Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.


Introduction
In this paper, we frequently use the notation from [36]. A "rational semigroup" G is a semigroup generated by a family of non-constant rational maps g :Ĉ →Ĉ, whereĈ denotes the Riemann sphere, with the semigroup operation being functional composition. For a rational semigroup G, The work on the dynamics of rational semigroups was initiated by Hinkkanen and Martin ( [14]), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren's group ( [54]), who studied such semigroups from the perspective of random complex dynamics. The theory of the dynamics of rational semigroups onĈ has developed in many directions since the 1990s ( [14,54,15,28,29,30,31,34,35,36,37,38,39,40,41,49,42,43,44,45,32,46]).
Since the Julia set J(G) of a rational semigroup G generated by finitely many elements f 1 , . . ., f u has backward self-similarity, i.e., (1.1) J(G) = f −1 1 (J(G)) ∪ · · · ∪ f −1 u (J(G)) (see [36]), it can be viewed as a significant generalization and extension of both, the theory of iteration of rational maps (see [23]), and conformal iterated function systems (see [22]). For example, the Sierpiński gasket can be regarded as the Julia set of a rational semigroup. The theory of the dynamics of rational semigroups borrows and develops tools from both of these theories. It has also developed its own unique methods, notably the skew product approach (see [36,37,38,39,42,43,44,45,49], and [50]). We remark that by (1.1), the analysis of the Julia sets of rational semigroups somewhat resembles "backward iterated functions systems", however since each map f j is not in general injective (critical points), some qualitatively different extra effort in the cases of semigroups is needed.
The theory of the dynamics of rational semigroups is intimately related to that of the random dynamics of rational maps. For the study of random complex dynamics, the reader may consult [13,4,5,3,2,16,24]. We remark that the complex dynamical systems can be used to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamical system of a polynomial f (z) = az(1 − z) such that f preserves the unit interval and the postcritical set in the plane is bounded (cf. [10]). From this point of view, it is very important to consider the random dynamics of polynomials. For the random dynamics of polynomials on the unit interval, see [33].
In this paper, we investigate the Hausdorff, packing, and box dimension of the Julia sets of semi-hyperbolic rational semigroups G = f 1 , . . . , f u satisfying the nice open set condition. We will show that these dimensions coincide, that 0 < H h (J(G)), P h (J(G)) < ∞, where h is the Hausdorff dimension of J(G) and H h (resp. P h ) denotes the h-dimensional Hausdorff (resp. packing) measure, that h is equal to the critical exponent of the Poincaré series of the semigroup G, that there exists a unique h-conformal measurem h on the Julia set J(f ) of the "skew product map"f , that there exists a unique Borel probability measureμ h on J(f ) which is absolutely continuous with respect tom h , and thatμ h is metrically exact and equivalent withm h . The precise statements of these results are given in Theorem 1.11. In order to prove these results, we develop and combine the idea of usual iteration of non-recurrent critical point maps ( [51]), conformal iterated function systems ( [22]), and the dynamics of expanding rational semigroups ( [38]). However, as we mentioned before, since the generators may have critical points in the Julia set, we need some careful treatment on the critical points in the Julia set and some observation on the overlapping of the backward images of the Julia set under the elements of the semigroup.
Our approach develops the methods from [38], [51], and [52]. In order to prove that a conformal measure exists, is atomless, and, ultimately, geometric, we expand the concepts of estimability of measures, which originally appeared in [51], we introduce a partial order in the set of critical points, and a stratification of invariant subsets of the Julia set. As an entirely new tool to all [38], [51], and [52], we introduce the concept of essential families of inverse branches. This concept, supported by the notion of nice open set, is extremely useful in the realm of semihyperbolic rational semigroups, at it would also (without nice open set) substantially simplified considerations in the expanding case.
In the second part of the paper, devoted to proving the existence and uniqueness of an invariant (with respect to the canonical skew-product) probability measure equivalent with the h-conformal measure, the most challenging task is to prove the uniqueness of the latter. We do it by bringing up and elaborating the tool of Vitali relations due to Federer (see [12]), the tool which has not come up in [51], [52] nor [38]. We rely here heavily on deep results from [12]. The second tool, already employed in [52] and subsequent papers of the second author, is the Marco Martens method of producing σ-finite invariant measures absolutely continuous with respect to a given quasi invariant measure. We apply and develop this method, proving in particular its validity for abstract measure spaces and not only for σ-compact measure spaces. This is possible because of our use of Banach limits rather than weak convergence of measures.
We remark that as illustrated in [41,40,47], estimating the Hausdorff dimension of the Julia sets of rational semigroups plays an important role when we investigate random complex dynamics and its associated Markov process onĈ. For example, when we consider the random dynamics of a compact family Γ of polynomials of degree greater than or equal to two, then the function T ∞ :Ĉ → [0, 1] of probability of tending to ∞ ∈Ĉ varies only inside the Julia set of rational semigroup generated by Γ, and under some condition, this T ∞ :Ĉ → [0, 1] is continuous onĈ and varies precisely on J(G). If the Hausdorff dimension of the Julia set is strictly less than two, then it means that T ∞ :Ĉ → [0, 1] is a complex version of devil's staircase (Cantor function) ( [40,41,47,48]).
In order to present the precise statements of the main result, we give some basic notations. For each meromorphic function ϕ, we denote by |ϕ ′ (z)| s the norm of the derivative with respect to the spherical metric. Moreover, we denote by CV (ϕ) the set of critical values of ϕ.
Given a set A ⊂ C and r > 0, the symbol B(A, r) denotes the Euclidean open r-neighborhood of the set A. Moreover, diam(A) denotes the diameter of A with respect to the Euclidean distance. Moreover, given a subset A ofĈ, B s (A, r) denotes the spherical open r-neighborhood of the set B. Moreover, diam s (A) denotes the diameter of A with respect to the spherical distance.
Let u ∈ N. In this paper, an element of (Rat) u is called a multi-map.
is not needed if our semigroup G is expanding (see [38] or note that our proofs would use only (osc1) and (osc2) under this assumption). Condition (osc3) is satisfied in the theory of conformal infinite iterated function systems (see [21], comp. [22]), where it follows from the open set condition and the cone condition. Moreover, condition (osc3) holds for example if the boundary of U is smooth enough; piecewise smooth with no exterior cusps suffices. Furthermore, (osc3) holds if U is a John domain (see [6]). 38]). Let G be a countable rational semigroup. For any t ≥ 0 and z ∈Ĉ, we set S G (z, t) := g∈G g(y)=z |g ′ (y)| −t s , counting multiplicities. We also set S G (z) := inf{t ≥ 0 : S G (z, t) < ∞} (if no t exists with S G (z, t) < ∞, then we set S G (z) := ∞). Furthermore, we set s 0 (G) := inf{S G (z) : z ∈Ĉ}. This s 0 (G) is called the critical exponent of the Poincaré series of G.
