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CONFORMAL MARKOV SYSTEMS, PATTERSON-SULLIVAN MEASURE ON LIMIT SETS AND SPECTRAL TRIPLES

For conformal graph directed Markov systems, we construct a spectral triple 
from which one can recover the associated conformal measure via 
a Dixmier trace. As a particular case, we can recover the Patterson-Sullivan 
measure for a class of Kleinian groups.


Introduction
One theme in noncommutative geometry is the recovery of classical geometric data from operator algebraic descriptions of various spaces, formalized through the concept of a spectral triple. This is defined below but, roughly speaking, consists of a C * -algebra acting on a Hilbert space, giving co-ordinates for the space, and a Dirac operator, which encodes geometric information. As a prototypical example, Connes showed how to recover the volume measure and metric on a spin Riemannian manifold [11,12].
As well as manifolds, many fractal sets may be described within this framework. Using ideas of Lapidus and Pomerance [21], Connes gave a construction of a spectral triple for a Cantor subset of the real line, from which the Minkowski content may be recovered, and used this to study the limit set of Fuchsian Schottky groups ( [11], Chapter IV, §3. ). Using a different approach, Connes and Sullivan constructed Patterson-Sullivan measue for a quasi-Fuchsian group as a noncommutative measure ( [11], Chapter IV, §3.γ).
Connes's original approach was generalized and the subject considerably advanced in the last decade by Guido and Isola [16,17,18]. They defined spectral triples associated to fractal subsets of R d obtained as a result of a limit construction. In particular, this includes the limit sets of finite state iterated function systems (IFSs). If the transformations are similarities (so the limit set is self-similar), the δ-dimensional Hausdorff measure, where δ is the Hausdorff dimension, can be recovered as a noncommutative measure. Similar results have been obtained for more general Gibbs measures by Kesseböhmer and Samuel [20,32] and the author [35], where the underlying dynamics is an expanding map on a Cantor set or a (onesided) subshift of finite type. (These should be compared with the constructions in [10].) Falconer and Samuel [15] have begun to investigare multifractal phenomena for Cantor subsets of the real line.
In the self-similar setting, the Hausdorff measure arises as a conformal measure and the recovery of conformal measures will be the subject of this paper. We will do this in the context of conformal graph directed Markov systems (CGDMSs) (defined in section 2), which provide a natural generalization of IFSs.
In greater generality, Palmer [24], following work of Pearson and Bellissard [27], has shown how to construct spectral triples from which the Hausdorff measure (in the Hausdorff dimension) may be recovered. However, his construction is not explicit (in the sense that it depends on the choice of an appropriate resolving sequence of open subsets, the existence of which depends on a fairly arbitrary construction, unrelated to the geometric structure of the set under consideration -see Theorem 5.3.4 of [24]) and, in the cases where the conformal measure is Hausdorff measure, our results can be viewed as a more explicit (in the sense of being given by the Markov structure) version of the construction of spectral triples in [24].
A particularly important example of this is given by the limit sets of a class of Kleinian groups, described below, and the associated Patterson-Sullivan measure. As a special case of our results, we will show that Patterson-Sullivan measure may be obtained as noncommutative measures via a Dixmier trace associated to an appropriate Dirac operator.
We begin by defining spectral triples.
A is a C * -algebra equipped with a faithful * -representation π : A → B(H) (the bounded linear operators on H); (iii) D is an essentially self-adjoint unbounded linear operator on H with compact resolvent and such that {f ∈ A : . This D is called a Dirac operator.
In this introduction, we state our results in the special (but important) case of Kleinian limit sets and Patterson-Sullivan measure. Let Γ be a Kleinian group, i.e. a discrete group of isometries of d-dimensional hyperbolic space H d , d ≥ 2, which we may identify with the unit ball in R d equipped with the Poincaré metric. If d = 2, Γ is called a Fuchsian group. The limit set Λ Γ ⊂ S d−1 is defined to be the accumulation set of the orbit Γo for some o ∈ H d and is independent of the point o chosen. We assume that Γ is non-elementary, i.e. Λ Γ contains more than two (and hence infinitely many) elements. Write C(Γ) ⊂ H d ∪ S d−1 for the (hyperbolic) convex hull of Λ Γ . If the quotient space Γ\(C(Γ) ∩ H d ) is compact then we say that Γ is convex co-compact.
