SEMIGROUP ACTIONS OF EXPANDING MAPS

Transcription

1 SEMIGROUP ACTIONS OF EXPANDING MAPS MARIA CARVALHO, FAGNER B. RODRIGUES, AND PAULO VARANDAS Abstract. We consider semigrous of Ruelle-exanding mas, arameterized by random walks on the free semigrou, with the aim of examining their comlexity and exloring the relation between intrinsic roerties of the semigrou action and the thermodynamic formalism of the associated skew-roduct. In articular, we clarify the connection between the toological entroy of the semigrou action and the growth rate of the eriodic oints, establish the main roerties of the dynamical zeta function of the semigrou action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the semigrou unsettles the ergodic roerties of the action.. Introduction In the mid seventies the thermodynamic formalism was brought from statistical mechanics to dynamical systems by the ioneering work of Sinai, Ruelle and Bowen [20, 4, 5, 24]. The corresondence between one-dimensional lattices and uniformly hyerbolic dynamics conveyed several notions from one setting to the other, introducing, via Markov artitions, Gibbs measures and equilibrium states into the realm of dynamical systems. Within non-invertible dynamics, a comlete descrition of the thermodynamic formalism has been established for Ruelle-exanding mas [2] and for exansive mas with a secification roerty [23, 5]. In articular, it is known that, for every otential under some regularity condition, there exists a unique equilibrium state, which is a Gibbs measure and has exonential decay of correlations. The classical strategy to rove these roerties ultimately relies on the analysis of the sectral roerties of the Perron-Frobenius transfer oerator, and it is known how to extend this method to finitely generated grou actions. Yet, the attemts to generalize the revious results have so far been riddled with difficulties, and a global theory is still out of reach. Some success has been registered for continuous actions of finitely generated abelian grous. More recisely, the statistical mechanics of exansive Z d -actions satisfying a secification roerty has been studied by Ruelle in [9], after introducing a suitable notion of ressure and discussing its link with measure theoretical entroy and free energy. The key ingredient in this context has been the fact that continuous Z d -actions on comact saces admit robability measures invariant under every continuous ma involved in the grou action. With it, Ruelle roved a variational rincile for the toological ressure and built equilibrium states as the class of ressure maximizing invariant robability measures. This duality between toological and measure theoretical comlexity of the dynamical system has been later used by Eizenberg, Kifer and Weiss [3] to establish large deviations rinciles for Z d -actions satisfying a secification roerty. A unified aroach to the thermodynamic formalism for continuous grou actions in the absence of robability measures invariant under all elements of the grou is still unknown. Although these actions are not dynamical systems, a few definitions of toological ressure have been roosed, although most of them unrelated and assuming either abelianity, amenability or some growth rate of the corresonding grou. Insired by the notion of comlexity resented by Bufetov in [7] in the context of skew-roducts, where no commutativity or conditions on the semigrou growth rate are required, the second and third named authors introduced in [8] a notion of toological ressure for semigrou actions, and showed that it reflects the comlex behavior of the action. In the 2000 Mathematics Subject Classification. Primary: 37B05, 37B40 Secondary: 37D20 37D35; 37C85. Key words and hrases. Semigrou actions, exanding mas, skew-roduct, zeta function, equilibrium states.

2 2 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS resent aer we ush this analysis further, considering finitely generated semigrous of Ruelleexanding mas. We relate the notion of toological entroy of a semigrou action introduced in [8] with the growth of the set of eriodic orbits, the concets of fibred and relative entroies and the radius of convergence of a dynamical zeta function for the semigrou action. Moreover, we generalize the classical Ruelle-Perron-Frobenius transfer oerator and construct equilibrium states for semigrou actions of C 2 exanding mas. In the meantime, we will verify the imact of changing the random walk inside the semigrou on the dynamical and ergodic attributes of the semigrou action. 2. Setting Let M be a comact metric sace and C 0 (M) denote the sace of all continuous observable functions ψ : M R. Given a finite set of continuous mas g i : M M, i {, 2,..., },, and the finitely generated semigrou (G, ) with the finite set of generators G = {id, g, g 2,..., g }, write G = n N 0 G n, with G 0 = {id}, and g G n if and only if g = g in... g i2 g i, with g ij G (for notational simlicity s sake we will use g j g i instead of the comosition g j g i ). Set G = G \{id}. As a semigrou can have multile generating sets, we will assume that the generator set G is minimal, meaning that no function g j, for j =,...,, can be exressed as a comosition from the remaining generators. Free semigrous. In G, one considers the semigrou oeration of concatenation defined as usual: if g = g in... g i2 g i and h = h im... h i2 h i, where n = g and m = h, then g h = g in... g i2 g i h im... h i2 h i G m+n. Each element g of G n may be seen as a word that originates from the concatenation of n elements in G. Clearly, different concatenations may generate the same element in G. Nevertheless, in most of the comutations to be done, we shall consider different concatenations instead of the elements in G they create. One way to interret this statement is to consider the itinerary ma ι : F G i = i n... i g i := g in... g i where F is the free semigrou with generators, and to regard concatenations on G as images by ι of aths on F. Set G = G \{id} and, for every n, let G n denote the sace of concatenations of n elements in G. To summon each element g of G n, we will write g = n instead of g G n. Semigrou actions. We say that the finitely generated semigrou G induces a semigrou action S : G M M in M defined by S(g, x) = g(x) if, for any g, h G and all x M, we have S(g h, x) = S(g, S(h, x)). The action S is said to be continuous if, for any g G, the ma g : M M given by g(x) = S(g, x) is continuous. A oint x M is a fixed oint of g G if g(x) = x; the set of these fixed oints will be denoted by Fix(g). A oint x M is a eriodic oint with eriod n of the action S if there exists g G n such that g(x) = x. We let Per(G n ) = g =n Fix(g) denote the set of all these eriodic oints with eriod n. Accordingly, Per(G) = n Per(G n) will stand for the set of eriodic oints of the whole semigrou action. We observe that, when G = {f}, this definition coincides with the usual one for the single dynamical system f. Ruelle-exanding mas. Let (X, d) be a comact metric sace and T : X X a continuous ma. T is said to be exansive if there exists ε > 0 such that, for any x y X, su n N d(t n (x), T n (y)) > ε. The ma T is locally exanding if there exist λ > and δ > 0 such that d(t (x), T (y)) > λ d(x, y) for all x, y X with d(x, y) < δ These two notions are bonded: every locally exanding system is exansive and an exansive ma is locally exanding with resect to an adated metric (cf. [0]). Definition 2.. The system (X, T ) is Ruelle-exanding if T is locally exanding and oen, i.e.: (i) there exists c > 0 such that, for all x, y X with x y, T (x) = T (y) imlies d(x, y) > c. (ii) there are r > 0 and 0 < ρ < such that, for each x X and all a T ({x}) there is a