, and s 0 (G) ≤ t 0 (f ). Note that for almost every f ∈ (Rat) u with respect to the Lebesgue measure, G = f 1 , . . . , f u is a free semigroup and so we have S G (t, z) = T f (t, z), S G (z) = T f (z), and s 0 (G) = t 0 (f ). Definition 1.9. Let ϕ : J(f ) → R be a function. Let ν be a Borel probability measure on J(f ). We say that ν is a ϕ-conformal measure for the mapf : s -conformal measure ν is sometimes called a t-conformal measure. When J(G) ⊂ C, a |f ′ | t -conformal measure is also sometimes called a t-conformal measure. Definition 1. 10. Let G = f 1 , . . . , f u and let t ≥ 0. For all z ∈Ĉ \ G * ( u j=1 CV(f j )), we set P z (t) := lim sup n→∞ The main result of this paper is the following. Let f = (f 1 , . . . , f u ) ∈ (Rat) u be a multi-map. Let G = f 1 , . . . , f u . Suppose that there exists an element g of G such that deg(g) ≥ 2, that each element of Aut(Ĉ) ∩ G (if this is not empty) is loxodromic, that G is semi-hyperbolic, and that G satisfies the nice open set condition. Then, we have the following.
Then there exists a constant C ≥ 1 such that for all z ∈ J(G) and all r ∈ (0, 1]. (e) h(f ) = HD(J(G)) = PD(J(G)) = BD(J(G)), where HD, PD, BD denotes the Hausdorff dimension, packing dimension, and box dimension, respectively, with respect to the spherical distance inĈ. Moreover, for each There exists a unique Borel probabilityf -invariant measureμ h on J(f ) which is absolutely continuous with respect tom h . The measureμ h is metrically exact and equivalent withm h .
The proof of Theorem 1.11 will be given in the following Sections 2-8. In Section 9, we give some examples of semi-hyperbolic rational semigroups satisfying the nice open set condition.
ACKNOWLEDGMENT: We wish to thank the anonymous referee for his/her valuable comments and suggestions which improved the final exposition of our paper.

Distortion and Measures.
All the points (numbers) appearing in this paper are complex unless it is clear from the context that they are real. In particular x and y are always assumed to be complex numbers and not the real and imaginary parts of a complex number.
The following is a straightforward consequence of these two distortion theorems.
. We also use the following more geometric versions of Koebe's Distortion Theorems involving moduli of annuli.
Theorem 2.4. (Koebe's Distortion Theorem, II) There exists a function w : (0, +∞) → [1, ∞) such that for any two open topological disks Q 1 ⊂ Q 2 ⊂ C with Mod(Q 2 \ Q 1 ) ≥ t and any univalent analytic function H : Moreover, by taking R > 0 sufficiently small, we can ensure that the above two inequalities hold for every z ∈ B(c, (AR) 1/q ) and the ball B(c, (AR) 1/q ) can be expressed as a union of q closed topological disks with piecewise smooth boundaries and mutually disjoint interiors such that the map H restricted to each of these interiors, is injective.
In the sequel we require the following technical lemma proven in [51] as Lemma 2.11. Thus, using this and (b) we obtain Developing the appropriate concepts from [51] we now shall define the notions of estimabilities (upper, lower and strongly lower) of measures, and we shall prove some of its properties and consequences. for all 0 ≤ r ≤ R. The number L is referred to as the lower estimability constant of the measure m at x and the number R is referred to as the lower estimability radius of the measure m at x. If there exists an L > 0 and an R > 0 such that the measure m is lower t-estimable at each point of X with the lower estimability constant L and the lower estimability radius R, then the measure m is said to be uniformly lower t-estimable. (3) (Strongly Lower Estimability) The measure m is said to be strongly lower t-estimable at a point x ∈ X if there exists an L > 0, a λ ∈ (0, ∞), and an R > 0 such that m(B(y, λr)) ≥ Lr t for every y ∈ B(x, R) for all 0 ≤ r ≤ R. The number L is referred to as the lower estimability constant of the measure m at x, the number R is referred to as the lower estimability radius of the measure m at x, and λ is referred to as the lower estimability size of the measure m at x. If there exists an L > 0, a λ, and an R > 0 such that the measure m is strongly lower t-estimable at each point of X with the lower estimability constant L, the lower estimability radius R, and the lower estimability size λ, then the measure m is said to be uniformly strongly lower t-estimable.
for all Borel sets A ⊂ U such that the restriction H| A is injective. The pair (m 1 , m 2 ) is called t-conformal if the above inequality sign can be replaced by equality. We will need the following lemmas. Then the measure m 1 is strongly lower t-estimable at z with lower estimability constant L, lower estimability radius K −1 |H ′ (z)| −1 r 0 , and lower estimability size K 2 λ. Thus The proof is finished. 1−q q t }, lower estimability radius (A −1 T ) 1/q , and lower estimability sizeλ = (2 q+1 KA 2 λ) 1/q . Proof. As in the proof of the previous lemma put A = A(H, c). Let 0 < r ≤ T and let x ∈ B(c, (A −1 r) 1/q ). Ifλ(A −1 r) 1/q ≥ 2|x − c|, then It follows from the assumptions that m 2 is lower t-estimable at H(c) with lower estimability constant λ −t L and lower estimability radius λT . Therefore, in view of Lemma 2.9 the critical point c is lower t-estimable with lower estimability constant A −2t λ −t L and lower estimability radius (AλT ) 1/q . Thus So, suppose that Since c is a critical point we have which means that In view of (2.2) Since, by the assumptions, the measure m 2 is lower t-estimable at H(x) with lower estimability constant λ −t L and lower estimability radius λr, it follows from the proof of Lemma 2.8 that the measure m 1 is lower t-estimable at x with lower estimability constant K −2t λ −t L and lower estimability radius Kλr|H ′ (x)| −1 . Thus, using (2.3), we get In view of this and (2.1) the proof is completed.
By writing A B we mean that there exists a positive constant C such that A ≤ CB for all A and B under consideration. Then A B means that B A, and A ≍ B says that A B and B A.

Open Set Condition and Essential Families.
In this section, starting with the open set condition, we develop the machinery of essential families of inverse branches. We first prove the following two lemmas.
Proof. By conjugating G by an element of Aut(Ĉ), we may assume that and m 2 := l 2,e | U ∩W , where l 2,e denotes the Euclidian measure on C. Then (m 1 , m 2 ) is a 2-conformal pair for f j . By the nice open set condition, there exist constants C > 0 and 0 < R < ∞ such that for each y ∈ U ∩ W and for each 0 < r < R, m 2 (B(y, r)) ≥ Cr 2 . By using the method of the proof of Lemma 2.11, it is easy to see that there exist constants Thus, the statement of our lemma holds. We are done.
Combining Lemma 2.12 and Koebe's Distortion Theorem, we immediately obtain the following lemma.
. . , f u be a rational semigroup satisfying the nice open set condition with U. Then, there exist constants ξ > 0 and T > 0 such that for each j = 1, . . . , u and for each For every family F ⊂ Σ * u let F = {τ : τ ∈ F} and F * = {τ * : τ ∈ F}.
