PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS

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1 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS BERND STRATMANN AND MARIUSZ URBAŃSKI Abstract. In this paper we extend the theory of conformal graph directed Markov systems to what we call conformal pseudo-markov systems. Unlike graph directed Markov systems, pseudo-markov systems may have a countable infinite set of vertices as well as a countable infinite set of edges. After having developed appropriate symbolic dynamics for pseudo-markov systems, we analyse these systems giving extensions of various aspects of the thermodynamical formalism and of fractal geometry. Most important, by establishing the existence of conformal measures along with their invariant versions, we obtain a generalised form of Bowen s formula for the Hausdorff dimension of the limit set of a conformal pseudo-markov system, as well as a formula for the closure of the limit set. Finally, we apply our analysis to the theory of Kleinian groups. Here we obtain the existence of a rather exotic class of infinitely generated Schottky groups of the second kind acting on d + 1)-dimensional hyperbolic space), containing groups with limit sets of Hausdorff dimension equal to any given number t d, whereas their Poincaré exponent can be less than any given positive number s < t. Additionally, the dissipative part of the limit sets of these groups is shown to have further interesting properties. 1. Introduction In this paper we extend the theory of conformal graph directed Markov systems to what we call conformal pseudo-markov systems. These systems can have a countable infinite set of edges, and unlike graph directed Markov systems see [9] for a fairly complete exposition), they may also have a countable infinite set of vertices. Additionally, certain distortion conditions are significantly weakend in the present paper. After having developed suitable symbolic dynamics for pseudo-markov systems, we analyse these systems giving extensions of various aspects of the thermodynamical formalism and of fractal geometry. Most important, by establishing the existence of conformal measures along with their invariant versions, we obtain a generalised form of Bowen s formula [2] for the Hausdorff dimension of the limit set of a conformal pseudo- Markov system, as well as a formula for the closure of the limit set. Finally, we apply our formalism to the theory of Kleinian groups and obtain the existence of an interesting, rather exotic class of infinitely generated Schottky groups of the second kind. Note that there is a significant qualitative difference between pseudo-markov systems with a countable infinite set of vertices and graph directed Markov systems, and the step from a finite to an infinite set of vertices is certainly much more demanding than the step from iterated function systems one vertex) to graph directed Markov systems finitely many vertices). Also, we expect The research of the second author was supported in part by the NSF Grant DMS

2 2 BERND STRATMANN AND MARIUSZ URBAŃSKI that pseudo-markov systems will develop into a powerful geometric theory with interesting applications in for instance complex dynamics and non-commutative geometry. For our applications to Kleinian groups recall that Patterson [13], making use of a careful computational study of Poincaré series of free products of finitely generated Schottky groups acting on d+1)-dimensional hyperbolic space H d+1, showed that there are infinitely generated Schottky groups of the first kind with critical exponent of their Poincaré series arbitrarily close to zero. By applying the theory of conformal pseudo-markov systems developed in this paper, we extend this class of groups to infinitely generated Schottky groups G of the second kind which have the properties that the Hausdorff dimension HDLG)) of their limit sets LG) can be equal to any given number t 0, d], whereas the critical exponent δg) of their Poincaré series can be less than any given positive number s < t. Moreover, these groups have the remarkable additional property that HDLG)) is equal to the Hausdorff dimension of the set G) of accumulation points of the generating balls, whereas the complement within LG) of the G-orbit of G) has already Hausdorff dimension equal to δg). Nevertheless, the main goal of this paper is the development of the theory of conformal pseudo-markov systems. This development was originally motivated by the combinatorics and geometry of the type of Schottky groups just mentioned, where one should note that for these groups the theory of graph directed Markov systems as developed for instance in [8] and [9] is not applicable. More precisely, our analysis of conformal pseudo-markov systems is structured as follows. In Section 2 we extend various aspects of the thermodynamical formalism to shift spaces generated by an infinite alphabet. Most notable, by studying the Perron-Frobenius operator and some relevant pressure functions, we obtain for these systems the existence of pseudo-conformal measures and their invariant versions. In Section 3 we introduce the concept of pseudo-markov systems and give a first analysis of the limit sets of these systems. We then investigate certain families F of functions and introduce the notion pseudo-conformality for a pseudo-markov systems. Using the results of Section 2, we eventually obtain for pseudo-conformal pseudo-markov systems the existence of F -pseudoconformal measures under very weak conditions on the family F. Finally, in Section 4 we introduce conformal pseudo-markov systems. After having discussed some of their main basic properties, especially their pseudo-conformality, we introduce the concept of thinness for a conformal pseudo-markov system. Roughly speaking, thinness implies a rather rapid decay of the sizes of the generators. Our main results in Section 4 are a generalization of Bowen s Formula and the existence of suitable conformal measures. In particular, we obtain that for a thin conformal pseudo-markov system the Hausdorff dimension of its limit set is equal to its hyperbolic dimension, that is the supremum of the Hausdorff dimensions of the limit sets of all finite subsystems. Acknowledgement: We would like to thank I.H.E.S. at Bures-sur-Yvette for the warm hospitality and excellent working conditions while we started with the work towards this paper.

3 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 3 2. Thermodynamical Formalism for Systems with Countable Alphabets 2.1. Symbolic Dynamics. Throughout let E be a countable alphabet which is either finite or infinite, and let A : E E {0, 1} be some arbitrary function, called the incidence matrix. The set of infinite admissible words is given by EA := {ω E : A ωi ω i+1 = 1 for all i IN}. Let EA refer to the set of all finite admissable subwords of EA. Also, the left shift map σ : EA EA is given for ω 1 ω 2... EA by σω 1 ω 2 ω 3... ) := ω 2 ω Occasionally, we will consider this shift to be defined on words of finite length. Given ω = ω 1 ω 2... ω n E, let ω := n denote the length of the word ω, and let EA n refer to the set of all words in EA of length n. If ω = ω 1 ω 2... EA and n IN, then ω n := ω 1... ω n. refers to the initial segment of ω of length n. Finally, for τ E n A we define [τ] := {ω EA : ω n = τ}. Throughout the entire paper we will always assume that the matrix A is finitely irreducible, that is there exists a finite set Ξ EA such that for all a, b E there exists α Ξ such that aαb EA Topological Pressure on the Symbol Space. With the notation from the previous section, consider some arbitrary set D E and some arbitrary function f : DA IR. For n IN, the n-th partition function Z n D, f) is defined by Z n D, f) := exp sup Sn fτ) )), where ω D n A S n f := n 1 j=0 τ [ω] D A f σ j. In case D = E, then we simply write Z n f) instead of Z n E, f). One immediately verifies that the sequence log Z n D, f)) n IN is subadditive. This allows to define the topological pressure of f with respect to the shift map σ : DA DA by 1 P D f) := lim n n log Z nd, f) = inf { 1 n log Z nd, f) : n IN }. 2.1) In case D = E, then we simply write Pf) instead of P E f). Finally, we require the following notion of finite acceptability, which is weaker than the finite accepability introduced in [9].

