , so that the relation (2.2) is satisfied by the orthogonality of the projectionsP i iS iS * i = 1.
We also have which shows thatS * iS i =Q i , whereQ i is the domain projection given by multiplication by the characteristic function of D i . Using the relation (2.8) this then gives We then compute explicitly the Perron-Frobenius operator of the coding map σ : X → X acting on the space of half-densities.

Proposition 2.12
Let σ : X → X be the coding map of a semibranching function system as in Proposition 2.11 above. The Perron-Frobenius operatorP σ on the Hilbert space of half-densities is given byP Proof The translation operator associated to the shift map σ : X → X is acting on the space of half-densities bỹ The Perron-Frobenius operatorP σ onH is the adjoint This gives For example, in the case of the Cuntz-Krieger algebras considered in [5][6][7], where the representation comes from the action of a Schottky group on its limit set, the generators S i are associated to a symmetric set of generators A = {γ 1 , . . . , γ g , γ −1 1 , . . . , γ −1 g } of a Schottky group of genus g, and the matrix A of the Cuntz-Krieger algebra has A i j = 1 for |i − j| = g and zero otherwise, corresponding to the admissibility of the infinite sequences w = a 0 a 1 a 2 · · · of elements of A parameterizing points in the limit set , namely that a i+1 = a −1 i . In this particular class of examples, the Perron-Frobenius operator of Proposition 2.12 has the form This resembles closely a Harper operator for the group , save for the important difference that the operators associated to the symmetric set of generators of here are partial isometries and not unitaries as in the usual Harper operator.

Projection Valued Measures
We recall how one constructs projection-valued measures using subdivions of compact metric spaces and subdivisions of projections in Hilbert spaces. (We follow the notation and terminology of [18] for the standard material we recall.) We then show how this technique applies to the representations of Cuntz-Krieger algebras described above.
We begin by recalling the notion of partitions and N -adic systems of partitions of a metric space.

Definition 2.13
Let (X, d) be a compact metric space. For subsets A ⊂ X , define the diameter as For a given N ≥ 2, an N -adic system of partitions of X is a family (indexed by k ∈ N) of partitions P k of X into Borel subsets A k (a), indexed by elements of A k , where A = {0, . . . , N − 1} is the given alphabet on N letters, with the properties: We then recall the equally well known notion of partitions of projections in Hilbert spaces.
Definition 2.14 Let H be a complex separable Hilbert space. A partition of projections in H is a collection {P(i)} i∈I of projections P(i) = P(i) * = P(i) 2 such that We also recall the notion of operator valued measure. (2.26) An orthogonal projection valued measure is a positive operator-valued measure as above satisfying: Note that the values E(B i ) in (2.26) are positive operators, so we take the summation on the right hand side of (2.26) to be convergent in the strong operator topology.
We are interested here in a particular construction of N -adic partitions, for the metric Cantor set A defined in (2.10) above. As above, we consider the alphabet A = {0, . . . , N − 1}. For any k ∈ N, we denote by W k,A ⊂ A k the finite set of all admissible words of length k in the alphabet A, We also denote by k,A (a) the clopen subset of the Cantor set A given by all words that start with a given a ∈ W k,A , We then have the following partition and corresponding operator valued measure.

Proposition 2.16
The subsets k,A (a) of (2.28) define an N -adic system of partitions for A . There is a corresponding N -adic system of projections P k (a) on the Hilbert space H = L 2 ( A , µ A ) and an orthogonal projection valued measure E on B( A ) satisfying

29)
for all k ∈ N and for all a ∈ W k,A .
Proof To see that the k,A (a) form an N -adic system of partitions, notice that, when we identify A with the subset of [0, 1] of numbers with admissible N -adic digital expansion and we measure diameters in the Euclidean distance on [0, 1], we see that the set k,A (a), which consists of such numbers with fixed first k digits in the N -adic expansion have since the sets of all numbers with fixed k digits in the N -adic expansion are intervals of length N −k . Moreover, by construction we have inclusions We also have, for fixed k, Thus, we have an N -adic system of partitions. One knows from [9] that there is an * -isomorphism between the C * -algebra of continuous functions C( A ) and the maximal abelian subalgebra of the Cuntz-Krieger algebra O A generated by all the range projections We show that the P k (a) define an N -adic system of projections on the Hilbert space In the representation of O A described in Proposition 2.5, the operator P k (a) acts as the projection given by multiplication by the characteristic function of the set k,A (a).
Since the k,A (a) form an N -adic system of partitions, in particular, as we have seen above, there are inclusions (2.31). These imply that the corresponding projections satisfy P k−1 (a 1 , . . . , a k−1 )P k (a 1 , . . . , a k ) = P k (a 1 , . . . , a k ).
For every k ∈ Z + , let us denote by U k the finite dimensional subalgebra of C( A ) spanned by the finite linear combinations where we still denote as π the extension above. It follows that E(·) satisfies the properties of Definition 2.15 and is countably additive. It also satisfies E( k,A (a)) = P k (a), for every k ∈ Z + and for all a ∈ W k,A .