Associated to a Kleinian group Γ is its critical exponent which is independent of the choice of o ∈ H d . For a non-elementary convex cocompact group, 0 < δ = dim Λ Γ , the Hausdorff dimension of Λ Γ [37,38]. Our results will hold for Fuchsian groups and also for the following two classes of Kleinian groups. Definition 1.2. A Kleinian group Γ is called quasi-Fuchsian if it is a discrete faithful representation of the fundamental group of a compact surface [7]. These can be obtained by deforming a Fuchsian group Γ 0 < PSL(2, R) to a discrete group Γ < PSL(2, C). Definition 1.3. A Kleinian group Γ is said to satisfy the even corners condition if Γ admits a fundamental domain R which is a finite sided polyhedron (possibly with infinite volume) such that g∈Γ g∂R is a union of hyperplanes. (This definition was introduced by Bowen and Series [9] for the case d = 2 and studied by Bourdon [6] for d ≥ 3.) Example 1.4. Let K 1 , . . . , K 2k be 2k disjoint d − 1-dimensional spheres in R d , each meeting S d−1 at right angles. For i = 1, . . . , k, let g i be the isometry which maps the exterior of K i onto the interior of K k+i . Then the group Γ generated in this is called a Schottky group and satisfies the even corners condition. Viewed as an abstract group, it is the free group on k generators. In this case, Λ Γ is a Cantor set.
Example 1.5. Let R be a polyhedron in H d with a finite number of faces and with interior angles all equal to π/k, k ∈ N, k ≥ 2. Let Γ be the Kleinian group generated by reflections in the faces of R. Then Γ satisfies the even corners condition. For instance, let R be a regular tetrahedron in H 3 with infinite volume and with dihedral angles π/4. In this case, Λ Γ is a Sierpiński curve [3,6].
The limit set of a Kleinian group supports a natural probability measure µ called Patterson-Sullivan measures [26,36]. If Γ is convex co-compact, this is characterized as the unique non-atomic measures supported on Λ Γ satisfying g * µ = |g | δ µ, for all g ∈ Γ. It is also equal to the normalized restriction of the δ-dimensional Hausdorff measure to Λ Γ [37,38]. (ii) a quasi-Fuchsian group acting on H 3 ; or (iii) a convex co-compact Kleinian group acting on H d , d ≥ 3 with the even corners property and let Λ Γ ⊂ S n denote its limit set. Then there is a spectral triples (A, H, D) from which the Patterson-Sullivan measure on Λ Γ may be recovered.
For a more precise statement, see Corollary 3.2. In the next section, we introduce conformal graph directed Markov systems and conformal measures and define our spectral triple. In section 3, we discuss singular traces and state our main result: that conformal measures may be recovered from a spectral triple. In section 4, we discuss some material on transfer operators. In section 5, we prove Theorem 2.13. In section 6, we complete the paper by proving our result on noncommutative measures and conformal measures, Theorem 3.1.

Conformal Graph Directed Markov Systems
2.1. Graph directed systems. Conformal graph directed Markov systems were introduced by Mauldin and Urbanski as a natural generalization of conformal iterated function systems. The theory they developed is described in [23] and the definitions presented below may be found there.
A (finite) conformal graph directed Markov system (CGDMS) is defined as follows. Let (V, E) be a finite directed graph with vertices V and directed edges E, where we assume E has cardinality #E ≥ 2. For e ∈ E, o(e) ∈ V will denote the initial vertex (the origin) of e and t(e) ∈ V with denote the final vertex (the terminus) of e. We define a zero-one matrix A, indexed by E × E, by We suppose that the following condition holds.