4 4 BERND STRATMANN AND MARIUSZ URBAŃSKI Definition 2.1. A function f : EA uniformly continuous and IR is called finitely acceptable if and only if f is sup f [e] ) inf f [e] ) < for all e E. Note that the concept of acceptable functions considered in [9] is a slightly stronger. There we assumed additionally that sup{sup f [e] ) inf f [e] ) : e E} <. Theorem 2.2. If f : E A IR is finitely acceptable, then Pf) = sup{p D f) : D E finite}. Proof. The inequality Pf) sup{p D f)} is obvious. In order to prove the reverse inequality, first suppose that Pf) <. Put q := #Λ, p := max{ ω : ω Ξ} and T := min { inf S ω f ) : ω Ξ }. Let ɛ > 0 be fixed. Since f is uniformly continuituous, there exists l N such that fω) fτ) < ɛ, for all ω, τ EA with ω l = τ l. Now, let k IN be fixed. By subadditivity of the sequence log Z n D, f)) n IN, we have that 1 log Z k kf) Pf). Also, note that there exists a finite set D E such that 1 k log Z kd, f) > Pf) ɛ. 2.2) Without loss of generality we can assume that D Ξ. Since f is finitely acceptable, it follows M D := max { sup f [e] ) inf f [e] ) : e D } <. Now, for each τ = τ 1 τ 2... τ n D k A) n there exist elements α1, α 2,..., α n 1 Ξ such that τ = τ 1 α 1 τ 2 α 2... τ n 1 α n 1 τ n E A. Note that the function given by τ τ is at most q n 1 to

5 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 5 1. We then have for each n IN, kn+pn 1) q n 1 i=kn Z i F, f) τ sup f σ j exp τ DA k )n [τ:d] j=0 τ exp inf τ DA k [τ] )n j=0 n exp τ DA k )n = expt n 1)) expt n 1)) f σ j infs k f)) [τi ] + T n 1) i=1 n exp τ DA k )n i=1 n exp τ DA k )n = expt n 1) k l)ɛn M D ln) = e T exp nt k l)ɛ M D l) ) ) infs k f)) [τi ] sups k f)) [τi ] k l)ɛ M D l) i=1 n exp τ DA k )n i=1 Hence, there exists a sequence i n ) n IN such that kn i n k + p)n and sups k f)) [τi ] expsups k f)) [τ] ) τ DA k ) Z in D, f) 1 pn e T exp nt k l)ɛ M D l log q) ) Z k D, f) n. Therefore, using 2.2), we obtain for k sufficiently large, 1 P D f) = lim log Z in D, f) T ɛ + lɛ n i n k k + p M Dl + log p + Pf) 2ɛ Pf) 7ɛ. k By letting ɛ tend to 0, the assertion follows for the case Pf) <. The case Pf) = is obtained in a similar way, and we omit its proof. n. ) 2.3. Perron-Frobenius Operators and Pseudo-Conformal Measures. Throughout this section let f : EA IR be a finitely acceptable function, and assume that f is summable in the following sense. Definition 2.3. A function f : EA IR is called summable if expsupf [e] )) <. 2.3) e E

6 6 BERND STRATMANN AND MARIUSZ URBAŃSKI With C b EA ) referring to the space of bounded continuous functions equipped with the supremums norm, the Perron-Frobenius operator L f : C b EA ) C b EA ) is defined for g C b EA ) and ω = ω 1 ω 2... EA by L f g)ω) := e E Aeω 1 =1 exp feω)geω). Note that the summability of f gives that L f e E expsupf [e] )) <. Also, for each n IN we have L n f g)ω) = exp S n fτω) ) gτω). τ E n A Aτnω 1 =1 The dual operator L f : Cb EA ) Cb EA ) is given by L fµ)g) = µl f g)) = L f g)dµ. Throughout this section we will always assume that m is an f-pseudo-conformal measure. This means by definition that m is an eigenmeasure of L f which is in particular a Borel probability measure on EA. Since L f is a positive operator, we have for the eigenvalue λ = λf) associated with m that λ 0. Also, since L n f m) = λ n m, we clearly have for each g C b EA ), exp S n fτω) ) gτω)d mω) = λ n gd m. 2.4) τ E n A Aτnω 1 =1 Note that 2.4) extends immediately to the space of all bounded Borel functions on EA. Let ω EA, n let B EA be some Borel set such that A ωnτ1 = 1, for every τ = τ 1 τ 2... B, and consider g := 1 ωb. Then 2.4) implies λ n mωb) = exp S n fτρ) ) 1 ωb τρ)d mρ) = = τ E n A Aτnρ 1 =1 {ρ B:A ωnρ1 =1} B exp S n fωρ) ) d mρ). The following lemmata will turn out to be useful. exp S n fωρ) ) d mρ) 2.5) Lemma 2.4. We have that m[e]) > 0, for each e E. Proof. Since EA = e E[e], there exists a E such that m[a]) > 0. Since A is irreducible, for each e E there exists α E such that eαa EA. It then follows from 2.5) that m[e]) m[eαa]) = λ n exp S n feαρ) ) d mρ) > 0. [a]

7 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 7 Lemma 2.5. If f : E A IR is summable and finitely acceptable, then λf) = e Pf). Proof. In order to show that log λ Pf), we apply 2.4) for g = 1 and obtain, for each n IN, λ n = exp S n fτω) ) d mω) τ EA n e E Aτne=1 τ EA n e E τ E n A [e] exp sup S n f [τ] )) m[e]) exp sup S n f [τ] )) = Zn f). Inserting this estimate into the definition of Pf), the assertion follows. In order to show that log λ Pf), let D E be finite. Observe that by Lemma 2.4 we have Q D := min{ m[j]) : j D} > 0. Therefore, an application of 2.4) with g = 1 gives for each n IN, with M D defined as in the proof of Theorem 2.2, λ n exp S n fτω) ) d mω) τ D n A τ DA n = τ D n A e D Aτne=1 [e] exp M D ) exp M D ) exp M D )Q D e D Aτne=1 e D Aτne=1 [e] τ DA n τ DA n = exp M D ) Q D Z n f, D). exp sup S n f )) d m exp sup S n f [τ] )) m[e]) exp sup S n f [τ] )) It follows that log λ Pf), and by inserting this into the definition of Pf), the assertion follows. Our aim is to show that a pseudo-conformal measure exists already under very mild conditions. We start with the following well-known result which is essentially due to Bowen [2]). Lemma 2.6. If the alphabet E is finite and if f : EA IR is continuous and hence summable and finitely acceptable), then there exists an f-pseudo-conformal measure. Proof. Consider the map given by µ L fµ) L f µ) 1).

