Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

(1)

Ann. I. H. Poincaré – AN 30 (2013) 225–249

www.elsevier.com/locate/anihpc

Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

A. Castro, P. Varandas

^∗

Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil Received 13 May 2011; received in revised form 29 May 2012; accepted 16 July 2012

Available online 4 August 2012

Abstract

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle–Perron–Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

1. Introduction

The thermodynamical formalism was brought from statistical mechanics to dynamical systems by the pioneer- ing works of Sinai, Ruelle and Bowen [39,11,12]in the mid seventies. Indeed, the correspondence between one- dimensional lattices and uniformly hyperbolic maps, via Markov partitions, allowed to translate and introduce several notions of Gibbs measures and equilibrium states in the realm of dynamical systems. Nevertheless, although uniformly hyperbolic dynamics arise in physical systems (see e.g.[25]) they do not include some relevant classes of systems including the Manneville–Pomeau transformation (phenomena of intermittency), Hénon maps and billiards with convex scatterers. We note that all the previous systems present some non-uniformly hyperbolic behavior and its relevant measure satisfies some weak Gibbs property. Moreover, an extension of the thermodynamical formalism beyond the scope of uniform hyperbolicity reveals fundamental difficulties. Even in the non-uniformly hyperbolic context, where there are no zero Lyapunov exponents and there exists a non-uniform geometric theory of invariant manifolds, the absence of finite generating Markov partitions constitutes an obstruction to use the same strategy pushed forward before. Nevertheless, more recently there have been established many evidences that non-uniformly hyperbolic dynamical systems admit countable and generating Markov partitions. This is now parallel to the development of a thermodynamical formalism of gases with infinitely many states, a hard subject not yet completely understood. We refer the reader to[16,31,32]for recent progress in this direction.

* Corresponding author.

E-mail addresses:[email protected](A. Castro),[email protected](P. Varandas).

http://dx.doi.org/10.1016/j.anihpc.2012.07.004

(2)

So, despite the effort of many authors, a general picture is still far from complete. Some of the recent contributions concerning the existence and uniqueness of equilibrium states in a context of non-uniform hyperbolicity include[21, 13,38,15,16,43,26,14,33,29,30,40,37,17,37,41,34]. Many of these papers deal with dynamical systems with neutral periodic points, unimodal maps, perturbations of hyperbolic transformations and shifts with countable many symbols, some of the relevant sources of examples of non-hyperbolic systems. However, a deep study on the statistical properties of the equilibrium states, as the mixing properties, limit theorems, strong stability under deterministic and random perturbations or regularity of the topological pressure is usually obtained as a consequence of the spectral properties of the Ruelle–Perron–Frobenius operator. This functional analytic approach has gained special interest in the last few years and produced new and interesting results even in the uniformly hyperbolic setting (see e.g.[9,23,8]). Just for completeness let us mention that, since the (semi)conjugacy between uniformly hyperbolic dynamical systems and the symbolic dynamics is only Hölder continuous, the strategy developed in the seventies did not allowed to understand the statistical properties in the space of smooth observables. Important and recent extensions of this functional analytic approach to the setting of non-uniform hyperbolicity include e.g. the works[44,28,18–20,4,5,36].

In this article we study the strong statistical properties of some equilibrium states built in[41]for a large class of non-uniformly expanding local homeomorphisms that may not admit a Markov partition. Using a characterization of equilibrium states as weak Gibbs measures absolutely continuous with respect to conformal reference measures, the authors proved roughly that every local homeomorphism with coexistence of expanding and contraction exhibit a form of average expansion. This enables to use Birkhoff’s method of projective cones applied to the Ruelle–Perron–

Frobenius operator acting on suitable Banach spaces to obtain the existence of a unique equilibrium state for any Hölder continuous potential with low variation and that it satisfies strong statistical properties. Natural examples are obtained by bifurcation of expanding homeomorphisms and subshifts of finite type and allows intermittency phenomena. Even in the absense of Markov partition we establish that the Ruelle–Perron–Frobenius transfer operator has a spectral gap in the Banach spaces of both Hölder continuous and smooth observables. This was inspired and extends the work of Matheus and Arbieto[1]that considered local diffeomorphisms under some slightly different assumptions but where the existence of a finite Markov partition played an important role. In consequence, we get an alternative proof for the existence and uniqueness of equilibrium states in[41], obtain exponential decay of correlations and prove a central limit theorem. Moreover, we prove that in this non-uniformly expanding setting the topological pressure varies continuously with respect to the dynamics and the potential.