3 SEMIGROUP ACTIONS OF EXPANDING MAPS 3 ma φ : B r (x) X, defined on the oen ball centered at x with radius r, such that φ(x) = a, T φ(z) = z and, for all z, w B r (x), we have d(φ(z), φ(w)) ρ d(z, w). Examles of Ruelle-exanding mas include one-sided Markov subshifts of finite tye, determined by aeriodic square matrices with entries in {0, }, and C -exanding mas on comact manifolds. We remark that, due to the sectral decomosition, if the domain of a Ruelle-exanding ma is connected, then it is toologically mixing. For more details we refer the reader to [5, 2, 9]. 3. Main results We start with a toological descrition of finitely generated semigrous of uniformly exanding mas. Later, we will assume that G is either a finite subset of Ruelle-exanding mas acting on a comact connected metric sace M or a finite subset of the sace End 2 (M) of non-singular C 2 endomorhisms in a comact connected manifold M. In this setting, we will show that the set of eriodic oints with eriod n for such a semigrou dynamics has a definite exonential growth rate with the eriod n, which is given by the toological entroy of the semigrou h to (S) (see Definition 6.). Moreover, we will rove that this entroy is equal to the logarithm of the sectral radius of a suitable transfer oerator L 0 (cf. Subsections 5 and 6). Theorem A. Let G be the semigrou generated by G = {Id, g,..., g }, where G is a set of Ruelle-exanding mas on a comact connected metric sace M, and let S : G M M be its continuous semigrou action. Then ( 0 < h to (S) = lim n n log n g =n ) Fix(g) = log s (L 0 ). The second art of this work concerns the asymtotic growth of the eriodic oints of a semigrou action. Insired by the Artin-Mazur zeta function (cf. []), we define the zeta function associated to the continuous semigrou action S : G M M by the formal ower series z C ζ S (z) = ex ( n= N n (G) n z n ), () where N n (G) = g =n Fix(g). (We refer the reader to [8] for an account on random zeta n functions.) Our next result relates the regularity of ζ S to the exonential growth rate of eriodic oints, given by (S) = lim su n + n log (max{n n(g), }). Theorem B. Let G be the semigrou generated by G = {Id, g,..., g }, where G is a set of Ruelle-exanding mas on a comact connected metric sace M, and S : G M M be the corresonding continuous semigrou action on M. Then ζ S is rational and its radius of convergence ρ S is equal to e (S) = e hto(s). Afterwards, we will study the ergodic roerties of semigrou actions of mas in End 2 (M). Although one does not exect to find an absolutely continuous common invariant robability measure, we may hoe to discover some robability measure which reflects an averaged distribution of the Lebesgue measure under the action of the semigrou. We say that R θ is a random walk on G if R θ = ι θ N, where θ is a robability measure on ι (G ) = {,..., }, the set of generators of F. For instance, if θ(i) = for any i {,..., }, then η = θ N is the equally distributed Bernoulli robability measure on the Borel sets of the unilateral shift Σ + = {,..., } N. If, instead, θ(i) = a i > 0 for each i {,..., }, then a = (a, a 2,..., a ) is a non-trivial robability vector and η a = θ N will stand for the Bernoulli robability measure θ N on Σ +, while R a = ι (η a ) will denote the corresonding random walk R θ on G. More generally, we may take a σ-invariant robability measure η on Σ + and consider the associated random walk R η = ι (η). Given a σ-invariant robability measure η, a robability measure ν on M is said to be R η -stationary if ν = g ν dr η (g). (2)

4 4 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS As robability measures invariant under all elements of the semigrou are unlikely to exist, the concet of stationary measure is the most natural to be addressed while studying ergodic roerties of semigrou actions. However, there is evidence that this is not the suitable notion for describing maximal entroy measures for semigrou actions (cf. Section 9). Finally, we will discuss a thermodynamic formalism and find an adequate definition of measure of maximal entroy for a semigrou action with resect to a fixed random walk. Theorem C. Let G be the semigrou generated by G = {id, g,..., g }, where G is a set of C 2 exanding mas on a comact connected manifold M. Consider the corresonding continuous semigrou action S : G M M and the random walk R = ι (η ). Then the semigrou action has a robability measure of maximal entroy which can be comuted as the weak -limit of an averaged distribution of either re-images or eriodic oints. 4. Overview To comlement the classical aroach of random dynamical systems, we are mainly interested in determining intrinsic objects for the dynamics of the semigrou action. Ruelle-Perron-Frobenius transfer oerators suitable for the analysis of semigrou actions will be defined in Section 5. These oerators are in a strong relation with the classical transfer oerator for a locally constant skewroduct dynamics. For that reason, we also discuss the deendence of the ergodic roerties of the skew-roduct dynamics on the chosen robability measure in the underlying shift, which describes the random walk on the free semigrou. In Sections 6 and 7 we will justify the choice of the notion of toological entroy for semigrou actions: we shall rove not only that the entroy can be comuted by the growth rate of the mean number of eriodic orbits but also that it arises naturally as the radius of convergence of the zeta function of the semigrou action (Theorems A and B). The rationality of the zeta function for semigrou actions of exanding mas will follow as a consequence of Lemma 7.. In Section 8 we focus on building a bridge between the several concets of the thermodynamic formalism for skew-roducts and the intrinsic objects for semigrou actions. In articular, we comare the notion of classical toological entroy with the fibred and relative entroies, and relate the quenched and annealed equilibrium states for symmetric and non-symmetric random walks. For instance, we rove that the toological entroy of the semigrou coincides with Ledraier-Walters entroy, introduced in [7], if and only if all mas have the same degree (Proosition 8.2); and that this fibred entroy coincides with a quenched ressure in random dynamics (Proosition 8.8). Additionally, the toological entroy of the semigrou is showed to coincide with an annealed ressure in random dynamical systems (Corollary 8.7), and to be equal to the classical toological ressure of a suitable otential for the skew-roduct dynamics (cf. (8)). In Section 9 we will introduce a rocedure to select measures for the semigrou action using some variational insight. More recisely, from connections between the semigrou action, the classical thermodynamic formalism and the annealed ressure function, we will rovide an intrinsic construction of robability measures (not necessarily stationary) on the ambient sace which arise as marginals of equilibrium states. The two different aroaches will allow us to conclude that maximal entroy measures can be comuted using either an averaged equidistribution of eriodic oints or of re-images (Theorem C). 5. Ruelle-Perron-Frobenius oerators In this section we shall introduce the Ruelle-Perron-Frobenius transfer oerator to be assigned to a semigrou action which is a natural extension of the concet of transfer oerator for an individual dynamical system. The oerator will deend on the chosen set G of generators of G and on the selected random walk R η on G (we will come back to this subject in Subsection 8.2). Let G be a semigrou generated by a finite subset G of Ruelle-exanding mas acting on a comact connected metric sace M and S : G M M the corresonding continuous semigrou action. Given a continuous observable φ : M R, let L g,φ : C 0 (M) C 0 (M) denote the usual