8 8 BERND STRATMANN AND MARIUSZ URBAŃSKI Clearly, this is a continuous automorphism of the compact convex space of all Borel probability measures on EA into itself. The Schauder-Tychonov Theorem then guarantees the existence of a fixed point, and one immediately verifies that this fixed point is an f-pseudo-conformal measure. With these preparations we now turn to the main result of this section. Note that the proof of the following theorem is similar to the proof of the corresponding result for graph directed Markov systems in [9] Theorem 2.7.3). Theorem 2.7. Every finitely acceptable and summable function f : EA an f-pseudo-conformal measure. IR gives rise to Proof. For ease of exposition we can assume without loss of generality that E = IN. Since A is finitely irreducible, there exists a strictly increasing sequence {D n } n IN of finite sets D n E such that D n = E and Dn Ξ, for all n IN. n=1 Clearly, the incidence matrix restricted to D n D n is irreducible, for each n IN. By Lemma 2.6 we then have that there exists an eigenmeasure m n of the conjugate L n of the Perron-Frobenius operator L n : CD n ) A ) CD n ) A ) associated to the function f Dn). Occasionally, we will view L n as acting on CE A A ), and L n as acting on C EA ). Now, our first aim is to obtain tightness of the sequence { m n } n IN of Borel probability measures on EA. For this let P n := P σ Dn), f ) A D n) A. Obviously, we have P n P 1 for all n IN. Let π k : EA E be the projection onto the k-th coordinate, that is π k e 1 e 2... ) = e k for each k IN. By Lemma 2.5, e Pn is the eigenvalue of L n corresponding to the eigenmeasure m n. Therefore, by applying 2.5), we obtain for each n, k N and every e E, m n π 1 k e)) = ω Dn) k A ω k =e e Pnk m n [ω]) ω Dn) k A ω k =e ω Dn) k A ω k =e exp sups k f [ω] ) P n k ) exp sups k 1 f [ω] ) + supf [e] ) ) Therefore, e P 1k m n πk 1 [e + 1, ))) e P 1k k 1 e [i])) supf e supf [e]). i E e supf [i]) i E ) k 1 e supf [j]). j>e

9 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 9 Now, let ɛ > 0 be fixed. For each k IN, let n k IN be chosen such that ) k 1 e P 1k e supf [j]) ɛ j>n k 2. k e supf [i]) i E Then m n πk 1 [n k + 1, ))) ɛ/2 k, for all n, k IN, and hence, m n [1, n k ] 1 E A k IN k IN m n πk 1 [n k + 1, ))) 1 k IN ɛ 2 k = 1 ɛ. Since EA k IN[1, n k ] is a compact subset of EA, it follows that the sequence { m n } n IN is tight. We can now apply Prokhorov s Theorem, which gives that there exists a weak-limit point m of the sequence { m n } n IN. In order to show that L f m = e Pf) m, let g C b EA ) and ɛ > 0 be fixed. Also, define the normalized Perron-Frobenius operator L 0 := e Pf) L f. For the remainder of the proof let us assume that n IN is chosen sufficiently large such that the following four inequalities are satisfied. g 0 exp supf [i] ) Pf) ) ɛ 6, 2.6) and i>n g 0 exp supf [i] ) ) e P 1 e Pf) e Pn ɛ i E 6, 2.7) m n g) mg) ɛ 3, 2.8) L 0 g)d m L 0 g)d m n ɛ ) Note that 2.7) is satisfied, since by Theorem 2.2 we have that lim n P n = Pf). For n in this range, we define g n := g Dn) and consider the normalized Perron-Frobenius operator A L 0,n := e Pn L n. Let us make the following two observations. L 0,n m n g) = giω) expfiω) P n )d m n ω) and = = E D n) A D n) A m n g n ) m n g) = i n A iωn =1 i n A iωn =1 i n A iωn =1 = L 0,n m n g n ) = m n g n ) D n) A giω) expfiω) P n )d m n ω) g n iω) expfiω) P n )d m n ω) g n g)d m n = D n) A 2.10) 0 d m n = )

10 10 BERND STRATMANN AND MARIUSZ URBAŃSKI Hence, using the triangle inequality, it follows L 0 mg) mg) L 0 mg) L 0 m n g) + L 0 m n g) L 0,n m n g) + + L 0,n m n g) m n g n ) + m n g n ) m n g) + m n g) mg). 2.12) In here we obtain for the second summand, by applying 2.6) and 2.7), L 0 m n g) L 0,n m n g) = = giω) expfiω) Pf)) expfiω) P n ) ) d m n ω) + E A E A i n A iωn =1 i>n A iωn =1 giω) expfiω) Pf))d m n ω) g 0 e fiω) e Pn e Pf) e Pn + g 0 exp supf [i] ) Pf) ) i n i>n 2.13) g 0 exp supf [i] ) ) e P 1 e Pf) e Pn + ɛ i E 6 ɛ 6 + ɛ 6 = ɛ 3. Combining 2.12) with 2.8), 2.9), 2.10), 2.11) and 2.13), it now follows that L 0 mg) mg) ɛ 3 + ɛ 3 + ɛ 3 = ɛ. By letting ɛ tend to 0, we conclude that L 0 mg) = mg), or alternatively L f mg) = e Pf) mg). Since g C b E A ) was assumed to be arbitrary, we have now shown that L f m = e Pf) m. Definition 2.8. A summable and finitely acceptable function f : EA distorted if and only if there exists a constant Q > 0 such that IR is called boundedly if τ n+1 = ω n+1 for some ω, τ E A and n IN, then S n fω) S n fτ) Q. 2.14) Also, we require the following notation. If ω = ω 1... ω n E n A for some n IN, then we put Σ ω := {τ E A : A ωnτ 1 = 1}, and let σ n ω : Σ ω E A refer to the map which is given for τ Σ ω by σω n τ) := ωτ. Remark 2.9. Note that the notion boundedly distorted represents a slightly weaker condition than the type of bounded distortion used in connection with graph directed Markov systems. More precisely, for the analysis of graph directed Markov systems in [9] it was sufficient to assume that S n fω) S n fτ) Q as long as the first n entries of τ and ω coincide. Whereas