(C1) The matrix A is aperiodic, i.e. that there exists N ≥ 1 such that A N has strictly positive entries.
is the product of a scalar |φ (x)| with an isometry of R d . In addition, φ is a contraction if sup x∈U |φ (x)| < 1. If d = 1, any monotone C 1 map is conformal; in this case we will also need to assume that the derivative φ is Hölder

a map is conformal if and only if it is a Möbius transformation.
For each v ∈ V , we suppose there is a non-empty compact connected set X v ⊂ R d and, for each e ∈ E, a conformal contraction φ e : X t(e) → X o(e) , satisfying the following conditions.
. If d = 1 then, in addition, φ is Hölder continuous. (C5) (Cone property.) There exists 0 < γ < π/2 and l > 0 such that for every x ∈ v∈V X v there exists an open cone with vertex x, central angle of measure γ, and altitude l.
Example 2.1. In the special case where V consists of a single vertex (so that A(e, e ) = 1 for all e, e ∈ E), S is called a conformal Iterated Function System (IFS). There is a large literature devoted to the study of IFSs.
This is well-defined because the intersection is a singleton.
The set Σ + A may be made into a compact metric space by defining, for e = (e n ) ∞ n=1 and e = (e n ) ∞ n=1 , where n(e, e ) = 0 if (e n ) ∞ n=1 = (e n ) ∞ n=1 and n(e, e ) = min{n ≥ 1 : e n = e n } if e = e . We then have the following result, which is a simple consequence of the above definition and the fact that {φ e } e∈E consists of contractions.
We also write X ∅ = v∈V X v .
The following two lemmas may be found in [23] (pages 72-73) and are commonly known as the principle of bounded variation.
Lemma 2.5. There exists K ≥ 1 such that, for all w ∈ W * and all x, y ∈ X t(w) , The next result is an easy consequence of Lemma 2.5.
Lemma 2.6. There exists C > 0 such that, for all w ∈ W * and all x, y ∈ X t(w) , We shall be interested in probability measures on Λ. We have the following definition.
The set Λ = ∞ n=0 T −n U is the limit set of a CGDMS, where the set of vertices V = {1, . . . , k}, the set of edges E is defined by and {φ e : e ∈ E} consists of the local inverses of T between appropriate pairs of X i 's. Provided T is topologically mixing, the conditions (C1)-(C5) are satisfied.
Example 2.10. Generalizing the previous example, we can consider maps T : X i such that T is conformal and expanding on a neighbourhood of each X i and such that T satisfies the Markov property: if T (int(X i ))∩int(X j ) = ∅ then T (X i ) ⊃ X j . The CGDMS is defined as in Example 2.9 and, provided T is topologically mixing, the conditions (C1)-(C5) are satisfied. Example 2.12. The limit sets of convex co-compact Fuchsian groups, quasi-Fuchsian groups and Kleinian groups with the even angle property are examples of limit sets of CGDMS as above. The case of Fuchsian groups is well-known [1,33] and quasi-Fuchsian groups have symbolic dynamics of the same type. For even angle Kleinian groups, we follow [6]. Let R be a polyhedron as specified by the even angle property. Label the faces of R by {R 1 , . . . , R m } and let g i ∈ Γ denote the isometry for which g i R ∩ R = R i . For each i = 1, . . . , m, R i extends to a co-dimension one hyperbolic hyperplane, which divides H d ∪ S d−1 into two half-spaces. Let H i denote the halfspace which does not contain R and let U i = H i ∩ S d−1 . In general, the U i 's will overlap; to obtain a partition we let X = {X 1 , . . . , X k } denote the sets formed by taking the closure of all possible intersections of the interiors of the U i 's.
Choose an arbitrary ordering ≺ on S = {g 1 , . . . , g m }. The map T : and where a i is the ≺-smallest element of {g j1 , . . . , g j l }, is conformal and expanding. If necessary refining a finite number of times by considering intersections of sets in X , T −1 X , . . . , T −n X , for some n ≥ 0, T will satisfy the Markov property: if T (int(X i )) ∩ int(X j ) = ∅ then T (X i ) ⊃ X j . The CGDMS is defined as in Example 2.9 and satisfies (C1)-(C5). The limit set Λ Γ is the limit set of the CGDMS and the Patterson-Sullivan measure is the conformal measure.