Definition 2.14. Let G = f 1 , . . . , f u be a rational semigroup satisfying the nice open set condition. Suppose that J(G) ⊂ C. Fix a number M > 0, a number a > 0, and V , an open subset of Σ u . Suppose x ∈ J(G) and r ∈ (0, 1]. A family F ⊂ Σ * u is called (M, a, V )-essential for the pair (x, r) provided that the following conditions are satisfied.
(ess2) The family F consists of mutually incomparable words.
We shall prove the following. x ∈ J(G), r ∈ (0, 1], and F ⊂ Σ * u is an (M, a, V )-essential family for (x, r), then we have the following. (a) Proof. Item (a) follows immediately from Theorem 2.1 ( 1 4 -Koebe's Distortion Theorem), and Theorem 2.2. The equality part in item (b) is obvious. In order to prove the inclusion take is a family of mutually disjoint sets. Hence, using also (a), we get where C > 0 is a constant independent of r, M , and a. Let L a := ξ min{(T /a) 2 , 1}, where ξ and T come from Lemma 2.13. By Lemma 2.13, we obtain that for each j = 1, . . . , u , for each y ∈ f −1 j (U ), and for each 0 < b ≤ a, It follows from Theorem 2.2, (2.6), and (ess1) that for all τ ∈ F, we have Combining this with (2.5) we get that #F ≤ (16L a ) −1 CK 4 πM 2 . We are done.

Basic Properties of semi-hyperbolic Rational Semigroups
In this section we define semi-hyperbolic rational semigroups and collect their dynamical properties, with proofs, which will be needed in the sequel.
Definition 3.1. A rational semigroup G is called semi-hyperbolic if and only if there exists an N ∈ N and a δ > 0 such that for each x ∈ J(G) and g ∈ G, The crucial tool, which makes all further considerations possible, is given by the following semigroup version of Mane's Theorem proved in [37].
. . , f u be a finitely generated rational semigroup. Assume that there exists an element of G with degree at least two, that each element of Aut(Ĉ) ∩ G (if this is not empty) is loxodromic, and that F (G) = ∅. Then, G is semi-hyperbolic if and only if all of the following conditions are satisfied.
(a) For each z ∈ J(G) there exists a neighborhood U of z inĈ such that for any sequence {g n } ∞ n=1 in G, any domain V inĈ and any point ζ ∈ U , the sequence {g n } ∞ n=1 does not converge to ζ locally uniformly on V.
The first author proved in [37] the following.
. . , f u be a semi-hyperbolic finitely generated rational semigroup. Assume that there exists an element of G with degree at least two, that each element of Aut(Ĉ)∩G (if this is not empty) is loxodromic, and that F (G) = ∅. Then there exist R > 0, C > 0, and Throughout the rest of the paper, we assume the following: Assumption ( * ): • G satisfies the nice open set condition. In order to prove the main results (Theorem 1.11 etc.), in virtue of [51] and [52], we may assume that u ≥ 2. If u ≥ 2, then the open set condition implies that F (G) = ∅. Hence, conjugating G by some element of Aut(Ĉ) if necessary, we may assume that J(G) ⊂ C. Thus, throughout the rest of the paper, in addition to the above assumption, we also assume that • u ≥ 2 and J(G) ⊂ C. Note that in Theorem 1.11, we work with the spherical distance. However, throughout the rest of the paper, we will work with the Euclidian distance. If we want to get the results on the spherical distance (and this would include the case u = 1), then we have only to consider some minor modifications in our argument. We now give further notation.
For each (c, j) ∈ CP(f ) let q(c, j) be the local order of f j at c. For any set F ⊂Ĉ, set The latter is called the ω-limit set of F with respect to the semigroup G. Similarly, for every set B ⊂ Σ u ×Ĉ, and this set is called the ω-limit set of F with respect to the skew product mapf : Σ u ×Ĉ → Σ u ×Ĉ.
We shall prove the following.
, η is sticked to by at most one critical pair (c, j) of f ; and if a critical pair (c, j) sticks to a component Comp f ω| k (z), f ω| n k+1 , η , then f j (c) ∈ J(G). Furthermore, each critical pair of f sticks to at most one of all these components Comp f ω| k (z), f ω| n k+1 , η). Proof. The first part is obvious by the choice of β. In order to prove the second part suppose that , η with some 0 ≤ k < l ≤ n − 1 and ω k+1 = ω l+1 . Then both c and f ω| l k+1 (c) belong to Comp f ω| l (z), f ω| n l+1 , η , and therefore, contrary to the choice of β.
Proof. By Lemma 3.4 there exists a geometric annulus R ⊂ B(g(z), 2γ) \ B(g(z), γ) centered at g(z) and with modulus ≥ log 2/#CP(f ) and such that Since covering maps increase moduli of annuli by factors at most equal to their degrees, we conclude that .
As an immediate consequence of this lemma and Theorem 2.4 we get the following.
for all x, y ∈ Comp(z, f ω , γ), where const is a number depending only on #CP(f ) and κ.
with some universal constant Γ > 0 depending on f only.
n be all the integers k between 1 and n such that , then by Lemma 3.6 there exists a universal constant T > 0 such that Since, in view of Lemma 3.4, v ≤ #CP(f ), in order to conclude the proof it is enough to show the existence of a universal constant E > 0 such that for every 1 . Indeed, let c be the critical point in Comp(f ω| n−n i (z), f ω| n n i +1 , 2γ) and let q be its order. Since both sets Q (1) .
The proof is finished.

Partial Order in Crit(f ) ∩ J(G) and Stratification of J(G)
In this section we introduce a partial order in the critical set Crit(f ) ∩ J(G) and stratification of J(G). They will be used to do the inductive steps in the proofs of the main theorems of our paper. We start with the following.
Proof. Suppose on the contrary that the interior (relative to J(G)) of ω G ((Crit(f ) ∩ J(G)) + ) is not empty. Then, there exists a critical point c ∈ Crit(f ) ∩ J(G) such that ω G (c + ) has nonempty interior. But then, in virtue of Proposition 1.2 there would exist finitely many elements Hence c ∈ ω G (c + ), contrary to the non-recurrence condition (Theorem 3.2). Now we introduce in Crit(f )∩J(G) a relation < which, in view of Lemma 4.2 below, is an ordering relation. Put Proof. Indeed, c < c means that c ∈ ω G (c + ), contrary to the non-recurrence condition.
Since the set Crit(f ) ∩ J(G) is finite, as an immediate consequence of this lemma and Lemma 4.2 we get the following. Cr Lemma 4.5. The following four statements hold.
. Take p to be the minimal number satisfying (b) and suppose that ( Iterating this procedure we would obtain an infinite sequence c 1 = c > c ′ = c 2 > c 3 > . . ., contrary to Lemma 4.4. Now, part (d) follows from (c) and (4.1).
Proof. The left-hand inclusion is obvious regardless of what l(f ) is. The equality part of the assertion is obvious. In order to prove the right-hand inclusion fix i ∈ {0, 1, . . . , p − 1}. By the definition of the ω-limit sets of G there exists l i ≥ 0 such that for every c ∈ Cr i+1 (f ) we have Setting l(f ) = max{l i : i = 0, 1, . . . , p − 1} completes the proof.