5 SEMIGROUP ACTIONS OF EXPANDING MAPS 5 Ruelle-Perron-Frobenius oerator associated to the dynamical system g and the observable φ: L g,φ (ψ)(x) = e φ(y) ψ(y). (3) g(y)=x It is not hard to check that L g,φ = L gi n,φ L gin,φ L gi,φ for all g = g in...g i G n and n. Consider now the non-stationary dynamical system whose comlexity is indexed by the time n, corresonding to the ball of radius n in the semigrou. Such viewoint has turned to be very fruitful in the descrition of the toological entroy and the comlexity of semigrou actions [8], and motivates the definition of the following weighted mean sequence of transfer oerators. Definition 5.. Given a continuous otential φ : M R and a shift-invariant robability measure η on Σ +, the Ruelle-Perron-Frobenius sequence of bounded linear oerators (L n,η,φ ) n acting on C 0 (M) is defined, for every n, by L n,η,φ = L gi n,φ... L gi2,φl gi,φ () dη([i,..., i n ]). Σ + Given φ = (φ,..., φ ) C 0 (M) and a non-trivial robability vector a, we define the integrated transfer oerator L a,φ : C 0 (M) C 0 (M) by L a,φ ϕ(x) = i= a i g i(y)=x e φ i(y) ϕ(y). For instance, if η is the symmetric random walk η, then, for all n, L n,η,φ = n L g,φ = L n,φ. g =n Taking into account that the semigrou action is naturally associated to the skew-roduct dynamics F G : Σ + M Σ + M (4) (ω, x) (σ(ω), g ω (x)) where ω = (ω, ω 2,... ), given a Bernoulli robability measure η a on Σ + we assign to any φ = (φ,..., φ ) C 0 (Σ + M) the integrated transfer oerators L a,φ : C 0 (Σ + M) C 0 (Σ + M) for F G, defined by L a,φ ψ(ω, x) = a i L gi,φ i ψ(iω, x) = e φi(iω,y) ψ(iω, y) (5) i= i= a i g i(y)=x where ψ C 0 (Σ + M) and iω stands for the sequence (i, ω, ω 2,... ). For instance, if φ = (0,..., 0) and ψ =, one gets L a,φ = deg (g i ) da(i), where deg stands for the degree of the ma. Now, for each φ C 0 (M) we may consider the ma φ(ω, x) = (φ(x),..., φ(x)) in C 0 (Σ + M) and, dually, if ψ C 0 (Σ + M) does not deend on ω, we may take ϕ : M R defined by ϕ(x) = ψ(, x). Then, L a,φ ψ = L a,φ ϕ. As L a,φ and L a,φ are ositive oerators, the logarithm of each sectral radius is equal to the exonential growth rate of L n a,φ Σ + M 0 and L n a,φ M 0, resectively. Thus, s ( L a,φ ) = s ( L a,φ ). A similar link between transfer oerators on the hase sace and on the skew-roduct dynamics has been considered reviously in [26]. However, in this reference the oerators are built averaging normalized transfer oerators for individual dynamics. 6. Toological entroy of the semigrou action This section is devoted to the roof of Theorem A. First, let us recall the concet of searated oints and toological entroy of a semigrou action adoted in [7, 8]. Given ε > 0 and g := g in... g i2 g i G n, the dynamical ball B(x, g, ε) is the set B(x, g, ε) := {y X : d(g j (y), g j (x)) ε, for every 0 j n} where, as before, for every j n we denote by g j the concatenation

6 6 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS g ij... g i2 g i G j, and g 0 = id. We also assign a dynamical metric d g to M by setting d g (x, y) := max 0 j n d(g j (x), g j (y)). Notice that both the dynamical ball and the metric deend on the underlying concatenation of generators g in... g i and not on the grou element g, since the latter may have distinct reresentations. Given g = g in... g i G n, we say that a set K M is (g, n, ε)- searated if d g (x, y) > ε for any distinct x, y K. The maximal cardinality of a (g, ε, n)-searated set on M will be denoted by s(g, n, ε). The toological entroy of a semigrou action estimates the growth rate in n of the number of orbits of length n u to some small error ε. Definition 6.. The toological entroy of the semigrou action S : G M M is ( h to (S) = lim lim su ε 0 n n log ) n s(g, n, ε). (6) Ghys, Langevin and Walczak roosed in [4] another definition of toological entroy of a semigrou action using the asymtotic exonential growth rate of oints that are searated by some grou element. This corresonds to the largest exonential growth rate, while the definition we have adoted observes the growth rates of searated oints averaged along semigrou elements. In the context of a single Ruelle-exanding dynamics, the toological entroy is related to the largest exonential growth rate of eriodic oints (cf. [9]). This motivates considering the following asymtotic seed. g =n Definition 6.2. The eriodic entroy of a semigrou action S : G M M is (S) = lim su n + n log (max{n n(g), }) where N n (G) has been defined in (). Observe that in order for (S) to be a finite value, the set Fix(g) must be finite for each g G \ {id}, as haens when g is exansive (that is, when there exists ε g > 0 such that, whenever x y M, we have max {d(g l (x), g l (y)): l N 0 } ε g.) After [8], we know that an action of a finitely generated semigrou of C -exanding mas is even strongly δ-exansive for some δ > 0, a notion that we now recall and which will ease the task of comuting h to (S). Definition 6.3. Given δ > 0, we say that a continuous semigrou action S : G M M is δ-exansive if, whenever x y M, there exist κ N and g G κ such that d(g(x), g(y)) > δ. The action S is said to be strongly δ-exansive if, for any γ > 0, there exists κ γ such that, for every x y M with d(x, y) γ, for all κ κ γ and any g G κ, we have d g (x, y) = max 0 j n d(g j (x), g j (y)) > δ. Lemma 6.4. [8, Theorem 25] Let G be the semigrou generated by a set G = {Id, g,..., g }, where G is a finite set of Ruelle-exanding mas on a comact metric sace M, and S : G M M its continuous semigrou action. Take 0 < ε < δ. Then h to (S) = lim su n n log ( n g =n ) s(g, n, ε). Additionally, it was roved in [8, Theorem 28] that h to (S) (S). We are left to show the oosite inequality. For that urose, we will have to secify in advance what is the ath, or concatenation of elements, one is interested in tracing. Definition 6.5. We say that a continuous semigrou action S : G M M, associated to a finitely generated semigrou G, satisfies the (strong) orbital secification roerty if, for any δ > 0, there exists T (δ) > 0 such that, given k N, for any h j G j with j T (δ) for every j k, for each choice of k oints x,..., x k in M, for any natural numbers n,..., n k and any semigrou element g nj,j = g i nj,j... g i2,j g i,j G nj, where j {,..., k}, there exists x M such that d(g l, (x), g l, (x )) < δ for all l n and d(g l,j h j... g n2,2 h g n, (x), g l,j (x j)) < δ for all 2 j k, l n j, where g l,j = g il,j... g i,j. The semigrou action satisfies the eriodic orbital secification roerty if the oint x can be chosen eriodic.