2.3. Spectral triples. We will use 2 (W * ) to denote the 2 space Our Hilbert space will be where we write a typical element as ξ = w∈W * ξ 1 (w) ξ 2 (w) and our C * -algebra will be A S = C(Λ, C) with the uniform norm.
The action of A S on H S and the definition of the Dirac operator will depend on the choice of a finite number of points in v∈V X v . More precisely, for each v ∈ V , We define a * -representation π : A S → B(H S ) by setting π(a) to be the multiplication operator We define a Dirac operator D S : .
We have the following theorem, which will be proved in section 5.

Singular traces and noncommutative measures
In order to state our main result, we need to briefly discuss the theory of singular traces of compact operators. For more details, see [2] or [17]. Let B(H) denote the algebra of bounded linear operators on a separable Hilbert space H and let K(H) denote the ideal of compact operators. A singular trace on a two-sided ideal I ⊂ K(H) is a positive linear functional Tr : I → R such that Tr is unitary invariant (the trace property) and vanishes on finite rank operators.
The most important singular traces are the so-called Dixmier traces [14]. These are defined on an ideal I = L 1,∞ (H), the Dixmier ideal, given by where this is a generalized limit corresponding to a state ω on l ∞ , and extended to L 1,∞ (H) by linearity. If the limit lim n→+∞ 1 log n n k=1 a k exists then we say that A is measurable and call the value of the limit the noncommutative integral of A. (There are more general definitions of the Dixmier tracesee, for example, Chapter IV, §2.β of [11], [22] or Chapter 5 of [39]. Correspondingly, there are more general definitions of measurability. It is shown in [22] that these are equivalent to the definition given here. ) We now return to the spectral triple (A S , H S , D S ) defined in the preceding section. For a ∈ A S , the operator π(a)|D S | −s is given by the formula The following is our main result. In particular, it tells us that the conformal measure µ may be recovered from the spectral triple (A S , H S , D S ). The proof will be given in section 6. Theorem 3.1. Let Λ be the limit set of a CGDMS S on R d satisfying conditions (C1)-(C5), and let δ > 0 denote the Hausdorff dimension of Λ. For any a ∈ C(Λ, C), π(a)|D S | −δ ∈ L 1,∞ (H) and, furthermore, π(a)|D S | −δ is measurable and where µ is the δ-conformal measure for S and where c > 0 is independent of a.
In fact, the constant c takes the form We state the special case of limit sets of Kleinian groups and Patterson-Sullivan measure as a separate result.

Corollary 3.2. Let Γ be a non-elementary Kleinian group which is either
(a) a convex co-compact Fuchsian group acting on H 2 ; (b) a quasi-Fuchsian group acting on H 3 ; or (c) a convex co-compact Kleinian group acting on H d , d ≥ 3 with the even corners property, let Λ Γ ⊂ S d−1 denote its limit set and let δ be its critical exponent. Let (A S , H S , D S ) be the spectral triple defined above (depending on the representation of Λ Γ as the limit set of a CGDMS). For any a ∈ C(Λ Γ , C), π(a)|D S | −δ ∈ L 1,∞ (H) and, furthermore, π(a)|D S | −δ is measurable and where µ is the Patterson-Sullivan measure for Γ and where c > 0 is independent of a.
The next result recovers a special case of Theorem 2.10 of [18]. Let φ i (x) = a i x + b i , i = 1, . . . , k, with 0 < a i < 1 and b i ∈ R d , be a finite family of contracting similarities on R d , satisfying the Open Set Condition. Define δ > 0 by k i=1 a δ i = 1. Then δ is equal to the Hausdorff dimension of the self-similar limit set Λ. Furthermore, the conformal measure µ is the solution of and is equal to the normalized δ-dimensional Hausdorff measure.