Hausdorff Dimension
We consider again the space A of numbers in the interval [0, 1] whose N -adic expansion is admissible according to the matrix A, that is, x = 0.a 0 a 1 · · · a n · · · with A a i ,a i+1 = 1.
We know that in this case the maps σ i are defined on domains D i ⊂ A satisfy-

38)
with r (A) the spectral radius of the matrix A. (4) The measure µ satisfies the self-similarity condition Proof (1) The maps σ i act as the restrictions to the set D i of the linear maps defined on the interval [0, 1]. Thus, we see directly that the Radon-Nikodym derivative of the Hausdorff measure µ H,s will give In particular for s = δ A = dim H ( A ) this gives (2.35).
(2) We first show that setting with p the normalized Perron-Frobenius eigenvector of A, defines a probability measure on A . The Perron-Frobenius theorem for the matrix A shows that, if r (A) denotes the spectral radius of A, then r (A) is an eigenvalue which has an eigenvector p = ( p i ) with non-negative entries. We can normalize it so that i p i = 1. Setting ν(R i ) = p i defines a measure on A . In fact, it suffices to see that we can define ν( k,A (a)) compatibly, for all a ∈ W k,A . We set where a = (a 1 , . . . , a k ) ∈ W k,A . To see that (2.42) consistently defines a measure on A we need to check that where we used the Perron-Frobenius relation The measure ν thus satisfies the self-similarity property Indeed, it suffices to check it on sets of the form E = k,A (a), for which σ −1 j ( k,A (a)) is non-empty for a 1 = j, in which case it is k−1,A (σ (a)). Then we have 1,A (σ (a))), which gives (2.44).
We then compare this with the Hausdorff measure µ = µ A . This satisfies In fact, this follows simply from the fact shown in (1) that the Radon-Nikodym derivatives are constant, Note that it then follows that the measure µ also satisfies for a = (a 1 , . . . , a k ). This follows directly from (2.45) and the fact that A (a j)).
Notice then that (2.45) is saying that the vector q = (q i ) with q i = µ(R i ) is also an eigenvector of the matrix A, with eigenvalue N −δ A ≤ r (A), with the normalization Under the assumption that the non-negative matrix A is irreducible, the Perron-Frobenius theorem for A ensures that the eigenvalue r (A) is simple and that if q = (q i ) is another eigenvector, Aq = λq with q i ≥ 0, then λ = r (A) and q is a scalar multiple of p. Since both vectors are normalized, this implies that (2.47) By (2.46) and (2.42), this implies that the measures µ and ν agree.
As a particular case, if the matrix A has the property that the value α = j A i j is the same for all i = 0, . . . , N −1, then one has uniform probability for all the R i , equal to µ(R i ) = 1/N , and the set A has then Hausdorff dimension δ A = log(α)/ log(N ).
We return to consider now in particular the representation of the Cuntz-Krieger algebra O A on the space L 2 ( A , dµ A ) as in Sect. 2.1.

Corollary 2.18
The Perron-Frobenius operator P σ on the Hilbert space Proof As we have seen in Proposition 2.17, in this case the σ i are locally constant and equal to N −δ A , with δ A the Hausdorff dimension, which in turn is given in terms of the spectral radius of A. Then we have from Proposition 2.10 that the Perron-Frobenius operator P σ on L 2 ( A , dµ A ) is simply given by (2.48), where the S i generate the representation of the Cuntz-Krieger algebra on L 2 ( A , dµ A ).
We then see that one can use the result of Theorem 2.17 to construct a fixed point for the Perron-Frobenius operator P σ .