7 SEMIGROUP ACTIONS OF EXPANDING MAPS Proof of Theorem A. We start roving that the class of Ruelle-exanding mas is closed under concatenation, and so forms a semigrou. Lemma 6.6. If each ma in the finite set G is Ruelle-exanding, then g is Ruelle-exanding for any g G {Id}. Moreover, there exists δ > 0 such that the semigrou action S : G M M is strongly δ-exansive. Proof. If g and g 2 are Ruelle-exanding mas on a comact metric sace, then it is not hard to use uniform continuity to rove that the comosition g 2 g is a Ruelle-exanding ma: if ρ i (0, ) denotes the backward contraction rate for g i G and r i > 0 is so that all inverse branches for g i are defined in balls of radius r i, then every ma g i2 g i is Ruelle-exanding and its inverse branches are defined in balls of radius r with backward contraction rates ρ, where r = min{r i : i } and ρ = min{ρ i : i }. (7) We now roceed by induction on n. If, for a fixed ositive integer n, the concatenation of n Ruelleexanding mas is Ruelle-exanding, then, considering n + such mas, say g in+ g in g i, we may slit their comosition into the concatenation of two Ruelle-exanding mas g in+ (g in g i ) and aly what we have just roved. Concerning the strong δ-exansiveness of the action S for some δ > 0, take δ = r 2 and, given γ > 0, let κ γ be such that ρ κγ δ < γ, where ρ is defined by (7). Now, for any oints x y X with d(x, y) γ and any g G κ with κ κ γ, clearly d g (x, y) > δ, otherwise we would get d(x, y) ρ κγ d(g(x), g(y)) ρ κγ d g (x, y) < γ which leads to a contradiction. Recall now, from Lemma 6.4 and the fact that the semigrou action is strongly exansive, that the comutation of the toological entroy of the semigrou action may be done with a well chosen, but fixed, ε. More recisely, if δ ( > 0 is given by the roof of Lemma 6.6 and we fix 0 < ε < δ, then h to (S) = lim su n n log n g =n ). s(g, n, ε) We claim that for every g G such that g = n, the set Fix(g) is (g, n, ε)-searated. Otherwise, there would exist P Q Fix(g) which were not (g, n, ε)-searated, that is, such that d(g m (P ), g m (Q)) < ε for every 0 m n. But then, if γ = d(p, Q)/2 and k γ is given by Lemma 6.6, we have g κ γ = nκ γ κ γ and d g κγ (P, Q) = d g (P, Q) < ε, a contradiction. Therefore, s(g, n, ε) Fix(g), for every g G with g = n, and, consequently, h to (S) = lim su n n log n g =n s(g, n, ε) lim su n log n n g =n Fix(g) = (S). To comlete the roof we are left to show that the lim su in the definition of (S) is indeed a limit. First notice that, as G is finitely generated by Ruelle-exanding mas, by [8, Theorem 6] it satisfies the eriodic orbital secification roerty. Fix ε (0, δ), let T (ε/2) N be given by this roerty and take g G m+n+t (ε/2), to which there exist a G n, b G T (ε/2) and c G m such that g = a b c. Let Fix(c) = {P,..., P r } and Fix(a) = {Q,..., Q s } be the sets of fixed oints of c and a, resectively. By the eriodic secification roerty, for the semigrou elements c, a and the oints P i Fix(c) and Q j Fix(a) there exists x ij Fix(a b c) such that d(c l (x ij ), c l (P i )) < ε 2 and d(a u b c (x ij ), a u (Q j )) < ε 2 for every l = 0,..., m and every u = 0,..., n. As the set Fix(c) is (c, m, ε)-searated, we have x i j x i2 j 2 for (i, j ) (i 2, j 2 ). This imlies that Fix(g) Fix(a) Fix(c) and so, Fix(g) Fix(c) Fix(a) = Fix(c) Fix(a). g =m+n+t (ε/2) This yields m+n+t (ε/2) g =m+n+t (ε/2) c =m, a =n Fix(g) T (ε/2) m c =m c =m Fix(c) n a =m a =n Fix(a). (8)

8 8 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS ( Write α n = log n a =n ). Fix(a) The inequality (8) imlies that α m+n+t (ε/2) α n + α m for all m, n. As T (ε/2) is a fixed constant, by a simle adatation of the roof of Fekete s Lemma ([28, Theorem 4.9]), it follows that the sequence ( α n n ) n N converges to its suremum. Therefore, the lim su in the definition of (S) may be relaced by a limit. Finally, by Lemma 6.6, each g G is a Ruelle-exanding ma and Fix(g) = deg (g). So, i= deg (gi) taking the observable φ 0, we get (S) = log s ( L,0 ) = log ( ). 7. The zeta function of the semigrou action Recall that ( the dynamical Artin-Mazur s zeta function of a dynamical system f comutes ) Fix(f ζ f (z) = ex n ) n= n z n (cf. []). The function ζ S we associate to the action of a semigrou G, generated by a finite set G of Ruelle-exanding mas, is linked to the notion of the annealed zeta function introduced in [2, 22] within the context of random families of C 2 exanding mas. The aim of this section is to show that, when we consider Ruelle-exanding mas and the random walk R, this annealed zeta function is rational and its radius of convergence is ex ( h to (S)). 7.. Proof of Theorem B. We will start estimating the radius ρ S of convergence of ζ S and relating it with h to (S). Notice that, as lim n n n =, then n Nn (G) n = lim su = lim su max{nn (G), } = ex( (S)). (9) ρ S n n n Thus, whenever (S) > 0, the zeta function ζ S has a ositive radius of convergence and is well defined in {z C: z < ex( (S))}. Under the assumtions of Theorem A, one also has ρ S = ex( h to (S)). We are left to show that the zeta function of a semigrou of Ruelle-exanding mas is rational. Lemma 7.. The skew-roduct F G is Ruelle-exanding and toologically mixing. Proof. Denote by d M and d the metrics in M and Σ +, resectively. We are considering in Σ + M the roduct toology, which is metrizable; its toology is given, for instance, by the metric D((ω 0, x 0 ), (ω, x )) = max {d M (x 0, x ), d (ω 0, ω )}. As σ and each g i G are Ruelleexanding, there exist ositive constants c σ and c i, for i {,..., }, such that: (a) if x, y M, x y, g i (x) = g i (y) then d M (x, y) > c i ; and (b) ω 0, ω Σ +, ω 0 ω, σ(ω 0 ) = σ(ω ) imlies d (ω 0, ω ) > c σ. Let (ω 0, x 0 ) (ω, x ) be such that F G ((ω 0, x 0 )) = F G ((ω, x )), that is, σ(ω 0 ) = σ(ω ) and g ω 0 (x 0 ) = g ω (x ). Then, either ω 0 ω, in which case we have D((ω 0, x 0 ), (ω, x )) > c σ ; or else ω 0 = ω, and then D((ω 0, x 0 ), (ω, x )) > c ω 0. Therefore, if c = min {c σ, c, c 2,..., c d } and (ω 0, x 0 ) (ω, x ) are so that F G ((ω 0, x 0 )) = F G ((ω, x )) then D((ω 0, x 0 ), (ω, x ) > c. The second roerty that characterizes Ruelle-exanding mas (cf. Definition 2.) is roved similarly by considering the minimal exansion rates and minimal radius where convergence holds. We now roceed showing that F G is toologically mixing. Consider a non-emty oen subset W of Σ + M, and take a cylinder U = C(; a a 2..., a k ) and an oen set V of M such that U V W. As the mas σ and g ak... g a are toologically mixing and Ruelle-exanding, there exist ositive integers m U and m V such that σ l (U) = Σ + and (g ak... g a ) l = M, for all l m = max {m U, m V }. Hence, for all l m we have FG l (U V ) = ( σ l (U), ω U f ω(v l ) ) = Σ + M since V contains all the sequences of + d whose k first entries are a a 2..., a k, in articular those which start with this block reeated l times for every l N. After Lemma 7., we conclude that the Artin-Mazur zeta function of F G, say ζ FG, is rational (cf. [2, 9]). Moreover, given n N, if Per n (F G ) denotes the number of eriodic oints with eriod n of F G, then it is straightforward to check that Per n (F G ) = Fix(fω n ) = Fix(g) = n N n (G). σ n (ω)=ω g =n