11 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 11 here, where we allow an infinite set of vertices, it is necessary to demand that τ and ω coincide at their first n + 1 entries. The following theorem gives the final result of this section. We remark that a similar result was obtained in [9] for graph directed Markov systems. The difference is that here we do not necessarily have that the Radon-Nikodym derivative of m with respect to µ is bounded from below and above. Theorem Let m be an f-pseudo-conformal measure, for some summable, finitely acceptable and boundedly distorted function f : EA IR. Then there exists a shift-invariant Borel probability measure µ which is equivalent to m. Proof. Fix e E, n IN, and τ = τ 1... τ n EA n such that A τne = 1. Since m is f-pseudo-conformal, it follows for each Borel set B [e], mσ τ n B)) = exp S n fτρ) Pf)n ) d mρ) e Q exp inf )) S n f [τe] mb) B 2.15) and mσ τ n B)) = exp S n fτρ) Pf)n ) d mρ) exp inf )) S n f [τe] mb). B 2.16) If in particular B = [e], then 2.15) reads as mσ τ n [e])) e Q exp inf S n f [τ [e])) n m[e]). Therefore, in this case we have exp inf S n f [τ n [e]) Similarly, using 2.16), we obtain )) e Q mσ n τ [e])) m[e]) exp inf S n f [τ n [e]) Inserting 2.18) into 2.15) and 2.17) into 2.16), we obtain and Q m[τe]) = e m[e]). 2.17) )) m[τe]) m[e]). 2.18) mσ τ n B)) eq m[τe]) mb) 2.19) m[e]) mσ τ n B)) e Q m[τe]) mb). 2.20) m[e]) By summing 2.19) and 2.20) over all τ E n A for which A τne = 1, we conclude that e Q m[e]) m[τe]) mb) mσ n B)) eq m[τe]) mb). 2.21) m[e])

12 12 BERND STRATMANN AND MARIUSZ URBAŃSKI Now, with L referring to a Banach limit defined on the Banach space of all bounded sequences of real numbers, let µ be defined by µa) := L mσ n A)) ) ), for each Borel set A n IN EA. One immediately verifies that µ is a finite, non-trivial, σ-invariant and finitely additive function defined on the σ-algebra of all Borel sets of EA. Furthermore, using 2.21), we obtain for all e E and arbitrary Borel sets B [e], e Q m[τe]) mb) µb) eq m[e]) m[τe]) mb). m[e]) Hence, using the fact that m is a countably additive measure, it follows that µ must also be countably additive. This finishes the proof. 3. Pseudo-Markov Systems 3.1. Preliminaries for Pseudo-Markov Systems. In this section we introduce the concept pseudo-markov system and then give a first analysis of the limit sets of these systems. Note that pseudo-markov systems may have an infinite set of vertices, and therefore these systems represent a significant extension of graph directed Markov systems. As in the previous section, we always assume that E is a countable alphabet, which is either finite or infinite, and that A : E E {0, 1} is some given finitely irreducible incidence matrix. Definition 3.1. Let Y, ρ) be a bounded metric space and let {X e : e E} be a family of compact subsets of Y. A set S = {φ e : e E} of continuous injections φ e : X e Y is called a pseudo-markov system if the following conditions are satisfied. a) The maps in S are uniform contractions. That is, there exists a constant 0 < s < 1 such that ρφ e y), φ e x)) sρy, x) for all e E, x, y X e. b) Open Set Condition) If a, b E and a b, then c) Markov Property) If a, b E, then φ a IntX a )) φ b IntX b )) =. either φ a IntX a )) IntX b )) = or φ a X a ) X b. d) If a, b E such that A ab = 1, then φ b X b ) X a. For a finite word ω = ω 1 ω 2... ω n E A, we write φ ω := φ ω1 φ ω2... φ ωn : X ωn φ ω1 Xω1 ).

13 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 13 In order to define the limit set of a pseudo-markov system S, observe that by condition d) in Definition 3.1, we have for ω = ω 1 ω 2... E A that the sequence φ ω n Xωn ))n IN is a descending family of compact subsets of Y. Also, by condition a) in Definition 3.1, we have lim n diam φ ω n Xωn )) = 0. Combining these two observations, it follows that the intersection ) n=1 φ ω n Xωn is a singleton which will be denoted by πω). Clearly, this defines the coding map π : EA Y. Note that by assuming that EA is equipped with a suitable metric, one immediately verifies, using a), that π is Hölder continuous. The limit set J S of a pseudo-markov system S is then defined by J S := π ) EA. The remainder of this section is devoted to investigations of the geometry, dynamics and topology of this set. In the following a pseudo-markov system S is said to be of finite multiplicity if and only if T := sup {card{e E : x φ e X e ))}} <. x Y The following lemma shows that finite multiplicity is sufficient to ensure that the geometry of the iterations of a pseudo-markov system is compatible with the geometry arising from the coding. For ease of exposition, in here we use the notation X ω := X ωn for ω = ω 1... ω n E n A, n IN. Lemma 3.2. If S is a pseudo-markov system of finite multiplicity, then J S = φ ω X ω ). n IN ω EA n Proof. Let x J S be fixed. Then there exists ω EA such that x = πω). By definition of J S, it hence follows that x φ ω n X ω ) τ EA n φ τx τ ), for all n IN. This implies that J S φ ω X ω ). n IN ω EA n Note that so far we did not use the assumption finite multiplicity ). For the opposite inclusion, let y n=1 ω EA n φ ω X ω ), and for each n IN consider the set E n y) := {ω E n A : y φ ω X ω )}. We then have that ω n E n y), for every ω E n+1 y). This shows that the set {E n y) : n IN} can be viewed as a tree. Since carde n y)) T n <, Ramsey s Theorem gives that there exists ω E A such that ω n E n y), for each n IN. This implies that ω E A and that y = πω).

14 14 BERND STRATMANN AND MARIUSZ URBAŃSKI Finally, we now briefly comment on a topological aspect of the limit set of a pseudo-markov system S. For this we introduce the boundary set S) := { lim φ en x n ) : lim e n = for some arbitrary elements x n X en }. n n The following result will be important in the final section where we will use pseudo-markov systems to study limit sets of infinitely generated Schottky groups. Proposition 3.3. If S is a pseudo-markov system such that lim e E diam φ e X e ) ) = 0, then J S = J S ) φ ω S) Xω. ω E A Proof. Since the matrix A is finitely irreducible, for each e E there exists ω EA such that eω EA. Therefore, using the assumption lim e E diam φ e X te) ) ) = 0, we deduce that S) J S. Combining this observation and the continuity of the maps φ ω which implies ) ) ) that φ ω S) Xω φω JS X ω φω JS X ω JS ), it follows that J S J S ) φ ω S) Xω. ω E A In order to prove the reverse inclusion, let x J S be fixed. Then there exists a sequence ω n) ) n IN of elements ωn) E A such that x = lim n πω n) ). For each k IN, define E kx) := {ω n) k : n k} and E 0x) :=, and note that if τ E k+1x) then there exists γ E kx) such that τ k = γ. This shows that the set {E kx) : k IN} can be viewed as a tree rooted at the vertex E 0x). We now distinguish the following two cases. First, suppose that there exists k IN such that E kx) has infinitely many elements, and define q := min{k 0 : E kx) is infinite}. Note that q 1, and that the set E q 1x) is finite and non-empty although it might be equal to the singleton { }). Then there exists τ E q 1x) EA and an infinite sequence ) n ω j ) q of distinct elements of E such that j IN τωn j) q = ω nj) q, for all j IN. By passing to a subsequence if necessary, we may assume without loss of generality that the sequence πσ q 1 ω nj) )) ) converges to a point y X τ. It follows that y S), and furthermore, j IN x = lim j π ω n j) ) = lim = φ τ lim j πσq 1 ω n j) )) π τσ q 1 ω nj) )) ) = lim φ τ πσ q 1 ω nj) )) ) j j ) = φ τ y) φ τ S) X τ ). This gives the assertion in this first case. Now suppose that the set E kx) is finite for each k IN. Since, as mentioned before, these sets form a tree rooted at E 0x), Ramsey s Theorem implies that there exists an infinite path ω E A such that ω k E kx), for each k