Proposition 2.19 Assume that the matrix A is irreducible, and let ω be the Perron-Frobenius eigenvector for A t . Then f = i ω i χ R i is a fixed point of the Perron-Frobenius operator P σ .
Proof Let ω be the Perron-Frobenius eigenvector The Perron-Frobenius operator P σ acting on the function f = i ω i χ R i gives by Corollary 2.18. We have where we used the fact that r (A) = N δ A as in (2.47).
There is a well known relation for Cuntz-Krieger algebras between the fixed points of the dual Perron-Frobenius (or Ruelle transfer operator) acting on measures and KMS states with respect to associated time evolutions, see [22]. We discuss the more general case of the Ruelle transfer operators later, but we comment here on the case that follows directly from Theorem 2.17.

Corollary 2.20 On the Cuntz-Krieger algebra O A consider the time evolution defined by setting
(2.49) The measure µ = µ A on A defines a KMS state for the system (O A , σ t ) at inverse temperature β = δ A .
Proof We define a state ϕ on O A associated to the measure µ by setting (2.50) We use here the fact that all elements in O A can be approximated by linear combinations of elements of the form S a S * b . We then need to check that the state ϕ satisfies the KMS condition at inverse temperature β = δ A for the time evolution (2.49). Because of the form of the state (2.50), and the fact that the measure µ satisfies (2.46), it suffices to check that This follows since we have using the fact that p = ( p i ) with p i = µ(R i ) is the Perron-Frobenius eigenvector of the matrix A.

Real Valued Measures and Fourier Transforms
Given an element f ∈ H with norm f = 1, one can define a real valued measure on A ⊂ [0, 1] by setting (2.52) We then have the following result, which is analogous to the case of the Cuntz algebras O n discussed in [17].

Proposition 2.21 For every function f
The Fourier transform µ f (t) satisfies where π denotes the embedding π : C( A ) → O A , as in (2.33), which realizes C( A ) as an abelian * -subalgebra of O A , with π(χ k,A (a) ) = S a S * a . In the algebra O A we have the relations (2.56) Thus, we have π(χ D k ψ • σ k )S * k = S * k π(ψ) and we write (2.55) as which gives (2.53). We then proceed as in [17], and observe that (2.53), applied to ψ(x) = e it x , gives which gives (2.54).
We can equivalently see (2.53) as an immediate consequence of (2.39), since we have where for a = (a 1 , . . . , a k ) ∈ W k,A we denote by x(a) the expression As in [17], we then obtain an approximation of the measure µ f with a family of combinations of Dirac measures in the following way.
Passing to Fourier transforms, we have The Fourier transform of µ (k) f is clearly of the form with x(a) as in (2.59), and one can estimate as in [17] |μ This gives which gives the weak convergence µ (k) f → µ f .

Sierpinski Fractals
There is another fractal object, besides the limit set A , that is naturally associated to an N × N -matrix A with entries in {0, 1}. This is a Sierpinksi fractal constructed in the following way. Consider the square S = [0, 1] ×[ 0, 1] and write points (x, y) ∈ S in terms of the N -adic expansion We then consider the subset S A ⊂ S given by This is a Sierpinski fractal whose iterative construction starts by subdividing the unit square S into the N 2 subsquares of size N −2 consisting of points (x, y) with first digits of the N -adic expansion equal to given (i, j) ∈ A 2 . One then keeps among these only those for which A i j = 1. The procedure is then iterated by subdividing each of the remaining squares into N 2 subsquares of size N −4 and keeping only those for which the same condition A i j = 1 is satisfied, and so on. At each step a square is of size These satisfy d i ≤ N and D ≤ N 2 . Thus, the Hausdorff dimension of the Sierpinski fractal S A is simply One can then consider maps τ (i, j) : S A → S A , for (i, j) satisfying A i j = 1, given by Notice how, unlike the σ i acting on A that we considered before, here the τ (i, j) are everywhere defined on S A . Since we are only considering such maps for pairs The corresponding coding map τ : S A → S A is given by for (x, y) = (0.x 1 x 2 · · · x k · · · , 0.y 1 y 2 · · · y k · · · ). Proof Let (i, j) denote the Radon-Nikodym derivative of the measure µ with respect to composition by τ (i, j) . Since τ (i, j) is of the form (2.65), we have We consider the operators S (i, j) and S * (i, j) defined as in the general case of a semibranching function system in the form j) . The adjoint S * (i, j) in the inner product of L 2 (S A , µ) is given by Thus, one sees that is the range projection given by multiplication by χ R (i, j) . Thus, the S (i, j) generate a representation of the Cuntz algebra O D on L 2 (S A , µ).
Notice then that we can embed the limit set A inside the Sierpinski fractal S A in the following way.