Holomorphic Inverse Branches.
In this section we prove the existence of suitable holomorphic inverse branches, our basic tools throughout the paper. Set and Recall that according to the formula (3.1), given a point (τ, z) ∈ Σ u ×Ĉ, the set ω(τ, z) is the ω-limit set of (τ, z) with respect to the skew product mapf : Σ u ×Ĉ → Σ u ×Ĉ.
Then there exists an infinite increasing sequence (n j ) ∞ j=1 of positive integers such that for all j ≥ 1. We claim that there exists η(τ, z) > 0 such that for all j ≥ 1 large enough Comp z, f τ |n j , η(τ, z) ∩ Crit(f τ |n j ) = ∅.
Indeed, otherwise we find an increasing subsequence (j i ) ∞ i=1 and a decreasing to zero sequence of positive numbers η i < η such that Then there exist 0 ≤ p i ≤ n j i − 1 and Since lim i→∞ η i = 0, it follows from Theorem 3.3 that lim i→∞ci = z. Since (τ, z) / ∈ n≥0f −n (Crit(f )), this implies that lim i→∞ p i = +∞. But then, making use of Theorem 3.3 again and of the formula (σ p i (τ ), c i ) = f p i (τ,c i ), we conclude that the set of accumulation points of the sequence ((σ p i (τ ), c i )) ∞ 1 is contained in ω(τ, z). Fix (τ ∞ , c) to be one of these accumulation points. Since Crit(f ) is closed we conclude that (5.4) (τ ∞ , c) ∈ Crit(τ, z).
Since that set Crit(f ) is finite, passing to a subsequence, we may assume without loss of generality that (c i ) ∞ 1 is a constant sequence, so equal to c. Since c = f τ |p i (c i ), we get But, looking at (5.3) and (5.4), we conclude that f τ | n j i p i +1 (c) ∈ G * (Crit(τ, z) + ). We thus arrived at a contradiction with (5.2), and the proof is finished.
Proof. Let (n j ) ∞ j=1 and η(τ, z) be produced by Proposition 5.1. Then, by this proposition and Theorem 3.3, the family f −1 τ |n j ,z : of holomorphic inverse branches of f τ |n j sending f τ |n j (z) to z is well defined and normal. As a matter of fact we mean here this family restricted to the disk B(ẑ, η(τ, z)/2) and j ≥ 1 large enough. Therefore by Theorem 3.3 again, lim j→∞ |f ′ τ |n j (z)| −1 = 0 and we are done.
We end this section with the following. Let f ′ = sup w∈J(f ) |f ′ (w)|.

Geometric Measures Theory and Conformal Measures; Preliminaries
In this section we deal in detail with Hausdorff and packing measures and we also establish some geometrical properties of conformal measures.
Given t ≥ 0, the t-dimensional outer Hausdorff measure H t (A) of the set A is defined as where infimum is taken over all countable covers {A i } ∞ i=1 of the set A by sets whose diameters do not exceed ε.
The t-dimensional outer packing measure Π t (A) of the set A is defined as Here the second supremum is taken over all packings {B(x i , r i )} ∞ i=1 of the set A consisting of open balls whose radii do not exceed ε. These two outer measures define countable additive measures on the Borel σ-algebra of X. Let ν be a Borel probability measure on X. Define the function ρ = ρ t,ν : X × (0, ∞) → (0, ∞) by ρ(x, r) = ν(B(x, r)) r t The following two theorems (see [26,11], and [20]) are for our aims the key facts from geometric measure theory. Their proofs are an easy consequence of Besicovič covering theorem (see [26]) or a more elementary 4r-covering theorem (see [20]). Theorem 6.1. Let ν be a Borel probability measure on R n with some n ≥ 1. Then there exists a constant b(n) depending only on n with the following properties. If A is a Borel subset of R n and C > 0 is a positive constant such that then for every Borel subset E ⊂ A we have H t (E) ≥ Cν(E). (1)' If t > 0 then (1) holds under the weaker assumption that the hypothesis of part (1) is satisfied on the complement of a countable set.
Theorem 6.2. Let ν be a Borel probability measure on R n with some n ≥ 1. Then there exists a constant b(n) depending only on n with the following properties. If A is a Borel subset of R n and C > 0 is a positive constant such that then Π t (E) ≤ Cν(E) and, consequently, Π t (A) < ∞. (1') If ρ is non-atomic then (1) holds under the weaker assumption that the hypothesis of part (1) is satisfied on the complement of a countable set.

Conformal Measures; Existence, Uniqueness, and Continuity
For every t ≥ 0 and every function φ : J(f ) → C let L t φ : J(f ) → C be defined by the following formula: L t φ(y) is finite if and only if y / ∈ Crit(f ). Otherwise L t φ(y) is declared to be ∞. Iterating this formula we get for all n ≥ 1 that If y ∈ J(f ) \ p −1 2 (G * (Crit(f ) + )), then L n t 1 1(y) is finite for all n ≥ 0. If ψ :Ĉ → C, then define L t ψ :Ĉ → C by the formula It will be always clear from the context whether L t is applied to a function defined on J(f ) or on a compact neighborhood A of J(G). Iterating this formula we get for all n ≥ 1 that Note that ifψ : J(f ) → C is defined by the formulaψ(τ, z) = ψ(z), then L n tψ (τ, z) = L n t ψ(z) for all (τ, z) ∈ J(f ). Without confusion we put1 1 = 1 1. Note that L n t ψ(z) is finite for all z ∈ A \ G * (Crit(f ) + ). For all z ∈ A \ G * (Crit(f ) + ) set P z (t) = lim sup Proof. Since, by Lemma 4.1, ω G (Crit(f ) + ) ∩ J(G) is nowhere dense in J(G) and since the set G * (Crit(f ) + ) is countable, it follows from the Baire Category Theorem the set G * (Crit(f ) + )∩J(G) is nowhere dense. In order to prove the second part of our lemma, suppose that PCV(f ) is not nowhere dense in J(f ). This means that PCV(f ) has non-empty interior, and therefore, because of it forward invariance and topological exactness of the mapf : J(f ) → J(f ), we have PCV(f ) = J(f ). Hence J(G) = p 2 (J(f )) = p 2 (PCV(f )) ⊂ G * (Crit(f ) + ) ∩ J(G), contrary to, the already proved, first part of the lemma.
We shall prove the following. Proof. For every z ∈ J(G) \ G * (Crit(f ) + ) fix U z = {w | |w − z| < r}, an open round disk centered at z and such that {w | |w − z| < 2r} is disjoint from G * (Crit(f ) + ). It then directly follows from Koebe's Distortion Theorem that the function w → P w (t) is constant on U z . Now, fix z 1 , z 2 ∈ J(G) \ G * (Crit(f ) + ). By [14,Lemma 3 . Then x ∈ J(G) and for every n ≥ 1, L n+|ω| t 1 1(g(x)) ≥ |g ′ (x)| −t L n t 1 1(x). Therefore, P g(x) (t) ≥ P x (t). Hence P z 2 (t) ≥ P z 1 (t). Exchanging the roles of z 1 and z 2 , we get P z 1 (t) ≥ P z 2 (t), and we are done. By Lemma 7.2 the set J(G) \ G * (Crit(f ) + ) is not empty. Denote by P(t) the constant common value of the function z → P z (t) on J(G) \ G * (Crit(f ) + ). P(t) is called the topological pressure of t. Its basic properties are contained in the following.