15 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 15 IN. Hence, there exists an increasing sequence n k ) k IN such that ω k = ω n k) k, for all k IN. It follows that πω), πω n k) ) φ ω k X ωk ). Combining this with the fact that x = lim k π ω n k) ) and lim k diam φ ω k X ωk ) ) = 0, we conclude that x = πω) J S. This completes the proof of the proposition Summable Families of Functions. Definition 3.4. A family F = {f e) : X e IR e E} of continuous functions f e) is called acceptable if the function F : {a} φ b X b ) IR, given by F e, x) := f e) x), a E b E A ab =1 is uniformly continuous with respect to the metric inherited from the Cartesian product E Y, where E is assumed to be equipped with the discrete metric. Since X e is compact for each e E, it follows that each function f e) of an acceptable family F is uniformly continuous and of bounded oscilation, where bounded oscilation means that supf e) ) inff e) ) < ) Note that the analogous concept of bounded oscilation for graph directed Markov systems in [9] required that in the formula above the supremum over all e E is finite. To link the current setting with the previous chapter we now introduce the following special potential function. A function f : EA IR is called amalgamated function induced by F if and only if fω) = f ω 1) πσω))) for all ω = ω 1 ω 2... E A. Our convention will be to use lower case letters for the amalgamated function induced by a given summable family of functions. By combining acceptability of the family F and formula 3.1), we immediately obtain the following result. Proposition 3.5. If a family of funtions F is acceptable, then the amalgamated function f induced by F is finitely acceptable. Throughout, a family F = {f e) : X e IR e E} is called summable if and only if exp supf e) ) ) <. 3.2) e E The topological pressure PF ) of the family F is given by 1 PF ) := lim n n log exp sup S ω F ) )), ω EA n

16 16 BERND STRATMANN AND MARIUSZ URBAŃSKI where the function S ω F ) : X ω IR is defined by n S ω F ) := f ωj) φ σ j ω for ω = ω 1 ω 2... EA. A straightforward calculation shows that if F is summable then j=1 PF ) log e E expsupf e) )) <. A further link between the symbolic dynamics and pseudo-markov systems is given by the following. Proposition 3.6. If F is an acceptable family of functions and f is the amalgamated function induced by F, then PF ) = Pf). Proof. First note that for each ω = ω 1... ω n EA n with n IN arbitrary, and for every τ = τ 1 τ 2... EA with A ωnτ1 = 1, we have n n n 1 S ω F )πτ)) = f ωj) φ σ j ωπτ)) = fσ j 1 ωτ) = fσ j ωτ) = S n fωτ). j=1 j=1 j=0 3.3) Hence, exp sup S ω F ) )) exp sup{s n fωτ) : τ E A, A ωnτ 1 = 1} ) = exp sup S n f [ω] )). This implies that PF ) Pf). The opposite inequality is more delicate. Similar as in the proof of Theorem 2.2, one verifies that PF ) = sup{p D F ) : D E finite}. Fix a finite set D E. Since for each e D the function f e) : X e IR is uniformly continuous and of finite oscilation, and since diam φ ω X ω ) ) s ω diamy ), it follows that for every ɛ > 0 there exists a constant C ɛ > 0 such that, for all ω D A and x, y X ω, exp S ω F )y) ) C ɛ e ɛ ω exp S ω F )x) ). 3.4) Now fix ɛ > 0, ω = ω 1... ω n EA n with n IN arbitrary, and some τ = τ 1 τ 2... DA A ωnτ1 = 1. Using 3.3) and 3.4), we obtain for each x X ωn, exp S ω F )x) C ɛ e ɛn exp S ω F )πτ)) ) = C ɛ e ɛn exp S n fωτ) ) C ɛ e ɛn exp sups n f [ω] ) ). It follows that exp sup S ω F ) )) C ɛ e ɛn exp sups n f [ω] ) ). Hence, by recalling the definition of the pressure function and by letting ɛ tend to 0, we deduce PF ) Pf). This completes the proof. with Definition 3.7. With the notation above, a Borel probability measure m on Y is called an F -pseudo-conformal measure if m is supported on the limit set J S and the following two conditions are satisfied.

17 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 17 i) For every ω EA and each Borel set B X ω, we have mφ ω B)) = exp S ω F ) PF ) ω ) dm. 3.5) B ii) For all incomparable words ω, τ EA, we have m φ ω X ω ) φ τ X τ ) ) = ) A simple inductive argument shows that for m to be an F -pseudo-conformal measure, instead of 3.5) and 3.6) it is enough to require that for every e E and for each Borel set B X e, we have mφ e B)) = exp f e) PF ) ) dm, 3.7) as well as, for all a, b E such that a b, B mφ a X a ) φ b X b ) ) = ) In the following auxiliary result we use the notation ˆX a := b E A ab =1 φ b X b ) X a for a E. Lemma 3.8. With the notation above, let m be a Borel probability measure on the limit set J S of the pseudo-conformal Markov system S. If m satisfies 3.6) and if 3.5) holds for all Borel sets A ˆX e with e E, then m is an F -pseudo-conformal measure. Proof. Since X e \ ˆX e ) J S = for each e E, we have that mx e \ ˆX e ) = ) Let x φ e Xe \ ˆX e ) JS be fixed, for some e E. Then x = πω) = φ ω1 πστ))), for some ω EA with ω 1 e. Hence, it follows that J S φ e Xe \ ˆX ) ) e b e φ b X b ) φ e Xe, and consequently 3.8) implies m φ e Xe \ ˆX )) e = m JS φ e Xe \ ˆX )) e m φ b X b ) φ e X e ) ) = 0. Therefore, using 3.9), we obtain by induction, for each ω E A and every B X ω, b e mφ ω B)) = m )) φ ω B ˆXω = exp B X S ω F ) PF ) ω ) dm ω = exp S ω F ) PF ) ω ) dm. B 3.10)