Lemma 2.24 The map
gives an embedding A → S A .
This means that the point (x, y) = (0.x 1 x 2 · · · x n · · · , 0.x 2 x 3 · · · x n+1 · · · ) = (x, σ (x)) satisfies A x i ,y i = A x i x i+1 = 1 for all i ≥ 1, hence it is a point in S A . The map is clearly injective since it is the identity on the first coordinate. It is continuous since the preimage of a clopen set S A (i 1 · · · i k , j i · · · j k ) of S A , given by numbers with fixed first k digits of the N -adic expansion, is either empty, or else, when j r = i r +1 for r = 1, . . . , k − 1, it is equal to the clopen set A (i 1 , . . . , i k , j k ) of A .
One can then use this embedding together with the representation of the algebra O D on L 2 (S A , µ) to obtain an induced action of a Cuntz-Krieger algebra.

Proposition 2.25 The maps τ i, j restricts to maps defined on domains D i, j ⊂ ( A ).
These determine a semibranching function system on ( A ) which gives rise to a representation of the algebra OÃ, where the D × D-matrixÃ is given bỹ In fact, the condition that A i j = 1 implies that R j ⊂ D i in A , so that (D i ∩ R j ) = (R j ). We identify the restriction of continuous functions on S A to ( A ) with continuous functions on A and we write equivalently, with a slight abuse of notation, f (x, σ (x)) or f (x). One then sees that This induces an isometry on the Hilbert space L 2 ( ( A ), µ s ), where µ s is the Hausdorff measure of dimension s = dim H ( ( A )), since for a function f (x, σ (x)) on ( A ) we have This has adjointŜ * We then obtain so that we have the relation We also havê Using the fact that we then obtain the other relation in the form These correspond to the Cuntz-Krieger relations for the matrixÃ of (2.71).
A general construction of wavelets on self-similar fractals was described in [15], see also [2]. The cases considered there correspond, from the point of view of semibranching function systems, to the case where the σ i are defined on all of X , as in the case of the Cuntz algebra. To adapt these constructions of wavelets to the main case we are interested in, which is the Cantor sets A introduced above, one can use the representation of the Cuntz-Krieger algebra O A on L 2 ( A , dµ A ) that we considered in the previous sections, and again the Perron-Frobenius theory for the non-negative matrix A. We begin by recalling briefly how the construction of [15] works in the case of a semibranching function system on a measure space (X, µ) where the N maps σ i are defined on all of X . In this case one considers the (m + 1)-dimensional linear space P m of polynomials on R of degree ≤ m, and one denotes by S 0 the linear subspace of L 2 (X, dµ), generated by the restrictions P| A of polynomials in P m . Under the condition that X preserves Markov's inequality (see Sect. 4 of [15]), one knows that one still has dim S 0 = m + 1. One then considers the linear subspace S 1 ⊂ L 2 (X, dµ) of functions f ∈ L 2 (X, dµ) that are µ-almost everywhere on R i = σ i (X ) restrictions P| R i of some polynomial P ∈ P m . Clearly S 0 ⊂ S 1 and dim S 1 = N dim S 0 = N (m +1), and let φ , for = 1, . . . , m +1 be an orthonormal basis for S 0 . One then considers the orthogonal complement S 1 S 0 , with a fixed choice of an orthonormal basis ψ ρ , for ρ = 1, . . . , (N − 1)(m + 1). The functions φ r and ψ ρ provide the mother wavelets. One then considers the family of linear subspaces S k of L 2 (X, dµ), of functions whose restriction to each subset σ i 1 •· · ·•σ i k (X ), agrees µ-almost everywhere with the restriction to the same set of a polynomial in P m . These satisfy S 0 ⊂ S 1 ⊂ · · · S k ⊂ · · · L 2 (X, dµ). Moreover, any function in L 2 (X, dµ) can be approximated by elements in since in fact the polynomials of degree zero already suffice, as they give combinations of characteristic functions of the sets σ i 1 • · · · • σ i k (X ). The wavelets are then obtained in [15] as for a = (i 1 , . . . , i k ) and σ a = σ i 1 • · · · • σ i k .
We show now how to adapt this construction to the case of the Cantor sets A . For simplicity, we describe in full only the case where one only considers locally constant functions, that is, where one starts with the 1-dimensional space P 0 . This is the case that is closest to the classical construction based on the Haar wavelets [11].