Lemma 7.4. The function t → P(t), t ≥ 0, has the following properties.
(a) P(t) is non-increasing. In particular P(t) < +∞ as clearly P(0) < +∞. Proof. Fix z ∈ J(G) \ G * (Crit(f ) + ). Since the family of all analytic inverse branches of all elements of G is normal in some neighborhood of z (see [36,Lemma 4.5]) and all its limit functions are constant (see Theorem 3.3), lim n→∞ max{|f ′ ω (x)| : |ω| = n, x ∈ f −1 ω (z)} = ∞. So, item (a) follows directly from (7.1). Item (b), that is convexity of P(t) follows directly from (7.1) and Hölder inequality. Item (c) follows from the fact that max{u, max{deg(f j ) : 1 ≤ j ≤ u}} ≥ 2. For the proof of item (d) let U ⊂Ĉ be the set coming from the nice open set condition. Fix z ∈ J(G) \ G * (Crit(f ) + ). Let U z = B z, 1 2 dist(z, G * (Crit(f ) + )) . It follows from Koebe's Distortion Theorem that for all g ∈ G and all analytic inverse branches g −1 * of g defined on B (z, dist(z, G * (Crit(f ) + ))), where C > 0 is a constant independent of g. Since, by the open set condition, all the sets g −1 * (U z ∩ U ) are mutually disjoint, we thus get Hence P(2) = P z (2) ≤ 0 and we are done.
We say that a measurem t on J(f ) is e P(t) |f ′ | t -conformal provided that for all Borel sets A ⊂ J(f ) such thatf | A is injective. If P(t) = 0, the measurem t is simply referred to as t-conformal. Fix z ∈ J(G) \ G * (Crit(f ) + ). Observe that the critical parameter for the series is equal to the topological pressure P(t), i.e. S s (z) = +∞ if s < P (t) and S s (z) < +∞ if s > P(t). For every σ-finite Borel measure m on J(f ) let L * n t m be given by the formula In particular, L * n t δ (τ,ξ) (J(f )) ≤ L n t 1 1(ξ) < ∞. Hence, if s > P(t), then is a Borel probability measure on J(f ). Now, for every Borel set A ⊂ J(f ) we have So, L * n t δ (τ,ξ) (f −n (τ, ξ)) = 1. Hence, denoting ν s =ν s • p −1 2 , we get the following. In what follows that we are in the divergence type, i.e. S P(t) (ξ) = +∞. For the convergence type situation the usual modifications involving slowly varying functions have to be done, the details can be found in [9]. The following lemma is proved by a direct straightforward calculations.  ξ) ).
Now we can easily prove the following.
Proposition 7.7. For every t ≥ 0 there exists an e P(t) |f ′ | t -conformal measurem t for the map f : Proof. Since lim sցP(t) S s (ξ) = +∞, it suffices to take asm t any weak limit ofν s when s ց P(t), and to apply Lemma 7.6(c).
Consider now a Borel set A ⊂ J(f ) such thatf | A is injective. It then follows from Lemma 7.6(c) that Suppose now that (ω, x) ∈ J(f ) and there exists a (unique) continuous inverse branch φ −1 (ω,x) : . It then follows from (7.3) and Lemma 7.6(c) that for every set A ⊂ Σ u × B(f ω 1 (x), 2R), we have that From now on throughout the paper we assume that We also require that Our goal now is to show that the measure is uniformly upper t-estimable. For every critical point c ∈ Crit(f ) let where [u] is defined by formula (1.2). Now suppose that Γ is a closed subset of J(G) such that g(Γ) ∩ J(G) ⊂ Γ for each g ∈ Γ, and thatm is a Borel probability measure on J(f ).
Definition 7.8. The measurem is said to be nearly upper t-conformal respective to Γ provided that there exists an S > 0 such that the following conditions are satisfied.
(b) For every c ∈ Crit(f ) such that |τ |=l f τ (c + ) ∩ J(G) ⊂ Γ (the integer l = l(f ) ≥ 0 coming from Lemma 4.8) and every 1 ≤ j ≤ l + 1, The constant S is said to be the nearly upper conformality radius. If Γ = J(G), we simply say thatm is nearly upper t-conformal. In any case put Let us prove the following.
Lemma 7.10. There are two functions (R, S) → R * and L →L with the following property.
• Suppose that Γ is a closed subset of J(G) such that g(Γ) ∩ J(G) ⊂ Γ for each g ∈ G, and thatm is a Borel probability nearly upper t-conformal measure on J(f ) respective to Γ with nearly upper conformality radius S. Fix i ∈ {0, 1, . . . , p} and suppose that the measure m is uniformly upper t-estimable at all points z ∈ J i (G) ∩ Γ with corresponding estimability constant L and estimability radius R. Then the measurem| Σ(c)×Ĉ •p −1 2 is t-upper estimable, with upper estimability constantL and radius R * at every point c ∈ Cr i+1 (f ) such that Proof. Fix c ∈ Cr i+1 (f ) such that |ω|=l f ω (c + ) ⊂ Γ and also j ∈ {0, 1, . . . , u} such that f ′ j (c) = 0. Consider an arbitrary τ ∈ Σ u such that τ 1 = j and (τ, c) ∈ J(f ). In view of Lemma 4.8 Let R > 0 (sufficiently small) be the radius resulting from uniform t-upper estimability at all points

Applying nearly upper t-conformality ofm we get for every Borel set
It therefore follows from Lemma 2.10 and item (c) of Definition 7.8 that the measure ν τ | l+1 is upper t-estimable at c with upper estimability constant L 0 and radius R 0 independent ofm (but possibly R 0 depends on (R, S) and L 0 depends on L). Let Let D c = ω∈F D ω (c). Since #F ≤ u l+1 and sincẽ we conclude that the measurem| Σ(c)×Ĉ • p −1 2 is t-upper estimable at the point c with upper estimability constantL and radius R * independent ofm. We are done. Now, a straightforward inductive reasoning based on Lemma 7.9 and (7.9), (which also give the base of induction since S 0 (f ) = ∅), and Lemma 7.10 yields the following.
Lemma 7.11. Suppose that Γ is a closed subset of J(G) such that g(Γ) ∩ J(G) ⊂ Γ for each g ∈ G, and thatm is a Borel probability nearly upper t-conformal measure on J(f ) respective to Γ with nearly upper conformality radius S. Then the measure m =m • p −1 2 is uniformly upper t-estimable at every point of Γ andm| Σ(c)×Ĉ • p −1 2 is upper t-estimable, with upper estimability constants and radii independent of the measurem (but possibly dependent on S), at every point c ∈ Γ ∩ Crit(f ). Now we are in the position to prove the following.