18 18 BERND STRATMANN AND MARIUSZ URBAŃSKI In the following a family of functions F = {f e) : X e IR e E} is called boundedly distorted if and only if there exists a constant Q > 0 such that for all e E and ω E A with A ωe = 1, sup S ω F ) φex e)) inf Sω F ) φex e)) Q. 3.11) Lemma 3.9. If F is a summable and boundedly distorted family of functions, then the amalgamated function induced by F is finitely acceptable and boundedly distorted. Proof. The assertion is an immediate consequence of 3.3). A finite word ωτ E is called a pseudo-code of an element x Y if and only if ω, τ E A such that φ τ X τ ) X ω and x φ ω φ τ X τ )). Note that the word ωτ is not required to belong to E A. If in here we do not want to specify the element x, we simply say that ωτ is a pseudo-code. Similar as for elements of E A, two pseudo-codes are called comparable if one of them is an extension of the other. Also, two pseudo-codes ωτ and ωρ of an element x Y are said to form an essential pair of pseudo-codes of x if τ ρ and τ = ρ. Finally, the essential length of the essential pair of pseudo-codes ωτ and ωρ of x is defined to be equal to ω. With these preparations we can now define the notion of a conformal like pseudo-markov system. Namely, a pseudo-markov system S is said to be conformal-like if and only if no element of Y admits essential pairs of pseudo-codes of arbitrarily long essential lengths. One easily verifies that a system which satisfies the strong separation condition that is φ a X a ) φ b X b ) =, for all a, b E with a b) is always conformal-like. Also, in the next section we will see that every conformal pseudo-markov systems is conformal-like. Now, the following theorem gives the main result concerning conformal-like systems and summable families of functions. Theorem Let S = {φ e : e E} be a conformal-like pseudo-markov system. If F is a summable and boundedly distorted family of functions, then there exists an F -pseudoconformal measure m F on J S. Moreover, we have that m F = m F π 1, where m F is the f-pseudo-conformal measure whose existence follows from Theorem 2.6 and Lemma 3.9. Proof. Let m F = m F π 1. In order to show that m F is an F -pseudo-conformal measure, assume by way of contradiction that m F φ ρ X ρ ) φ τ X τ )) > 0, for two incomparable words ρ, τ EA. Without loss of generality we can assume that ρ and τ are of the same length, say q IN. Then define, for n IN, V := φ ρ X ρ ) φ τ X τ )) and V n := φ ω X ω V ). Since each element of V n admits at least one essential pair of pseudo-codes of essential length equal to n, and since the system S is confomal-like, it follows that V n =. k=1 n=k ω E n A

19 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 19 On the other hand, we have V n πσ n π 1 V ))), which implies that π 1 V n ) σ n π 1 V )). By Theorem 2.10 we know that there exists a σ-invariant Borel probability measure µ F equivalent to m F. Hence, µ F π 1 V n )) µ F σ n π 1 V ))) = µ F π 1 V ) > 0, and therefore, µ F π 1 k=1 n=k V n ) µ F π 1 V ) > 0. In particular, this gives that k=1 n=k V n, and hence we obtain a contradiction. This shows that property 3.6) is satisfied. In order to prove 3.5), fix ω = ω 1... ω n E n A as well as a Borel set B X ω, and define π 1 ω B) := {τ π 1 B) : A ωnτ 1 = 1}. Consider some element τ = τ 1 τ 2... π 1 B) \ π 1 ω B). We then have A ωnτ 1 = 0 and πτ) B X ω. This implies that πτ) = φ e X e ), for some e E with A ωne = 1. Hence, e τ 1 and πτ) φ e X e ) φ τ1 X τ1 ), and therefore, π π 1 B) \ π 1 ω B) ) e b φ e X e ) φ b X b ). By combining this inclusion with 3.6), we obtain m F π 1 B) \ π 1 ω B) ) m F π 1 φ e X e ) φ b X b ) e b = m F φ e X e ) φ b X b ) = 0. e b 3.12) Note that one immediately verifies that π 1 φ ω B)) σ n ω Now, consider some element ρ π 1 φ ω B)) \ σ n ω A ωne = 1, φ e x) B and ρ / σ n ω π 1 ω πω 1 B). πω 1 B)). Then πρ) = φ ωe x), where B). If ρ n+1 = ωe then x = πσ n+1 ρ), and it follows that πeσ n+1 ρ) = φ e πσ n+1 ρ)) = φ e x). Consequently, ρ = ωeσ n+1 ρ) σω n hence ρ n+1 ωi. This shows that π 1 φ ω B)) \ σ n ω π 1 ω B) ) π 1 τ,η E n+1 A τ η Using 3.6) and the definition of the measure m F, it follows that m F π 1 φ ω B)) \ σ n ω π 1 ω B) ) = 0. φ τ X τ ) φ η X η ). π 1 ω B), and

20 20 BERND STRATMANN AND MARIUSZ URBAŃSKI Eventually, by combining the latter equality, 3.12) and 2.5), we obtain m F φ ω B)) = m F π 1 φ ω B)) = m F σ n ω πω 1 B) ) = exp S n fωρ) Pf)n ) d m F ρ) = = = πω 1 B) πω 1 B) π 1 B) B exp S ω F πρ)) PF )n ) d m F ρ) exp S ω F πρ)) PF )n ) d m F ρ) exp S ω F x) PF )n ) d m π 1 x) = B exp S ω F PF )n ) dm F. This shows that 3.5) is satisfied, and hence an application of Lemma 3.8 then completes the proof of the theorem. 4. Conformal Pseudo-Markov Systems 4.1. Preliminaries for Conformal Pseudo-Markov Systems. In this section we consider a particular class of pseudo-markov systems which we call conformal pseudo-markov systems. These systems are represented in some Euclidean space IR d, and this allows to study the fractal and topological structure of their limit sets. Before giving the definition of a conformal pseudo-markov system, recall that a set X IR d is called K-quasiconvex, for some K 1, if for every pair of points x, y X there exists a piecewise smooth path in X joining x and y of length less than or equal to K x y. Definition 4.1. Let S = {φ e : X e Y e E} be a pseudo-markov system. Then S is called a conformal pseudo-markov system if and only if Y is a compact subset of the Euclidean space IR d, for some d IN, and if the following conditions are satisfied. 1) Quasi-Convexity) There exists a constant K 1 such that X e and φ e X e ) are K- quasi-convex, for all e E. 2) Derivative Decay) lim e E φ e = 0. 3) Cone Condition) There exists θ 0, π/2) such that for all e E and x X e, there exists an open cone Conex, θ) with vertex x and central angle θ such that Conex, θ) Intφ e X e ) ), the interior being taken with respect to the Euclidean topology of IR d. 4) Conformal Extension) For each e E, there exists an open connected set V e Y such that X e V e, and such that the map φ e : X e Y extends to a C 1+ɛ -conformal diffeomorphism from V e to Y. 5) Pre-Distortion) There exist L 1 and α > 0 such that for all ab EA 2 and all x, y φ b X b ), φ ay) φ y x α ax) L ) 1. φ a φb X b )