On the space A ⊂ [0, 1], with the Hausdorff measure µ = µ A , let S k denote the linear subspaces of L 2 ( A , dµ A ) obtained as above, starting from the 1-dimensional space P 0 . Let be a family of locally constant functions on A such that the support of f ,k is contained in R k and We also require that Proof To see that linear combinations of characteristic functions χ R k j suffice to construct the functions f ,k , notice first that the χ R k j give an orthogonal basis for the space S 2 , which is of dimension dim S 2 = k d k . We then write the f ,k in the form where the conditions (3.4) and (3.5) translate into conditions on the coefficients of the form where we use the notation where we again use (3.8).
Let us introduce the following notation for convenience. Consider on C d k ⊂ C N the inner product v, w k := j A k jv j w j p j . (3.10) Let V k denote the orthogonal complement, in the inner product (3.10) on C d k of the vector u = (1, 1, . . . , 1), and let {c ,k = (c ,k i )} =1,...,d k −1 be an orthonormal basis of V k , in the inner product (3.10), namely c ,k , u k = 0, and c ,k c ,k k = δ , . (3.11) Then for c ,k as above, one sees that the functions (3.6) are an orthonormal family satisfying the conditions (3.4) and (3.5). The space spanned by the f ,k is contained in S 2 by construction. The condition (3.5) ensures that the functions f ,k are orthogonal to all the χ R k , hence they are in for a = (a 1 , . . . , a k ) ∈ W k,A , give an orthonormal basis for the space S k+1 S k hence, for varying a ∈ W k,A and for all k ≥ 0, they give an orthonormal basis of wavelets for L 2 ( A , µ).
Proof We have shown in Lemma 3.1 that the functions f ,r , for r = 0, . . . , N -1 and = 1, . . . , d r , give an orthonormal basis of S 2 S 1 . We then check that the functions S a f ,r give an orthonormal basis for S k+1 S k . Since in the representation of O A on L 2 ( A , dµ A ) we have constant Radon-Nikodym derivatives σ i = N −δ A , this gives For a ∈ W k,A , we have where we write R a = k,A (a), for the range of σ a = σ a 1 • · · · • σ a k . Notice then that we have, for any function f ∈ L 2 ( A , dµ) and any a ∈ W k,A , Applied to the above this gives S a f ,r , S a f ,r = δ a,a δ r,r A a k r R r f ,r f ,r dµ = δ a,a δ r,r δ , .
Thus the S a f ,k form an orthonormal system. The space spanned by these functions is contained in S k+1 and a counting of dimensions shows that it has the dimension of S k+1 S k . To see that the S a f ,k are in fact orthogonal to the elements of S k it suffices to compute by (3.13) and (3.5). This shows that we obtained an orthonornal basis of S k+1 S k , hence a wavelet system for L 2 ( A , dµ).
It is useful to remark how the main difference in this case, as opposed to the similar constructions given for instance in [15] that we mentioned above, is that here we need to start from an orthonormal basis of S 2 S 1 instead of S 1 S 0 . This reflects the fact that our functions σ i are not everywhere defined and, while the choice of an orthonormal basis for S 1 S 0 gives the needed information on the ranges R i , in order to control both the ranges and the domains D i one needs to go one step further before starting the induction that constructs the wavelets, and consider S 2 S 1 . Thus, the wavelet decomposition of a function f ∈ L 2 ( A , µ) will be given by (3.14) The more general case where one starts the wavelet construction from the linear space of polynomials P m with m ≥ 1 can be done along the same lines as Lemma 3.1 and Theorem 3.2. We describe in the next section a different approach to wavelets constructions based on the Ruelle transfer operator for the coding map σ . This is closer to the point of view developed in [12].
A more general version of the Perron-Frobenius operator associated to the coding map σ : A → A is obtained by considering the Ruelle transfer operator. This depends on the choice of a potential function W , defined on A , and is defined as (4.1)