Corollary 3.3. Suppose φ 1 , . . . , φ k be a finite collection of contracting similarities on R d satisfying the Open Set Condition. Let Λ denote the associated self-similar set and let δ denote its Hausdorff dimension. For any a ∈ C(Λ, C), π(a)|D S | −δ ∈ L 1,∞ (H) and, furthermore, π(a)|D S | −δ is measurable and where µ is the normalized δ-dimensional Hausdorff measure on Λ and where c > 0 is independent of a.
In this special case, one can calculate that f (e) = log a e1 , h δf = 1 and m δf is the (a δ 1 , . . . , a δ k )-Bernoulli measure, so that where h(m δf ) is the measure theoretic entropy of the shift map σ with respect to m δf .

Transfer Operators
In this section we shall discuss some of the ergodic theory associated to the subshift of finite type (Σ + A , σ), where σ : Σ + A → Σ + A is the shift map defined by (σe) n = e n+1 , and hence to the CGDMS S. The main references are [8,25] and, for CGDMSs, [23]. We shall write M σ for the space of σ-invariant probability measures on Σ + A . Given m ∈ M σ , we write h σ (m) ≥ 0 for the entropy of σ as a measure preserving transformation of (Σ + A , m) (see [40] for the definition). For a continuous function f : Σ + A → R, we define its pressure P (f ) by If f is Hölder continuous, then there is a unique probability measure m f , called the equilibrium measure for f , for which this supremum is realized [4,8,25].
Let f ∈ C(Σ + A , R). We define the transfer operator L f : σe =e e f (e ) a(e ).
A key element of our approach will be to relate the eigenvalue asymptotics of our Dirac operators to the spectral properties of transfer operators. For this approach to work, we need a space on which L f acts quasi-compactly.
If f ∈ C α (Σ + A , R) then L f restricts to a bounded linear operator L f : is the Banach space of complex valued Hölder continuous functions of exponent α with the norm The basic spectral properties of L f on this space are contained in the following result, which is Ruelle's generalization of the classical Perron-Frobenius Theorem for non-negative matrices.
has a simple eigenvalue equal to e P (f ) with the rest of the spectrum contained in a disk {z ∈ C : |z| ≤ θe P (f ) }, for some 0 < θ < 1. Furthermore, there exist (i) a strictly positive eigenfunction h f ∈ C α (Σ + A , R) such that L f h f = e P (f ) h f ; and (ii) an eigenmeasure ν f ∈ C(Σ + A , R) * such that L * f ν f = e P (f ) ν f . If ν f is chosen to be a probability measure and the eigenfunction h f is chosen so that h f dν f = 1 then m f = h f ν f is the equilibrium measure for f . This result has two consequences, stated below as corollaries, which will be important for our analysis. There exists λ f < e P (f ) such that, for any a ∈ C α (Σ + A , C), we have Proof. We recall the following basic fact from spectral theory (see, for example, [5] or [19]). Let L : B → B be a bounded linear operator on a Banach space B with spectrum spec(L) = Σ ⊂ C. If Σ can be decomposed into two disjoint non-empty sets Σ 1 and Σ 2 and if γ is a simple closed curve which is disjoint from Σ and which has Σ 1 in its interior and Σ 2 in its exterior then P : B → B defined by is a projection (i.e. P = 1 and P 2 = P). Moreover, B = B 1 ⊕ B 2 , where B 1 = P(B) and B 2 = (I − P)(B) are closed and L-invariant subspaces with spec(L|B 1 ) = Σ 1 and spec(L|B 2 ) = Σ 2 . Now consider the operator L f : . By Theorem 4.1, we may decompose its spectrum into Σ 1 = {e P (f ) } and a disjoint set Σ 2 . Thus, we may decompose the operator L f as a sum where P = ν f (·)h f is the projection onto the eigenspace spanned by e P (f ) . Furthermore, since e P (f ) is strictly maximal in modulus, we have lim n→+∞ L n f (I − P) 1/n = sup{|z| : z ∈ Σ 2 } < e P (f ) .
Choosing λ f such that lim n→+∞ L n f (I − P) 1/n < λ f < e P (f ) completes the proof. Proof. First we note that L f depends analytically on f . The result is then a standard consequence of the fact that e P (f ) is an isolated simple eigenvalue for L f [5,19].