At this point one could think the stability of the equilibrium states under deterministic and random perturbations could follow directly from the spectral gap property. We refer the reader to[24]for perturbation theory of smooth families of quasi-compact operators. However this is not the case since the transfer operators acting on the space of Hölder continuous potentials may not vary continuously on the dynamical system as illustrated in Example4.14.

Nevertheless we prove that the densities of the equilibrium states with respect to the conformal measures are Hölder continuous and vary uniformly with the dynamics. Strong statistical and stochastic stability results hold in the space of differentiable observables and are proved after careful analysis of the action of the transfer operators in those functional spaces. We obtain a spectral stability under random perturbations. Namely, the spectral components of the Ruelle–Perron–Frobenius operator associated to general random perturbations of the transformation and the potential varies continuously and converges to the spectral components of Ruelle–Perron–Frobenius of the unperturbed dynamical system outside of a disk containing zero in the spectrum.

Finally, let us also mention that the program to understand to statistical and stochastic properties of the equilibrium states for this class of multidimensional non-uniformly expanding transformations is under way. Some of the very interesting remaining questions are to understand if one can obtain further regularity of the topological pressure and the density of the equilibrium states with respect to conformal measures along parametrized families of potentials (e.g.

real analytic) and the study of zeta functions. Such program has been carried out with success for uniformly hyperbolic and some partially hyperbolic and one-dimensional non-uniformly expanding dynamical systems. See e.g.[2,35,22, 6,7] and the references therein. Just to mention some recent developments, in a joint work with T. Bomfim[10], we prove the differentiability of thermodynamical quantities as topological pressure, invariant densities, conformal measures and measures of maximal entropy despite the lack of continuity of the Ruelle–Perron–Frobenius operator with respect to the dynamics.

This paper is organized as follows. In Section2, we recall some definitions and make the precise statements of our main results and some preliminary results are given in Section3. The proof of the spectral gap for the Ruelle–

Perron–Frobenius operator in the space of Hölder continuous observables, continuity of the topological pressure,

(3)

uniform continuity of the densities of equilibrium states with respect to conformal measures and exponential decay of correlations are given in Section4. In Section 5 we show that Ruelle–Perron–Frobenius operator acting on the space of smooth observables also admits a spectral gap and obtain the strong stability of the equilibrium states under deterministic and random perturbations. Finally, some examples are given in Section6.

2. Statement of the main results

2.1. Setting

LetM be compact and connected Riemannian manifold of dimensionmwith distanced. Letf :M→M be a local homeomorphismand assume that there exists a continuous functionx→L(x)such that, for everyx∈Mthere is a neighborhoodU_xofxso thatf_x:U_x→f (U_x)is invertible and

d

f_x⁻¹(y), f_x⁻¹(z)

L(x)d(y, z), ∀y, z∈f (Ux).

In particular every point has the same finite number of preimages deg(f )which coincides with the degree off. For all our results we assume thatf andφsatisfy conditions (H1), (H2), and (P) stated below. Assume there exist constants σ >1 andL1, and an open regionA⊂Msuch that

(H1) L(x)Lfor everyx∈AandL(x) < σ⁻¹for allx /∈A, andLis close to 1: the precise condition is given in (3.1) and (3.2).

(H2) There exists a finite covering U of M by open domains of injectivity forf such that A can be covered by q <deg(f ).

The first condition means that we allow expanding and contracting behavior to coexist inM:fis uniformly expanding outsideAand not too contracting insideA. In the case thatAis empty thenf is uniformly expanding. The second one requires that every point has at least one preimage in the expanding region. An observableg:M→Risα-Hölder continuous if the Hölder constant

|g|α=sup

x=y

|g(x)−g(y)| d(x, y)^α

is finite. As usual, we endow the spaceC^α(M,R)of Hölder continuous observables with the norm · α= · 0+|·|α. We assume that the potentialφ:M→Ris Hölder continuous and that

(P) supφ−infφ < εφand|e^φ|α< ε_φe^infφ

for someεφ>0 satisfying Eq. (4.1), depending on the constantsL,σ,qand deg(f ). The previous is an open condition on the potential, relative to the Hölder norm, and it is satisfied e.g. by constant functions. In particular we consider measures of maximal entropy. The second condition above means that exp(φ)is contained in a small cone of Hölder continuous as discussed after Theorem4.1.