Lemma 4.1
If the function W is real valued, one can describe the operator R σ,W as the adjoint of the operator Proof We have using the fact that the Radon-Nikodym derivative dµ • σ i /dµ = N −δ A . We then write the above as We also have Since the set of preimages of the point x under the coding map is given by we see that the above is in fact This shows that T W f, h = f, R σ,W (h) .
We assume that the potential W of the Ruelle transfer operator satisfies the Keane condition, namely that it has non-negative real values W : A → R + , and satisfies y:σ (y)=x W (y) = 1. (4.3) Equivalently, this means

Random Processes
In the same way as described in [12], we relate here harmonic functions for the Ruelle transfer operator, that is, functions satisfying R σ,W h = h to random processes defined by transition probabilities for paths from a given point x to the image under the σ j and their iterates. Let A t be the transpose of the matrix A. Then we have a t = (a k , . . . , a 1 ) ∈ W k,A if and only if a = (a 1 , . . . , a k ) ∈ W k,A t . We construct probability measures on the limit set A t that are related to fixed points of the Ruelle transfer operator for the coding σ : A → A . In the following we denote by R i and D i , as before, the ranges and domains of the maps σ i in A and by R t i and D t i the corresponding sets in A t . For a given potential W on A satisfying the Keane condition (4.3), consider a func- is a non-negative function on the Borel subsets of A t defined by assigning to the k,A t (a) the values P W x ( k,A t (a)) = A a 1 x 1 W (σ a 1 (x))W (σ a 2 σ a 1 (x)) · · · W (σ a k · · · σ a 1 (x)), (4.5) for a t = (a k , . . . , a 1 ) ∈ W k,A and for x ∈ D a 1 ⊂ A .

Lemma 4.2
The assignment (4.5), for x ∈ D i ⊂ A , defines a measure on R t i ⊂ A t . Proof Similarly, to the case of A seen in (2.43), to check that (4.5) defines a measure one has to check the compatibility condition for all x ∈ D a 1 ⊂ A . We have P W x ( k+1,A (a j)) = A a 1 ,x 1 W (σ a 1 (x)) · · · W (σ a k · · · σ a 1 (x))W (σ j σ a k · · · σ a 1 (x))) Moreover, the Keane condition for W on A gives j A ja k W (σ j σ a k · · · σ a 1 (x))) = 1, so we obtain (4.6).
One can think of the values of the potential W as defining a probability of transition, or walk, from x to σ a 1 (x), so that (4.5) can be regarded as the probability of a random walk from x to σ a k · · · σ a 1 (x). We then see that the random process P W x is related to the fixed points of the Ruelle transfer operator. Proposition 4. 3 The random process x → P W x introduced above is related to fixed points of the Ruelle transfer operator in the following ways.
converges, then the function h(x) is a fixed point of the Ruelle transfer operator with potential W on A .
Proof (1) We check that this condition is equivalent to the fixed point condition under the Ruelle transfer operator. For a given set k,A t (a), we have By construction of the measures P W x , we know that P W x (σ j (E ∩ R t i )) is non-trivial provides that x ∈ D j , so that A j x 1 = 1. Thus, for σ −1 (E) = E, we have which shows that P W x (E) is a fixed point for R σ,W . (2) Assuming that the series (4.7) converges, we have A a 1 j W (σ a 1 σ j (x)) · · · W (σ a k · · · σ a 1 σ j (x)) = k b= ja∈W k+1,A t A j x 1 W (σ j (x))W (σ a 1 σ j (x)) · · · W (σ a k · · · σ a 1 σ j (x)).

A Trigonometric Example
We give an example of a potential W satisfying the Keane condition, constructed using trigonometric functions.

Lemma 4.4 The function
since σ j (x) = (x + j)/N and the above becomes a sum over all the N 1 th roots of unity. It follows directly from this that the real valued trigonometric version also satisfies from which it follows that the potential of (4.8) satisfies Moreover, the function W (x) takes non-negative real values, so it gives a potential with the Keane condition.