9 SEMIGROUP ACTIONS OF EXPANDING MAPS 9 This imlies that, for any z C, ( + ζ S (z) = ex n= and so ζ S is a rational function. Per n (F G ) n n z n) ( + = ex n= Per n (F G ) n ( ) n z ) = ζ FG ( ) z 8. Intrinsic objects vs skew-roduct dynamics In this section we will establish a bridge between toological Markov chains and semigrou actions regarding the notions of fibred and relative toological entroies and the concets of annealed and quenched toological ressures. 8.. Thermodynamic formalism for the skew-roduct. There have been several aroaches to study the thermodynamic formalism of skew-roduct dynamics. For instance: (i) Ruelle exanding skew-roduct mas. (ii) Fibred entroy for factor mas. (iii) Quenched and annealed equilibrium states for random dynamical systems. (iv) Relative measures for skew-roducts. For future use, we briefly collect some of them, referring the reader to [2,, 2, 6, 25]. Toological entroy of the skew-roduct. Since the skew roduct F G is a Ruelle-exanding ma (cf. Lemma 7.), for any Hölder continuous otential φ on Σ + M the ma F G admits a unique equilibrium state. In articular, if φ 0, there is a unique measure µ m of maximal entroy of F G, which is equally distributed along the i= deg (g i) elements of the natural Markov artition Q on Σ + M and may be comuted by the limit rocess using distribution by re-images (cf. [2]). From [26], the rojection of µ m in Σ + is η m, where ( deg (g ) m = k= deg (g k), deg (g 2 ) k= deg (g k),..., deg (g ) ) k= deg (g. (0) k) Moreover, the toological entroy of the skew-roduct F G is given by h to (F G ) = h to (S) + log () (cf. [7]), so h to (F G ) = log( i= deg (g i= deg (gi) i)) and h to (S) = log( ). Notice that the last equality for h to (S) deends only on the ingredients that set u the semigrou action. More generally (see [2]), if φ is iecewise constant along the i= deg (g i) elements of Markov artition Q, then there exists a unique equilibrium state µ φ for F G with resect to φ, it is a Bernoulli measure and satisfies ( P to (F G, φ) = log e φ(q)) and µ φ (Q) = e φ(q) Q Q e φ( Q), Q Q. Q Q Fibred entroy of the skew-roduct. Following [7], consider the skew-roduct F G and the rojection on the first coordinate, say π : Σ + M Σ +, π(ω, x) = ω. We say that a subset E of π (ω) is (n, ε)-searated if there exists i {0,..., n} such that d(g ωi... g ω (x), g ωi... g ω (y)) ε where g = g ωn... g ω G and g = n. Therefore, if s(n, ε, π (ω)) is the maximal cardinality of a (n, ε)-searated subset of π (ω), then s(n, ε, π (ω)) = s(g, n, ε). Given a σ-invariant robability measure η on Σ +, the ma ω h to (F G, π (ω)) := lim ε 0 lim su n n log s(n, ε, π (ω)) is measurable and (cf. [7]) su h µ (F G ) = h η (σ) + h to (F G, π (ω)) dη(ω). (2) {µ : F G (µ)=µ, π (µ) = η} We will refer to h to (F G, π (ω)) as the relative entroy on the fiber π (ω). The fibred entroy of the semigrou action S with resect to η is Σ h to(f + G, π (ω)) dη(ω). Σ +