21 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 21 Let us first collect a few geometric consequences which arise from the Pre-Distortion condition. Lemma 4.2. If S = {φ e : X e Y e E} is a conformal pseudo-markov system, then for all ω EA and b E such that ωb EA, log φ ωy) log φ ωx) L1 s α ) 1 y x α, for all x, y φ b X b ). Proof. Let n := ω, and define z 0 := z and z k := φ ωn k+1 φ ωn k+2 φ ωn z), for arbitrary z φ b X b ) and for all 1 k n. Using the Pre-Distortion condition 5) in Definition 4.1, it follows for x, y φ b X b ), log φ ωy) ) log φ ωx) ) = n log 1 + φ ω j y n j ) φ ) ω j x n j ) j=1 φ ω j x n j ) n φ ω j ) 1 φ ωj y n j ) φ ω j x n j ) j=1 n L y n j x n j α j=1 n L s αn j) y x α L1 s α ) 1 y x α. j=1 4.1) Let us define Q := L1 s α ) 1 diamy )) α. An immediate consequence of the previous lemma is that each conformal pseudo-markov system has the following property. Bounded Distortion Property. For all ω EA and b E such that ωb EA, we have for all x, y φ b X b ), e Q φ ωy) φ ωx) eq. 4.2) The following lemma gives a sufficient condition for the Pre-Distortion property to hold. Lemma 4.3. Suppose that S = {φ e : X e Y e E} is a pseudo-markov system, with d 2, satisfying conditions 1)-4) in Definition 4.1, and suppose that there exists κ > 0 such that dist φ b X b ), IR d \ X a ) κ diam φb X b ) ) for all ab E 2 A. Then S satisfies the Pre-Distortion property 5), and hence S is a conformal pseudo-markov system.

22 22 BERND STRATMANN AND MARIUSZ URBAŃSKI Proof. The assertion follows immediately from Theorem and Theorem in [9]. Proposition 4.4. If S is a conformal pseudo-markov system, then J S = J S ) φ ω S) Xω. ω E A Proof. Using the Mean Value Inequality, conditions 1) and 2) in Definition 4.1, and the assumption that Y is bounded, we immediately obtain that lim diam φ e X e ) ) = ) e E Hence, we can apply Proposition 3.3, which then gives the assertion. Lemma 4.5. There exists β > 0 such that φ ω φτ Xτ )) contains an open central cone with vertex φ ω φ τ x)) and angle measure β, for each x Y and every pseudo-code ωτ of x. Proof. By using the Cone Condition 3) in Definition 4.1, the proof is exactly the same as the proof of formula 4.14) in [9], and will therefore be omitted. One easily verifies that φ ω φ ρ Int Xρ )) φτ φ γ Int Xγ )) =, for any two incomparable pseudo-codes ωρ and τ γ. Hence, as an immediate consequence of the previous lemma we obtain the following corollary. In here λ d 1 refers to the d 1)-dimensional Lebesgue on the unit sphere S d 1 IR d. Corollary 4.6. With β as in the previous lemma, we have that for each x Y there are at most λ d 1 S d 1 )/β mutually incomparable pseudo-codes of x. Lemma 4.7. Every conformal pseudo-markov system S is a conformal-like system. Proof. Suppose on the contrary that there exists a point x Y with the following properties. There exists an infinite increasing sequence n k ) k IN of positive integers, and there exist words ω k), τ k), ρ k) E A, for each k IN, such that φ τ k) Xτ k)) φρ k) Xτ k)) Xω k), such that the words τ k) and ρ k) are incomparable, and as well as lim k ωk) =, 4.4) x φ ω k) φ τ k) Xτ k)) φω k) φ ρ k) Xρ k)) for all k IN. We now construct by induction for each n IN a set C n which contains at least n + 1 mutually incomparable pseudo-codes of x. The existence of such a set for large n will then clearly contradict the statement in Corollary 4.6, and hence will finish the proof. Define C 1 := {ω 1) τ 1), ω 1) ρ 1) },

23 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 23 and suppose that the set C n has been obtained, for some n IN. In view of 4.4), there exists k n IN such that ω kn) > max{ ξ : ξ C n }. 4.5) If ω kn) ρ kn) does not extend any word from C n, then we have by 4.5) that ω kn) ρ kn) is not comparable with any element of C n. We then obtain C n+1 from C n by adding the word ω kn) ρ kn) to C n. Similarly, if ω kn) τ kn) does not extend any word from C n, then form C n+1 by adding ω kn) τ kn) to C n. On the other hand, if ω kn) ρ kn) extends an element α C n and ω kn) τ kn) extends an element β C n, then we obtain from 4.5) that α = ω kn) α and β = ω kn) β. Since C n consists of mutually incomparable words, this implies that α = β. Now, form C n+1 by removing α = β from C n and then adding both ω kn) ρ kn) and ω kn) τ kn). Note that no element γ C n \ {α} is comparable with ω kn) ρ kn) or ω kn) τ kn), since otherwise γ = ω kn) γ, and consequently, γ would be comparable with α. Since also ω kn) ρ kn) and ω kn) τ kn) are not comparable, it follows that C n+1 consists of mutually incomparable pseudocodes of x. This completes our inductive construction, and hence finishes the proof Pressure Functions, Conformal Measures and Bowen s Formula. Throughout this section let S = {φ e : X e Y : e E} be a given conformal pseudo-markov system. For each t 0, consider the following family of functions tlog := {t log φ e : e E}. Lemma 4.8. tlog is an acceptable and boundedly distorted family of functions, for each t 0. Proof. The assertion is an immediate consequence of Lemma 4.2. Let ζ : EA IR be the amalgamated function induced by the family Log, that is for every ω = ω 1 ω 2... EA we have ζω) := log φ ω 1 πσω)). Then tζ is the amalgamated function induced by the family tlog, for each t 0. Hence, by combining Proposition 3.6 and Lemma 4.8, we have Ptζ) = PtLog), and in the sequel we will simply write Pt) to denote their common value. For the following theorem note that by condition 2) Derivative Decay) in Definition 4.1, we have that there exists a unique number θs) [0, ) such that the family tlog is summable for all t > θs), and not summable for all t < θs). Clearly, PθS)) < if and only if θs)log is summable. Finally, recall that a Borel probability measure m t on J S is a tlog-pseudo-conformal measure if and only if m t φω X ω ) φ τ X τ ) ) = 0 for all incomparable words ω, τ E A, 4.6)