Hensley Cantor Sets and Continued Fraction Expansion
In [23] the coding of geodesics on the modular curves X = H/ , for ⊂ PGL 2 (Z) a finite index subgroup and H the hyperbolic upper half plane, was related to a generalization of the shift map of the continued fraction expansion T : [0, 1]×P → [0, 1]×P, where P = PGL 2 (Z)/ is the finite coset set. It was then shown in [24,25], that the restriction of this dynamical system to the Hensley Cantor sets, that is, those subsets This can be written in the form (n,t) where the matrix A is defined by the condition A (n,t),(k,s) = 1 M n s = t, 0 otherwise, where the matrix M n ∈ GL 2 (Z), acting on the left on the coset P, is The shift invariant measure µ N ,P on E N × P constructed in [24] using the fixed point of the Ruelle transfer operator can then be also seen as in [22] as KMS β state for the time evolution on the Cuntz-Krieger algebra O A given by where we identify W −it , for fixed t, with an element in C(E N × P). The KMS state is then of the form for a = ((k 1 , s 1 ), . . . , (k r , s r )) ∈ W r,A and f a the element in C(E N × P) that corresponds to S a S * a . The Ruelle operator can correspondingly be written as in term of generators of the Cuntz-Krieger algebra.

Graph Wavelets from Cuntz-Krieger Algebras
It was recently shown, see for instance [8], that the crucial problem of spatial traffic analysis on networks can be addressed using a form of wavelet analysis which is adapted to the topology of the network graph. These graph wavelets are constructed as families of functions α (v) on the set of vertices V (G) of a given finite graph G, localized with respect to certain scaling indices α, and with the property that where µ(v) is a given measure that weights the nodes of the network with assigned probabilities. We show here how to construct families of graph wavelets using the representations of Cuntz-Krieger algebras and the corresponding wavelets on A constructed in Sect. 3. Let G be a finite directed graph with no sinks. It is well known that one can associate to such a graph a Cuntz-Krieger algebra in the following way. One considers a collection of projections P v associated to the vertices v ∈ V (G) and a collection of partial isometries S e associated to the oriented edges e ∈ E(G), with the relations for all edges e ∈ E(G). Assuming that the graph has no sinks, so that all vertices are sources, one has v P v = 1 so that the isometries S e satisfy the relation (2.2), e S e S * e = 1. As before, let A be the limit set associated to the algebra O A of the graph G. Let d e = #{e | r (e) = s(e )} = #{e | A ee = 1}. Consider as in Sect. 3 the orthonormal family of functions { f ,e } with e ∈ E(G) and = 1, . . . , d e . As we have seen in Sect. 3 these are the mother wavelets for the orthonormal basis of L 2 ( A , µ A ) given by the functions {S a f ,e }, for varying a ∈ W k,A and k ∈ N. Here an element a = (e 1 , . . . , e k ) ∈ W k,A is a path in the graph G of length k starting at the vertex s(e 1 ). Here we use the same mother functions to construct a family of graph wavelets.
Recall where we keep the same notation for these rescaled coefficients. The p e are the components of the Perron-Frobenius eigenvector Ap = r (A) p.
After fixing a choice of a base vertex v 0 ∈ E(G), we define a measure on the set of vertices of the graph by µ G,v 0 (v 0 ) = 0 and µ G,v 0 (v) := p e 1 · · · p e k , (5.12) where e 1 · · · e k is the shortest path in the graph G starting at v 0 and ending at v. This means that we are considering a random walk on the graph starting at v 0 , where at the first step one has probability p e of moving to the nearby vertex r (e) and probability zero of remaining at v 0 . The measure (5.12) gives the probability of reaching at time k one of the vertices that are k steps away from v 0 .
In addition to fixing the base vertex v 0 , we also fix a choice of an edge e 0 with r (e 0 ) = v 0 . We then define functions · · · c k ,e k−1 e k p e 1 · · · p e k = δ 1 , 1 · · · δ k , k . (5.18) The functions 1 ,..., k , for k ≥ 1, constructed in this way, are supported on concentric regions U k (v 0 ) made of vertices at a distance k from a chosen base vertex v 0 . Unlike other types of graph wavelets constructions where the functions are constant on such concentric regions U k (v 0 ) and average to zero over different k, the ones we obtain here are supported on a single U k (v 0 ) with zero average. In terms of traffic analysis on networks, while one type of graph wavelets may be more suitable in analyzing radial propagation from a vertex, the other may be preferable for directional propagation away from a chosen vertex. In [6,7] one considered, in the setting of Mumford curves with p-adic Schottky uniformization, the Cuntz-Krieger algebras associated to the finite graphs with no sinks obtained from the action of a p-adic Schottky group on the subtree of the Bruhat-Tits tree spanned by geodesics with boundary points on the limit set in P 1 (Q p ). In that context it would be interesting to compare the wavelet constructions described in this paper with the p-adic wavelet theory (see for instance [1]).