10 0 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS Quenched and annealed equilibrium states for the skew-roduct. Following [2], given a continuous otential φ : Σ + M R and a robability measure a on {,..., }, the annealed toological ressure of F G with resect to φ and a is defined by P (a) to(f G, φ, a) = su {µ : F G µ=µ} { h µ (F G ) h π (µ)(σ) + h a (π (µ)) + } φ(ω, x) dµ(ω, x) where ω = (ω, ω 2,...), π (µ) is the marginal of µ in Σ + and the entroy er site h a (π (µ)) with resect to η a is given by h ηa (π (µ)) = log ψ π (µ)(ω) dπ (µ)(ω) = ψ π (µ)(ω) log ψ π (µ)(ω) da(ω ) dπ (µ)(σ(ω)) Σ + Σ + dπ (µ) da dπ (µ) σ if dπ (µ)(ω, ω 2,... ) da(ω ) dπ (µ)(ω 2, ω 3,... ) and ψ π (µ) := denotes the Radon- Nykodin derivative of π (µ) with resect to a π (µ) σ; and h a (π (µ)) = otherwise. Recall from [2, age 676] that h a (π (µ)) = 0 if and only if π (µ) = η a. According to [2, Equation (2.28)], the annealed ressure can also be evaluated by P (a) to(f G, φ, a) = su {µ : F G µ=µ} { h µ (F G ) + Σ + M log ( a(ω )e φ(ω,x)) } dµ(ω, x). (3) The quenched toological ressure of F G with resect to φ and a is defined by { } P to(f (q) G, φ, a) = su {µ : F G µ=µ, π (µ)=η a } h µ (F G ) h ηa (σ) + φ(ω, x) dµ(ω, x). (4) It follows from the definitions that we always have P to(f (a) G, φ, a) P to(f (q) G, φ, a). An F G -invariant robability measure is said to be an annealed (res. quenched) equilibrium state for F G with resect to φ and a if it attains the suremum in equation (3) (res. (4)). In the case of finitely generated semigrous of C 2 exanding mas, there exists a unique quenched and a unique annealed equilibrium state for every Hölder continuous observable φ and every a, and they exhibit an exonential decay of correlations (cf. [2]). For instance, for a semigrou G with generators G = {id, g,..., g } where each g i is a C 2 exanding ma, we may consider the Hölder otential φ(ω, x) = log det Dg ω (x) ; its annealed equilibrium state µ (a) φ,a was described in [2, Proosition 2] and is also the quenched equilibrium state for this otential. In less demanding setting of a semigrou action of Ruelle exanding mas, relative (or quenched) equilibrium states have been constructed by Denker, Gordin and Heinemann in [, 2]. Beyond uniform hyerbolicity, Benedicks and Young [3] constructed absolutely continuous stationary measures using aroriate skew-roduct dynamics to model the random erturbations of quadratic mas. Given a continuous otential φ : Σ + M R and a σ-invariant robability measure η on Σ +, the notion of relative ressure of φ on the fiber π (ω), denoted by P to (F G, φ, π (ω)), was also studied in [7, Section 2]. In this reference, the authors showed that the relative ressure satisfies the following relative variational rincile: { } P to (F G, φ, π (ω)) dη(ω) = h µ (F G ) h η (σ) + φ dµ. (5) Σ + su {µ : F G (µ)=µ, π (µ)=η} In articular, when φ 0 and η = η a, the equations (4) and (2) imly that the fibred entroy coincides with the quenched ressure, that is, h to (F G, π (ω)) dη a (ω) = P to(f (q) G, 0, a). Σ + Examle 8.. Let G be a semigrou with generators G = {id, g,..., g }, where each g i is a C 2 exanding ma. For the otential φ 0 we will analyze how the annealed and quenched ressures vary with a. When a = = (,..., ), we obtain { P to(f (a) G, 0, ) = h µ (F G ) + log ( (ω ) ) } dµ(ω, x) = h to (F G ) log = h to (S). su {µ : F G µ=µ} (6)

11 SEMIGROUP ACTIONS OF EXPANDING MAPS The corresonding annealed equilibrium state µ (a) is the measure of maximal entroy µ m of F G. Moreover, by [2, Equation (2.28)], for any non-trivial robability vector a and the corresonding annealed equilibrium state µ (a) a we have so, h (π (µ (a) h a (π (µ (a) a )) = h π (µ (a) ) (σ) + )) = h π (µ (a) ) (σ) + Σ + M Concerning the quenched oerator, we get P (q) to(f G, 0, ) = a Σ + M log ( a(ω ) ) dµ (a) a (ω, x) log ( (ω ) ) dµ (a) (ω, x) = h ηm (σ) log = h ηm (σ) h η (σ). su {µ : F G µ=µ, π (µ)=η } { } h µ (F G ) log. Yet, as π (µ m ) = η m, if m (which haens if the degrees of g i are not equal; cf. (0)), then the quenched equilibrium state differs from µ (a). In general, for a non-trivial robability vector a, we obtain { P to(f (a) G, 0, a) = su h µ (F G ) + log ( a(ω ) ) } dµ(ω, x) = P to (F G, φ a ) (8) {µ : F G µ=µ} Σ + M where φ a : Σ + M R is the locally constant otential given by φ a (ω, x) = log a(ω 0 ). Therefore, a quenched equilibrium state µ (q) a for φ 0 and a satisfies π (µ (q) a ) = η a and In articular, when a = m, we conclude that µ (q) m (7) h (q) µ (F G ) = su h µ (F G ). (9) a {µ : F G µ=µ, π (µ)=η a } = µ m = µ (a) and P (q) to(f G, 0, m) = h to (F G ) h ηm (σ). (20) Relative measures for the skew-roduct. In [26, Theorem.3] it was shown that, having fixed a Bernoulli robability measure η a on Σ +, one may find a self-similar robability measure µ a which is invariant under the skew-roduct F G, whose rojection to the base sace is π (µ a ) = η a and which satisfies h µa (F G ) = su {µ : F G (µ)=µ, π (µ) = η a } h µ (F G ) = h ηa (σ) + a k log deg (g k ). (2) When a = m, the measure µ m is the unique robability of maximal entroy of the skew-roduct F G, and a simle comutation indicates that h to (F G ) = log ( i= deg (g i) ). A generalization for other otentials was obtained in [27] Toological entroy of the semigrou action with resect to a random walk. In what follows we will comare the notions of entroy for the semigrou action with the revious notions of fibred, quenched, annealed and relative ressures. Fibred entroy for the symmetric random walk. We first study the case of the symmetric random walk, that is, when each semigrou generator receives the same weight and η = η. Under this assumtion, the formula (2) becomes su {µ : F G (µ)=µ, π (µ) = η } h µ (F G ) = log + h to (F G, π (ω)) dη (ω). (22) Thus, taking into account that h to (F G ) = su µ h µ (F G ), we conclude that log + h to (F G, π (ω)) dη h to (F G ) log + su h to (F G, π (ω)). (23) Σ + ω Σ + Σ + k=