24 24 BERND STRATMANN AND MARIUSZ URBAŃSKI and m t φ ω B)) = If Pt) = 0, then the measure m t is called t-conformal. B e Pt) φ ω t dm t for all ω E A, and B X ω Borel. 4.7) Theorem 4.9. If tlog is a summable family, and hence in particular if t > θs), then there exists a tlog-pseudo-conformal measure m t on J S. Proof. The assertion follows immediately by combining Theorem 3.10, Lemma 4.7 and Lemma 4.8. The following proposition collects some of the basic properties of the pressure function t Pt). These properties are immediate consequences of the Hölder inequality and the fact that φ e s < 1 for all e E, and we omit the proof. Proposition For the pressure function P : [0, ) IR of a conformal pseudo-markov system S, the following holds. a) P is decreasing. b) P [θs), ) is strictly decreasing. c) lim t Pt) =. d) P [θs), ) is convex, and hence, continuous. For ease of exposition, let us introduce the following two parameters. h = hs) := HDJ S ) and h = h S) := sup{hdj SD )) : D E finite}, where S D := {φ e : e D}. Also, we introduce the concept of thinness, which will turn out to be crucial in the following. Definition A conformal pseudo-markov system S is called thin if and only if θs) < h S). We remark that an equivalent formulation of thinnes is that there exists a positive constant κ = κs) < h S) such that φ e κ <. e E Note that thinness implies in particular that φ e h <. 4.8) e E Furthermore, if for instance θs) = 0, then the underlying pseudo-markov system is thin. The following auxiliary result will turn out to be useful in the proof of the next theorem.

25 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 25 Lemma If S is a finite irreducible) conformal pseudo-markov system, then PhS)) = 0. Proof. Since 0 P0) <, Proposition 4.10 implies that there exists a unique η 0 such that Pη) = 0. Let m η be the associated η-conformal measure see Theorem 4.9). Note that the system S = {φ a : φ b X b ) φ a X a ) a, b E, A ab = 1} satisfies all the requirements of a graph directed Markov system see [9], page 71-72). Also note that the limit sets of S and S coincide, and that the measure m η is an η-conformal measure also for S. The proof now follows from Theorem in [9], where primitivity can easily be replaced by irreducibility. The following theorem represents the main result of this section. The theorem gives a extension of Bowen s formula [2] to the setting of thin conformal pseudo-markov systems. Theorem If S is a thin conformal pseudo-markov system, then hs) is the unique zero of the pressure function P, and hs) = h S). Proof. Put h := hs), h := h S), and u := inf{s 0 : Ps) 0}. 4.9) Our first aim is to show that h u. For this, fix t > u and observe that Proposition 4.10 implies that Pt) < 0. Since X e is K-quasi-convex for all e E see Definition 4.1 1)), we have by definition of P, for all n IN sufficiently large, diam φω X ω ) )) t K t φ ω t diamx ω ) ) t ω E n A ω E n A K t diamy ) ) t φ ω t K t diamy ) ) t exp 1 2 Pt)n ). ω E n A Letting n tend to infinity and using the fact that by Lemma 3.2 we have that {φ ω X ω ) : ω E n A} is a covering of J S, we conclude that the t-dimensional Hausdorff measure of J S vanishes. This shows that h t, and hence by letting t tend to u from above, it follows that h u. 4.10) Our next aim is to show that u h. For ease of exposition and without loss of generality, let us assume that E = IN. Since A is finitely irreducible, there exists p IN sufficiently large such that Ξ {1, 2,..., p}. For k p, let S k := S {1,...,k} and let h k := hs k ). One immediately verifies that lim k h k = h. Note that by Lemma 4.12 and Theorem 4.9, for every k p there exists an h k -conformal measure associated with S k. Also, since the system S is thin, there exists 0 < κ < h and q p such that φ e κ < 1 2. j=q+1

26 26 BERND STRATMANN AND MARIUSZ URBAŃSKI Finally, by choosing p sufficiently large, we can always assume without loss of generality that p min{m : h m κ}. With these preparations we now obtain for k q, using the h k -conformality of the measure m k, 1 = k q m k φ j X j )) = m k φ j X j )) + k m k φ j X j )) j=1 j=1 j=q+1 q m k φ j X j )) + k φ j h k m k X j ) j=1 j=q+1 q k m k φ j X j )) + φ j κ 1 q j=1 j=q m k φ j X j )). j=1 This implies that q j=1 m k φ j X j )) 1/2, and hence there exists a {1, 2,..., q} such that m k φ a X a )) 1 2q. 4.11) Since k q p, for each e {1, 2,..., k} there exists ρ Ξ such that eρa is an admissable word in S k. Therefore, using 4.2) and 4.11), we obtain m k φ e X e ) m k φeρ φa X a ) )) e Qh k φ eρ h k m k φa X a ) ) 2q) 1 e Qh k φ eρ d. Since for every e E there are at most #Ξ possible paths of the form eρ, we deduce that inf{m k φ e X e )) : k q} > 0. Consequently, T q := min{m k φ e X e )) : 1 e q, k q} > ) For n IN { } and k q, let E n k denote the set of all A-admissible words of length n over the alphabet {1, 2,..., k}. For all t 0, set Z n t, k) := φ ω t and Σ t := Z 1 t, ). ω E n k

27 PSEUDO-MARKOV SYSTEMS AND INFINITELY GENERATED SCHOTTKY GROUPS 27 Using the h k -conformality of m k as well as 4.2) and the definition of T q, we now obtain for all n IN and k q, Z n+1 h, k) = ω Ek n ω E n k e Qh k φ ωe h e q A ωne=1 + ω E n k φ ω hk φ e h e q A ωne=1 ω E n k Tq 1 e Qh k Tq 1 e Qh k = Tq 1 e Qh k = Tq 1 e Qh k φ mk e h e q A ωne=1 ω E n k ω E n k φ ωe h e>q A ωne=1 + ω Ek n e>q φω φ e X e )) ) m k φe X e ) ) φ ω h φ e h + ω E n k φ e h m k φω φ e X e )) ) + 1 e q 2 A ωne=1 φ e h m k φω X ω ) ) + 1 e q 2 A ωne=1 φ e h e q A ωne=1 φ e h e q A ωne=1 ω E n k ω E n k φ ω h φ e h k ω E n k φ ω h m k φω X ω ) ) Z nh, k) Z nh, k) e>q φ ω h Tq 1 e Qd Σ h Z nh, k). A straightforward inductive argument then gives Z n h, k) 2Σ h Tq 1 e Qd + 2 n 1) Z 1 h, k) 2Σ h Tq 1 e Qd + Z 1 h, ) = Σ h 2Tq 1 e Qd + 1) <, where the last inequality follows from 4.8). Hence, we obtain Ph ) = lim sup n 1 n log Z nh 1, ) lim sup n n log Σ h 2Tq 1 e Qd + 1) ) = 0. It follows that h u. By combining this with 4.10), we conclude that h u h. Since obviously h h, we now obtain u = h = h. Finally, using 4.9) and Proposition 4.10, we conclude that Ph) = 0, and also that h is the only zero of the pressure function P.