Proof. Fix s > P(t) ≥ 0 and consider the measureν s defined in (7.2). We want to apply Lemma 7.11 with Γ = G * (Crit(f ) + ∩ J(G)) ∩ J(G) andm =ν s . For this we have to check thatν s is nearly upper t-conformal respective to Γ. Condition (c) of Definition 7.8 follows directly from Lemma 7.5 and the fact that ξ / ∈ G(Crit(f )) (see (7.6)). Since ξ / ∈ Γ and G(Γ) ∩ J(G) ⊂ Γ, there exists an S 0 > 0 such that ξ / ∈ u j=1 f j (B(Γ, S 0 )) ∩ J(G). Formula (7.4) then yields that for every z ∈ Γ,ν for every Borel set A ⊂ Σ u × B(z, S 0 ) such thatf | A is injective. Thus, condition (a) of Definition 7.8 is also verified. Condition (b) of this definition follows by iterating the above argument l + 1 times and keeping in mind that ξ / ∈ G * (Crit(f ) + ). Hence, there exists a constant S such that for each s > P (t),ν s is nearly upper t-conformal respective to Γ with nearly upper conformality radius S. Therefore, Lemma 7.11 applies and we conclude that all measuresν s | Σ(c)×Ĉ • p −1 2 are upper t-estimable at respective points c ∈ Crit(f ) ∩ J(G) with estimability constants and radii independent of s > P(t). Therefore,m t , a weak limit of measuresν s , s > P(t), (see the proof of Proposition 7.7)) also enjoys the property thatm t | Σ(c)×Ĉ • p −1 2 is upper t-estimable at respective points c ∈ Crit(f )∩J(G). Consequentlym t (Σ(c)×{c}) = 0. Having this we immediately see from Proposition 7.7 that the measurem t is nearly upper t-conformal, i.e. respective to Γ = J(G). So, applying Lemma 7.11, we conclude that the measure m t =m t •p −1 2 is uniformly upper t-estimable at every point of Γ = J(G). We are done. Now we assume that t = h, i.e. P(t) = 0 and we deal with the problem of lower estimability. It is easier than the upper one. We start with the following. Lemma 7.13. Fix i ∈ {0, 1, . . . , p} and suppose that for every critical point c ∈ S i (f ) and every j ∈ I(c) the measurem h | [j]×Ĉ • p −1 2 is strongly lower h-estimable at c with sufficiently small lower estimability size. Then m h is uniformly strongly lower h-estimable at all points of J i (G).
where all λ(c) are lower estimability sizes at respective critical points c. Fix z ∈ J i (G) \ S i (f ) and take τ ∈ Σ u such that (τ, z) ∈ J(f ). Assume r > 0 to be sufficiently small. Let s = s(θ, (τ, z), αr) ≥ 0 be the integer produced in Proposition 5.3 for the point z and radius r. A straightforward calculation based on Proposition 7.7 shows that form an h-conformal pair of measures with respect to the map . By Koebe's Distortion Theorem we also get (with small enough λ) In virtue of Koebe's Distortion Theorem and t-conformality of the pair (ν 1 , ν 2 ), we get as a consequence of all of this that Suppose now that the first alternative in Proposition 5.3(b) holds. We then can continue the above estimate as follows.
Now we shall prove the following.
Lemma 7.14. Fix i ∈ {0, 1, . . . , p} and suppose that the measure m h is uniformly strongly lower h-estimable at all points of J i (G). Then the measurem h | [j]×Ĉ • p −1 2 is strongly lower h-estimable at every critical point c ∈ Cr i+1 (f ) and every j ∈ I(c).
Proof. Fix c ∈ Cr i+1 (f ) and then an arbitrary j ∈ I(c). Next consider an arbitrary τ ∈ Σ u such that τ 1 = j and (τ, c) ∈ J(f ). Now, ignoring Γ, follow the proof of Lemma 7.10 up to the definition of the measure ν τ | l+1 . It follows from conformality ofm h that the measure ν τ | l+1 on . So the measure ν τ | l+1 is strongly lower h-estimable at c in virtue of our assumption and Lemma 2.11. Since D τ | l+1 (c) is an open neighborhood of c and The second main result of this section is this. Recall that two measures are said to be equivalent if they are absolutely continuous one with the other. Since every uniformly strongly lower h-estimable measure is uniformly lower h-estimable, as an immediate consequence of Lemma 7.12, Lemma 7.15, and [11,19,26], we obtain the following main result of this section and one of the two main results of the entire paper. (a) The measure m h =m h • p −1 2 is geometric meaning that there exists a constant C ≥ 1 such that for all z ∈ J(G) and all r ∈ (0, 1]. Consequently, (b) h = HD(J(G)) = PD(J(G)) = BD(J(G)).
(c) HD(J(G)) is the unique zero of t → P (t).
(d) All the measures H h , P h , and m h are equivalent one with each other with Radon-Nikodym derivatives uniformly separated away from zero and infinity. In particular (e) 0 < H h (J(G))), P h (J(G))) < +∞. Proof. Let z ∈ J(G) \ G * (Crit(f ) + ). Since G satisfies the open set condition, G is a free semigroup. Hence T f (z) = S G (z) and t 0 (f ) = s 0 (G). Moreover, by [37, Theorem 5.7], we have HD(J(G)) ≤ s 0 (G) ≤ S G (z). We now let a > h(f ). Since h(f ) is the unique zero of P (t) and since t → P (t) is non-increasing function, we have P (a) < 0. Hence there exists a number v < 0 such that for each n ∈ N, |ω|=n x∈f −1 Since h(f ) = HD(J(G)), it follows that h(f ) = T f (z) = t 0 (f ) = S G (z) = s 0 (G) = HD(J(G)) = PD(J(G)) = BD(J(G)). We are done.
It follows from Theorem 7.16 that the measure m h is atomless. We thus get the following. Proof. Indeed, the set Crit(f ) is finite and so, G −1 (Crit(f )) is countable. For all n ≥ 0 we havẽ −n (Crit(f )), we are thus done.

Invariant Measures
In this section we prove that there exists a unique Borel probabilityf -invariant measure on J(f ) which is absolutely continuous with respect tom h . This measure is proved to be metrically exact, in particular ergodic.
Frequently in order to denote that a Borel measure µ is absolutely continuous with respect to ν we write µ ≺ ν. We do not use any special symbol however to record equivalence of measures. We use some notations from [1]. Let (X, F, µ) be a σ-finite measure space and let T : X → X be a measurable almost everywhere defined transformation. T is said to be nonsingular if and only if for any A ∈ F, µ(T −1 (A)) ⇔ µ(A) = 0. T is said to be ergodic with respect to µ, or µ is said to be ergodic with respect to T , if and only if µ(A) = 0 or µ(X \ A) = 0 whenever the measurable set A is T -invariant, meaning that T −1 (A) = A. For a nonsingular transformation T : X → X, the measure µ is said to be conservative with respect to T or T conservative with respect to µ if and only if for every measurable set A with µ(A) > 0, Note that by [1,Proposition 1.2.2], for a nonsingular transformation T : X → X, µ is ergodic and conservative with respect to T if and only if for any A ∈ F with µ(A) > 0, Finally, the measure µ is said to be T -invariant, or T is said to preserve the measure µ if and only if µ • T −1 = µ. It follows from Birkhoff's Ergodic Theorem that every finite ergodic Tinvariant measure µ is conservative, for infinite measures this is no longer true. Finally, two ergodic invariant measures defined on the same σ-algebra are either singular or they coincide up to a multiplicative constant.