12 2 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS Besides, () together with (23) imly that h to (F G, π (ω)) dη h to (S) su h to (F G, π (ω)). Σ + ω Σ + So, it makes sense to ask under what conditions h to (S) = Σ + h to(f G, π (ω)) dη. Proosition 8.2. Let G = {id, g,, g }, 2, be a finite set of exanding mas in End 2 (M), G be the semigrou generated by G and F G : Σ + M Σ + M be the corresonding skewroduct. Then the following conditions are equivalent: () h to (S) = Σ h to(f + G, π (ω)) dη. (2) η = π (µ m ), where µ m is the unique maximal entroy measure for F G. (3) The degrees of the mas g i are the same for all i. Moreover, if any of these conditions holds, then h to (S) = Σ log deg (g ω) dη + (ω). Proof. Recall that F G is a toologically mixing Ruelle-exanding ma (cf. Lemma 7.). Hence, taking φ 0, we deduce from [2] that F G has a unique maximal entroy measure µ m. Furthermore, it follows from the variational rincile for F G and the relations (22) and () that h to (S) = h to (F G ) log su [h µ (F G ) log ] = h to (F G, π (ω)) dη (ω). F G (µ)=µ Σ + π (µ) = η Clearly the revious inequality becomes an equality if an only if π (µ m ) = η, which roves that () is equivalent to (2). The remaining of the roof relies on the roerty of µ m stated in (0). It imlies that h to (S) = Σ h to(f + G, π (ω)) dη if and only the degrees of the mas g i are the same for all i. This roves that (2) is equivalent to (3). Finally, as η is a Bernoulli measure, h η (σ) = log and h µm (F G ) = h to (F G ), then h to (S) = Σ log deg (g + ω ) dη (ω). Relative entroy for non-symmetric random walks. If, instead of η, we take another σ- invariant Borel robability measure η on Σ +, then η ortrays an asymmetric random walk on G and it suggests a generalization of the concet of toological entroy of S. Definition 8.3. The relative toological entroy of the semigrou action S with resect to η is given by h to (S, η) = lim ϵ 0 lim su n n log Σ s(g + ω n... g ω, n, ϵ) dη(ω), where s(g ωn... g ω, n, ϵ) is the maximum cardinality of a (g, n, ε)-searated set (cf. Section 6). The revious notion is well defined since the ma ω s(g ωn... g ω, n, ϵ) is constant on n- cylinders (hence measurable), bounded by e n max i {,...,} {h to (g i )} and s(g ωn... g ω, n, ϵ) is monotonic in the variable ϵ. For instance, h to (S, η ) = h to (S). In view of Definition 8.3, h to (S, η) is also given by h to (S, η) = lim su n n log ( g =n ι (η)(g) L g,0 () ). (Notice that this formula makes sense since g L g,0 () is bounded, its values are away from 0 and and ι (η) is a robability measure.) Following an argument analogous to the one used to rove [8, Theorem 25], we obtain: Corollary 8.4. Assume that the continuous action of G on the comact metric sace M is strongly δ -exansive. Then, if 0 < ε < δ, we have h to (S, η) = lim n n log Σ s(g + ω n... g ω, n, ϵ) dη(ω). A variational rincile for the relative entroy. Denote by M B (Σ + ) the sace of Bernoulli measures on Σ +, that is, the robability measures η = η a for some robability vector a = (a,..., a ), where some of the entries may be zero. This sace of σ-invariant measures, which encodes all random walks on the semigrou G we have considered so far, is homeomorhic to the finite dimensional simlex {a = (a,..., a ) R : a i 0 and i= a i = }, and therefore it is a closed subset of M σ (Σ + ). Let H be the entroy ma with resect to the random walks in M B (Σ + ), given by H : M B (Σ + ) [0, + ] η h to (S, η).

13 SEMIGROUP ACTIONS OF EXPANDING MAPS 3 Lemma 8.5. The ma H is continuous. Proof. Firstly, recall that the action of G on the comact metric sace M is strongly δ -exansive. To rove that H is lower semicontinuous, we go back to the roof of Theorem A where it was established that, having fixed ε (0, δ ), the eriodic secification roerty roduces T (ε/2) N such that very g G m+n+t (ε/2) can be written as g = a b c (for a G n and c G m) and Fix(g) Fix(a) Fix(c). This roves that Fix(g) dη ( Fix(c) dη ) ( Fix(a) dη ) and so, as we are restricting to M B (Σ + ), the ma H(η) = su n n log Fix (g ωn... g ω ) dη(ω) (24) is the suremum of the continuous functions η n log Fix (g ωn... g ω ) dη, hence lower semicontinuous. We roceed by showing that H is uer semicontinuous as well. This is due to the characterization of of the toological entroy via generating sets. Indeed, the same stes of the roof of Theorem 25 in [8] (where integration is considered with resect to the equidistributed Bernouli measure η ) imly that, as S is a strongly δ -exansive continuous semigrou action, then h to (S, η) = lim su n n log Σ c(g + ω n... g ω, n, ϵ)dη(ω) for every 0 < ε < δ, where c(g ωn... g ω, n, ϵ) = inf { U : U is a (g ωn... g ω, ε) cover}. Take ε > 0, n N and g = g ωn+m... g ω G, and consider l = g ωn+m... g ωn+ and k = g ωn... g ω. Given an (l, ε)- cover U and a (k, ε)-cover V, the set W = k (U) V is a (g, ε)-cover with W U V. This imlies that c(g ωn+m... g ω, n + m, ϵ) c(g ωn+m... g ωm+, m, ϵ) c(g ωn... g ω, n, ϵ). Since all measures in M B (Σ + ) are Bernoulli, the revious inequality imlies that the sequence (β n ) n N = ( log Σ c(g + ω n... g ω, n, ϵ) dη(ω) ) is subadditive and H is uer semicontinuous as it can be n N exressed as the infimum of continuous functions: H(η) = inf n n log c(g ωn... g ω ) dη(ω). Proosition 8.6. There exists η 0 M B (Σ + ) such that su η MB (Σ + ) h to(s, η) = h to (S, η 0 ). Moreover, ( ) su h to (S, η) = log max deg (g i) (25) η M B (Σ + i ) and η attains the suremum if and only if the degrees of the mas g i are the same for all i. Proof. The first assertion is a direct consequence of the comactness of M B (Σ + ) together with the continuity of the function H. We are left to rove (25). Let j be such that deg (g j ) = max i deg (g i ). Take a = (a i ) i, where a i = δ ij is the Kronecker delta function, and η a = δ jjj.... Then su η MB (Σ + ) h to(s, η) h to (S, η a ) = h to (g j ) = log deg (g j ) = log ( max i deg (g i ) ). Conversely, assume that su η MB (Σ + ) h to(s, η) > log Using (24), we may find δ > 0 and η M B (Σ + ) satisfying ( ) lim n n log Fix(g ωn... g ω ) dη(ω) > log max deg (g i) + 2δ. i ( max i deg (g i ) Then, for every large n, there exists ω Σ + such that Fix (g ωn... g ω ) > e ( δn max i deg (g i ) ) n which contradicts the growth rate( of the fixed oint sets of the exanding mas g ωn... g ω. Finally, recall that h to (S, η ) = log deg (g i) ) i= so η is a maximizing measure for H if and only if the mas g i, for i, have equal degrees. From equality (25) we also conclude that any robability measure in M B (Σ + ) that attains the maximum of H is of the form η a M B (Σ + ) for some a in the simlex } = {a = (a...., a ) R + 0 : a i = and a k = 0 whenever deg (g k ) max deg (g i). i i= In articular, we have uniqueness of such maximizing measures if and only if there exists a unique exanding ma with largest degree. ).