In order to use the transfer operator formalism to study CGDMSs and, in particular, their conformal measures, we need to introduce a one-parameter family of functions of Σ A which capture information related to the set of contractions {φ e } e∈E . We define a function f : Σ + A → R by the formula f (e) = log |φ e1 (Π(σe))|.
Clearly, f is strictly negative. By Lemma 2.4 and the Hölder continuity of Π, f is Hölder continuous. Then the δ-conformal measure for S is µ = Π * ν δf .
To prove Theorem 3.1, we shall need to consider a family of transfer operators L tf , for t ∈ R. By Proposition 4.1, these will have a maximal eigenvalue equal to e P (tf ) . We end the section with a result on the regularity and derivative of the function t → P (tf ). Proof. The analyticity of P (tf ) follows from Corollary 4.3. From Proposition 4.10 of [25] we have that Since f is strictly negative, it follows that P (tf ) is strictly decreasing and the formula is obtained by evaluating at t = δ.

Proof of Theorem 2.13
In this section we prove that the the triple (A S , H S , D S ) constructed above is a spectral triple. The key point is that the Lipschitz functions give a dense subalgebra of A S = C(Λ, C) on which [D S , π(·)] is finite.
Next we prove that D S is self-adjoint. Note that D −1 S is defined by and is a bounded operator. Let ξ, η ∈ H S . Then so that D −1 S is symmetric and, since it is bounded, self-adjoint. By Proposition 2.4 of chapter X of [13], if D −1 S is injective then D S is self-adjoint. However, it is clear that D −1 S ξ = 0 if and only if ξ = 0, so self-adjointness holds. The eigenvalues of D S are the numbers 1 |φ w (x t(w) )| : w ∈ W * (counted with the appropriate multiplicity). In particular, 0 is not an eigenvalue. Thus, the resolvent of D S is compact provided D −1 S is compact and it is clear that D −1 S is a compact operator.
For a ∈ C(Λ, C), Let A 0 denote the subalgebra of Lipschitz functions on Λ, i.e.
6. Proof of Theorem 3.1 We will use the following version of the Hardy-Littlewood Tauberian Theorem, adapted from Proposition 4 in chapter IV, §2.β of [11].
We will use the results of section 4 to analyse ζ a,+ (t) (and hence ζ a,− (t) and ζ a (t)) as t approaches δ. Lemma 6.2. For a ∈ C Lip (Λ, R) with a > 0, ζ a (t) converges for t > δ and where, for each v ∈ V , x v ∈ Σ + A is chosen so that Π(x v ) = x v . Proof. First, we shall obtain the corresponding result for ζ a,+ (t). Write a = a • Π. Then a ∈ C α (Σ + , R), for some α > 0. Provided ζ a,+ (t) converges, we may use the definitions of x t(w) and L tf to write By Theorem 4.1 and Lemma 4.4, for t > δ, L tf has spectral radius e P (tf ) < 1. Thus, using the spectral radius formula, it is easy to see that ζ a,+ (t) converges. Furthermore, by Corollary 4.2, we have  where h t and ν t are the eigenfunction and eigenmeasure for L tf given by Theorem 4.1 and where q n (t) = O(λ n tf ) (with λ tf < e P (tf ) ). Since First, suppose that a ∈ C Lip (Λ, R) and that a ≥ 0. Lemma 6.2 shows that ζ a (t) converges for t > δ and diverges for t = δ. Thus π(a)|D S | −δ ∈ L 1,∞ (H). It follows immediately from Lemma 6.1 and Lemma 6.2 that lim n→+∞ 1 log n n k=1 β k (a) = δ −1 lim t→δ+ (t − δ)ζ a (t) = c a dµ. Now suppose a ∈ C(Λ, R) and a ≥ 0. Given > 0, we may choose a 1 , a 2 ∈ C Lip (Λ, C) such that 0 ≤ a 1 ≤ a ≤ a 2 and a dµ − ≤ a 1 dµ ≤ a 2 dµ ≤ a dµ + . Since > 0 is arbitrary, the required convergence result holds for a.