Definition 8.1. Suppose that (X, F, ν) is a probability space and T : X → X is a measurable map such that T (A) ∈ F whenever A ∈ F. The map T : X → X is said to be weakly metrically exact provided that lim sup n→∞ µ(T n (A)) = 1 whenever A ∈ F and µ(A) > 0.
We need the following two facts about weak metrical exactness, the first being straightforward (see the argument in [1, page 15]), the latter more involved (see [26]).
Fact 8.2. If a nonsingular measurable transformation T : X → X of a probability space (X, F, ν) is weakly metrically exact, then it is ergodic and conservative. Fact 8.3. A measure-preserving transformation T : X → X of a probability space (X, F, µ) is weakly metrically exact if and only if it is exact, which means that lim n→∞ µ(T n (A)) = 1 whenever A ∈ F and µ(A) > 0, or equivalently, the σ-algebra n≥0 T −n (F) consists of sets of measure 0 and 1 only. Note that if T : X → X is exact, then the Rokhlin's natural extension (T ,X,μ) of (T, X, µ) is K-mixing.
The precise formulation of our main result in this section is the following. The proof of this theorem will consist of several steps. We start with the following.
Combining inequalities (8.4) and (8.5) (with ν =m h ) from the proof of Lemma 8.5, and letting ε ց 0 in (8.5), we get for every Borel set E ⊂ I v , v ≥ 1, such that p 2 (E) is measurable, that I v andm h (Sing(f )) = 0, we get the following.
Lemma 8.6. If E is a Borel subset of J(f ) such that p 2 (E) is measurable andm h (E) = 0, then m h (p 2 (E)) = 0. So, by Lemma 8.5, for any h-conformal measure ν forf : We now shall recall the concept of Vitali relations defined on the page 151 of Federer's book [12]. Let X be an arbitrary set. By a covering relation on X one means a subset of If C is a covering relation on X and Z ⊂ X, one puts One then says that C is fine at x if inf{diam(S) : (x, S) ∈ C} = 0.
If in addition X is a metric space and a Borel measure µ is given on X, then a covering relation V on X is called a Vitali relation if (a) All elements of V (X) are Borel sets. and, following notation from Federer's book [12], let We shall prove the following. Proof. Fix (τ, z) ∈ J(f ) \ Sing(f ). Since p 2 (B j (τ, z)) ⊂ B(z, r j (τ, z)) and since This means that the relation B is fine at the point (τ, z). Aiming to apply Theorem 2.8.17 from [12], we set δ((B j (ω, x))) = r j (ω, x) for every B j (ω, x) ∈ B 2 . Fix 1 < κ < +∞ (a different notation for 1 < τ < +∞ appearing in Theorem 2.8.17 from [12]). With the notation from page 144 in [12] we havê So, in virtue of Theorem 7.16 and (8.2), we obtain where C > 0 is a constant independent of j. Hence, using (8.8), we get Thus, all the hypothesis of Theorem 2.8.17 in [12], p. 151 are verified and the proof of our lemma is complete.
As an immediate consequence of this lemma and Theorem 2.9.11, p. 158 in [12] we get the following.
Now, we shall prove the following.
Lemma 8.9. The measurem h is weakly metrically exact for the mapf : J(f ) → J(f ). In particular it is ergodic and conservative.
Proof. Fix a Borel set F ⊂ J(f ) \ Sing(f ) withm h (F ) > 0. By Proposition 8.8 there exists at least one point (τ, z) ∈ F h . Our first goal is to show that where, we recall η = η(τ, z) > 0 is the number produced in Proposition 5.1 and (n j ) ∞ 1 is the corresponding sequence produced there. Indeed, suppose for the contrary that Then, disregarding finitely many ns we may assume that Hence, making use of Theorem 7.16, we obtaiñ Letting j → ∞ this contradicts the fact that (τ, z) ∈ F h and finishes the proof of (8.9). Now sincẽ f : J(f ) → J(f ) is topologically exact, there exists q ≥ 0 such thatf q (p −1 2 (B(w, η/2))) ⊃ J(f ) for all w ∈ J(G). It then easily follows from (8.9) and conformality ofm h that lim sup k→∞m h (f k (F )) ≥ lim sup j→∞m h (f q+n j )(F )) = 1.
Noting also thatm h (Sing(f )) = 0 (by Corollary 7.19), the weak metric exactness ofm h is proved. Ergodicity and conservativity follow then from Fact 8.2. We are done. Proof. Let ν be an arbitrary h-conformal measure on J(f ) for the mapf : J(f ) → J(f ). Since, by Lemma 8.5 the measure ν is absolutely continuous with respectm h , it follows from Theorem 2.9.7 in [12], p. 155 and Lemma 8.7 that form h -a.e. (τ, z) ∈ J(f ) \ Sing(f ), Since, by Lemma 8.9, the measurem h is ergodic, it follows that the Radon-Nikodym derivative dν dm h ism h -almost everywhere constant. Since ν andm h are equivalent (by Lemma 8.5) this derivative must be almost everywhere, with respect tom h as well as ν, equal to 1. Thus ν =m h and we are done.
In order to prove the existence of a Borel probabilityf -invariant measure on J(f ) equivalent tõ m h , we will use Marco-Martens method originated in [18]. This means that we shall first produce a σ-finitef -invariant measure equivalent tom h (this is the Marco-Martens method) and then we will prove this measure to be finite. The heart of the Martens' method is the following theorem which is a generalization of Proposition 2.6 from [18]. It is a generalization in the sense that we do not assume our probability space (X, B, m) below to be a σ-compact metric space, neither assume we that our map is conservative, instead, we merely assume that item (6) in Definition 8.11 holds. Also, the proof we provide below is based on the concept of Banach limits rather than (see [18]) on the notion of weak limits.
Definition 8.11. Suppose (X, B, m) is a probability space. Suppose T : X → X is a measurable mapping, such that T (A) ∈ B whenever A ∈ B, and such that the measure m is quasi-invariant with respect to T , meaning that m • T −1 ≺ m. Suppose further that there exists a countable family {X n } ∞ n=0 of subsets of X with the following properties. Remark 8.12. Note that (6) is satisfied if the map T : X → X is finite-to-one. For, if T is finite-to-one, then ∞ l=1 T ( ∞ j=l Y j ) = ∅.