14 4 M. CARVALHO, F. RODRIGUES, AND P.VARANDAS Relative entroy vs. annealed and quenched ressure. In order to comare the relative entroies of the semigrou action with the several notions of ressure for skew-roduct dynamics we shall use the transfer oerators defined in Section 5. As reviously remarked, when η = η a, we have h to (S, η a ) = lim su n n log L n a,φ () 0 = lim su n n log L n a,φ () 0. It follows from [2, Proosition 3.2] that the sectral radius of L a,φ coincides with ex( P to(f (a) G, φ, a)) and, for that reason, h to (S, η a ) = P to(f (a) G, 0, a). (26) This suggests the following generalization of Bufetov s formula (). Corollary 8.7. Let G = {id, g,, g }, 2, where G is a finite set of exanding mas in End 2 (M). If h a (π (µ (a) a )) = h π (µ (a) ) (σ) h η a (σ), then h (a) µ (F G ) = h to (S, η a ) + h ηa (σ). a Proof. It is a direct consequence of the equality (26) since, under the assumtion on h a (π (µ a )), we have { } P to(f (a) G, 0, a) = su h µ (F G ) h π (µ)(σ) + h a (π (µ)) = h (a) µ (F G ) h ηa (σ). a {µ : F G µ=µ} Notice that the assumtion on h a (π (µ (a) a )) is fulfilled when a = (cf. (7)). Given a non-trivial robability vector a, for some otentials φ the transfer oerator L a,φ coincides with the averaged normalized transfer oerator used in [26]. Therefore, we may match the values of the corresonding ressures and their equilibrium states, and deduce the following thermodynamic criterium for the self-similar robability measures constructed in [26]. Proosition 8.8. Let G = {id, g,, g }, 2, where G is a finite set of exanding mas in End 2 (M). Given a Bernoulli robability measure η a on Σ + and the corresonding self-similar robability measure µ a, then the following assertions are equivalent: () h µa (F G ) = su {µ : FG (µ)=µ, π (µ) = η a } h µ (F G ). (2) P to(f (q) G, 0, a) = log deg (g i ) da(i). Moreover, Σ h to(f + G, π (ω)) dη a (ω) = k= a k log deg (g k ) = Σ log deg (g + ω ) dη a (ω). We observe that the condition () is equivalent to say that µ a = µ (q) a, where µ (q) a is the unique quenched equilibrium state of F G with resect to φ 0 and a; notice also that () and (2) haen when a = m. Proof. Fix a = (a, a 2,..., a ) and the otential φ = ( log deg (g ),..., log deg (g )). transfer oerator L a,φ is recisely the averaged normalized transfer oerator introduced in [26]. Therefore, L a,φ =, and consequently P to(f (a) G, φ, a) = 0. Moreover, µ a = µ (a) φ,a, which is the unique annealed equilibrium state for F G with resect to φ and a. So, equation (3) yields h µa (F G ) = log(a(ω )e φ(ω,x) ) dµ a. On the other hand, the quenched variational rincile indicates that { } su {µ : F G µ=µ, π (µ)=η a} h µ (F G ) h ηa (σ) = P to(f (q) G, 0, a). Thus the first condition in the statement of the roosition is equivalent to the equality P to(f (q) G, 0, a) = log(a(ω )e φ(ω,x) ) dµ a h ηa (σ) = a i log a i + i= a i log deg (g i ) + i= a a i log a i = log deg (g i ) da(i). Finally, equations (2) and (2) imly Σ h to(f + G, π (ω)) dη a = k= a k log deg (g k ). The second equality is immediate from the fact that η a is a Bernoulli robability measure. i= The

15 SEMIGROUP ACTIONS OF EXPANDING MAPS Examles. Let us analyze two (non-abelian) examles that illustrate the range of alications of our results on semigrou actions, transfer oerators and the dynamical zeta function. Examle 8.9. Consider the semigrou G generated by G = {id, g, g 2 } where g, g 2 are circle exanding mas given by g (z) = z 2 and g 2 (z) = z 3. A simle comutation shows that, for every n N, we have N n (G) = n 2 n k=0 (2k 3 n k ) = O ( 5 n; 2) consequently, hto (S) = (S) = log 5 log 2. If η m is the Bernoulli robability measure on Σ + 2 determined by the weights m = ( 2 5, 3 5 ) (0.4, 0.6), equation (2) becomes h µ m (F G ) = h to (F G ) = log 5 and equation (2) informs that h to (F G, π (ω)) dη m (ω) = 2 5 log log 3. 5 Σ + 2 So, h to (S) < h Σ + to (F G, π (ω)) dη m (ω). Concerning a = ( 2 2, 2 ) (0.5, 0.5) and η a = η 2, we deduce from Proosition 8.2 that h to (S) > h Σ + to (F G, π (ω)) dη 2 (ω). If µ 2 is the self-similar 2 log 2+log 3 measure assigned to η a in [26], then, again by (2) and (2), we have h µ2 (F G ) = log and su h µ (F G ) = log 2 + h to (F G, π (ω)) dη 2 (ω). µ : F G (µ)=µ, π (µ) = η 2 Σ + 2 Consequently, log 2 + log 3 2 Σ + 2 h to (F G, π (ω)) dη 2 (ω) < h to (S) = log 5 log 2. log 2+log 3 2 and, Moreover, from Proosition 8.8, we conclude that h Σ + to (F G, π (ω)) dη 2 (ω) = 2 more generally, that, for any choice of a = (a, a 2 ) with a i > 0 and a + a 2 =, we have h to (F G, π (ω)) dη a (ω) = a log 2 + a 2 log 3. Therefore, there is a (unique) vector a whose Σ + 2 corresonding robability measure η a on Σ + 2 satisfies h to(s) = h Σ + to (F G, π (ω)) dη a (ω), ( ) 2 log 6 5 namely a =, (0.45, 0.55). log 3 2 log 5 4 log 3 2 Examle 8.0. Given q N, let A i GL(q, Z) induce linear exanding endomorhisms g i := g Ai on T q, for i =,...,. Consider the locally constant otential φ : Σ + T q R given by φ(ω, x) = log det Dg ω (x) = log deg (g ω ). Then, by [2, Proosition 2], the quenched and the annealed equilibrium states of φ coincide and are SRB measures. In this setting, for any non-trivial robability vector a, the self-similar F G -invariant robability measure µ a constructed in [26] coincides with the annealed (hence quenched) SRB measure for F G with resect to F, φ and a. In articular, we have h µa (F G ) = P to(f (q) G, φ, a) + h ηa (σ) φ dµ a so, comaring this equality with (2), we obtain P (q) to(f G, φ, a) = i= a i log deg (g i ) + φ dµ a = log deg (g i ) da(i) + φ dµ a. 9. Selection of measures for semigrou actions The action of a semigrou generated by more than one dynamics is not a dynamical system, thus it is not straightforward how to define equilibrium states and establish a variational rincile that might relate toological and measure theoretical asects of the semigrou action. Yet, a semigrou action can be embodied into a dynamical system whose toological and measure theoretical roerties we may convey to the semigrou action. From the formulas (8) and (26) in Section 8, recall that h to (S, η a ) = P to(f (a) G, 0, a) = P to (F G, φ a ). These two different flavored equalities motivate the construction of maximal entroy measures for semigrou actions from the skew-roduct dynamics, in a way that discloses eriodic data and highlights the equidistribution of re-images.