Theorem 8.13. Let (X, B, m) be a probability space and let T : X → X be a Marco-Martens map with a Marco-Martens cover {X j } ∞ j=0 . Then, there exists a σ-finite T -invariant measure µ on X equivalent to m. In addition, 0 < µ(X j ) < +∞ for each j ≥ 0. The measure µ is constructed in the following way: Let l B : l ∞ → R be a Banach limit and let Y j := X j \ i<j X i for each j ≥ 0. . If A ∈ B and A ⊂ Y j with some j ≥ 0, then we obtain (m n (A)) ∞ n=1 ∈ l ∞ . We set µ(A) := l B ((m n (A)) ∞ n=1 ). For a general measurable subset A ⊂ X, set In addition, if for a measurable subset A ⊂ X, the sequence (m n (A)) ∞ n=1 is bounded, then we have the following formula.
Furthermore, if the transformation T : X → X is ergodic (equivalently with respect to the measure m or µ), then the T -invariant measure µ is unique up to a multiplicative constant.
In order to prove Theorem 8.13, we need several lemmas.
Lemma 8.14. If (Z, F) is a σ-algebra of sets, Z = ∞ j=0 Z j is a disjoint union of measurable sets (elements of F) , and for each j ≥ 0, ν j is a finite measure on Z j , then the function , is a σ-finite measure on Z. Proof. Let A ∈ F and let (A n ) ∞ n=1 be a partition of A into sets in F. Then where we could have changed the order of summation since all terms involved were non-negative. Thus, we have completed the proof of our lemma.
We now suppose that we have the assumption of Theorem 8.13.
Since X j = j i=0 Y i , we are therefore done. Now, for every j ≥ 0, set µ j := µ| Y j .
Lemma 8. 16. For every j ≥ 0 such that µ(Y j ) > 0, and for every measurable set A ⊂ Y j , we have Proof. This is an immediate consequence of (4) of Definition 8.11 and the definition of the measure µ.
Lemma 8.17. For any j ≥ 0, µ j is a (countably additive) measure on Y j .
Proof. Let j ≥ 0. We may assume without loss of generality that µ j (Y j ) > 0. Let A ⊂ Y j be a measurable set and let (A k ) ∞ k=1 be a countable partition of A into measurable sets. For every n ≥ 1 and for every l ≥ 1, we have It therefore follows from (4) of Definition 8.11 that Since, by Lemma 8.15, (m n (Y j )) ∞ n=1 ∈ l ∞ , and since lim l→∞ ∞ k=l+1 m(A k ) = 0, we conclude that lim l→∞ ( ∞ k=1 m n (A k )) ∞ n=1 − l k=1 (m n (A k )) ∞ n=1 ∞ = 0. This means that in the Banach space l ∞ , we have ( ∞ k=1 m n (A k )) ∞ n=1 = ∞ k=1 (m n (A k )) ∞ n=1 . Hence, using continuity of the Banach limit l B : l ∞ → R, we get, We are done.
Proof. Fix a measurable set A ⊂ X. Then, for every l ≥ 1 we have that Hence, letting l → ∞, we get We are done.
Proof. Let i ≥ 0 be such that m(Y i ) > 0. Fix a measurable set A ⊂ Y i . Fix l ≥ 1. We then have where the last inequality sign was written because of (4) of Definition 8.11 and since A ⊂ Y i . Since, the limit when n → ∞ at last quotient is 1, we get that Hence, in virtue of (6) of Definition 8.11, For an arbitrary A ⊂ X, write A = ∞ j=0 A ∩ Y j and observe that We are done.
We now give the proof of Theorem 8.13.
Applying Theorem 8.13 we shall prove Theorem 8.4.
Set A j := p −1 2 (B(z j , r j )). Verifying the conditions of Definition 8.11 (with X = J(f ), T =f , m = m h , X j = A j ),f is nonsingular because of Corollary 7.19 and h-conformality ofm h . We immediately see that condition (1) is satisfied, that (2) holds because of (8.12), and that (3) holds because of h-conformality ofm h and topological exactness of the mapf : J(f ) → J(f ). Condition (5) follows directly from ergodicity and conservativity of the measurem h . Condition (6) follows sincẽ f : J(f ) → J(f ) is finite-to-one (see Remark 8.12). Let us prove condition (4). Fix j ≥ 1 and two arbitrary Borel sets A, B ⊂ A j withm h (A),m h (B) > 0. Since B(z j , 2r z j ) ∩ p 2 (PCV(f )) = ∅, for all n ≥ 0 all continuous inverse branches where τ is an arbitrary element of Σ u . Hence, and consequently, condition (4) of Definition 8.11 is satisfied. Therefore, Theorem 8.13 produces a Borel σ-finitef -invariant measure µ on J(f ), equivalent tom h . Now, let us show that the measure µ is finite. Indeed, by Theorem 3.3, there exists a δ > 0 such that for all g ∈ G * and for all x ∈ J(G), every connected component W of g −1 (B(x, δ)) satisfies that diam(W ) < γ and that W is simply connected. Cover p 2 (PCV(f )) with finitely many open balls {B(z, δ) : z ∈ F }, where F is some finite subset of p 2 (PCV(f )). for all j ≥ 1. Since J(G) \ z∈F B(z, δ) is covered by finitely many balls B(z j , r z j ), j ≥ 1, it therefore suffices to show that µ(p −1 2 (B(z, δ))) < +∞ for all z ∈ F . So, fix z ∈ F . Since z ∈ p 2 (PCV(f )), there thus exists k ≥ 1 such that B(z k , r z k ) ⊂ B(z, δ). By Lemma 8.15 and the formula (8.10) of Theorem 8.13, it therefore suffices to show that (8.13) lim sup n→∞m h f −n (p −1 2 (B(z, δ))) m h (f −n (A k )) < +∞.
In order to do this let for every τ ∈ {1, 2, . . . , s} n , the symbol Γ τ denote the collection of all connected components of f −1 τ (B(z, δ)). It follows from Theorem 7.16, Lemma 3.7 and [37, Corollary 1.9] that for every V ∈ Γ τ , we have where C > 0 is a constant independent of n and τ , V k is a connected component of f −1 τ (B(z k , r z k )) contained in V , and Γ is the constant in Lemma 3.7. But, from conformality of the measurem h and from the fact that V k = f −1 τ * (B(z k , r z k )), where f −1 τ * : B(z k , 2r z k ) →Ĉ is an analytic inverse branch of f τ , we see that Combining this with (8.14) we get that . Thus, the upper limit in (8.13) is bounded above by C(2K 2 δΓ −1 ) h (m h (A k )) −1 < +∞, and finiteness of the measure µ is proved.
Dividing µ by µ(J(f )), we may assume without loss of generality that µ is a probability measure. Since for every Borel set F ⊂ J(f ) the sequence (µ(f n (F ))) ∞ n=1 is (weakly) increasing, the metric exactness of µ follows from weak metrical exactness ofm h (Lemma 8.9) and the fact that µ and m h are equivalent. Since, by metrical exactness, µ is ergodic, it is a unique Borel probability measure absolutely continuous with respect tom h . The proof is complete.

Examples
In this section, we give some examples of semi-hyperbolic rational semigroups with nice open set condition.    [39] implies that J(G λ ) is porous and HD(J(G λ )) < 2. Moreover, by Theorem 1.11, we have h(f λ ) = HD(J(G λ )). We are done.