2.2. Existence and uniqueness of equilibrium states

Let us first recall some necessary definitions. Given a continuous mapf:M→Mand a potentialφ:M→R, the variational principle for the pressure asserts that

P_top(f, φ)=sup

h_μ(f )+

φ dμ: μisf-invariant

whereP_top(f, φ)denotes the topological pressure off with respect toφandh_μ(f )denotes the metric entropy. An equilibrium statefor f with respect toφ is an invariant measure that attains the supremum in the right-hand side above.

The equilibrium states constructed in[41]are absolutely continuous with respect to an expanding, conformal and non-lacunary Gibbs measureν. Let us recall these definitions and the notions involved. A probability measureν, not

(4)

necessarily invariant, is conformalif there exists a functionψ:M→Rsuch thatν(f (A))=

Ae⁻^ψdν for every measurable setAsuch thatf |Ais injective. LetS_nφ=_n₋₁

j=0φ◦f^j denote thenth Birkhoff sum of a functionφ.

Thebasin of attractionof anf-invariant, ergodic probability measureμis the setB(μ)of pointsx∈Msuch that the probability measures ¹_n_n₋₁

j=0δ_fj(x)converge weakly toμwhenn→ ∞. We build over the following theorem which is a direct consequence of the results in[41].

Theorem 2.1. Letf :M→M be a local homeomorphism with Lipschitz continuous inverse satisfying(H1), (H2) and φ:M→R a Hölder continuous potential such thatsupφ−infφ <log deg(f )−logq. Then, there exists a finite number of ergodic equilibrium statesμ₁, μ₂, . . . , μ_kforf with respect toφ, and they are absolutely continuous with respect to some conformal expanding measureν. Moreover, the union of the basins of attractionB(μ_i)contains ν-almost every point.

Observe that despite the characterization that equilibrium states are absolutely continuous invariant measures no information was known e.g. on the continuity of the topological pressure and density functions. Here we shall address these questions, the uniqueness of the equilibrium states and also the strong stability of the equilibrium states. Since our assumption (P) implies that the potential φ has small variation then it fits in the assumption of the previous theorem. We will build over the aforementioned result with a completely different functional analytic approach.

2.3. Statement of the main results

In this section we recall some necessary definitions and state our main results. The Ruelle–Perron–Frobenius transfer operator Lφ associated to f :M→M andφ:M→Ris the linear operator defined on a Banach space X⊂C⁰(M,R)of continuous functionsϕ:M→Rby

Lφϕ(x)=

f (y)=x

e^{φ (y)}ϕ(y).

Sincef is a local homeomorphism it is clear thatLφϕ is continuous for every continuousϕ and, furthermore,Lφis indeed a bounded operator relative to the norm of uniform convergence inC⁰(M,R)because Lφ deg(f )e^sup^|^φ^|. Analogously,Lφpreserves the Banach spaceC^α(M,R), 0< α <1 of Hölder continuous observables. Moreover, it is not hard to check thatLφis a bounded linear operator in the Banach spaceC^r(M,R)⊂C⁰(M,R)(r1) endowed with the norm · r wheneverf is aC^r-local diffeomorphism andφ∈C^r(M,R). We say that the Ruelle–Perron–

Frobenius operatorLφacting on a Banach spaceXhas thespectral gap propertyif there exists a decomposition of its spectrumσ (Lφ)⊂Cas follows:σ (Lφ)= {λ₁} ∪Σ₁whereλ₁is a leading eigenvalue forLφwith one-dimensional associated eigenspace andΣ1{z∈C: |z|< λ1}.

The first result is a spectral gap for the Ruelle–Perron–Frobenius operator in the space of Hölder continuous observables, which is enough to derive the uniqueness and further regularity of the density of the equilibrium state with respect to the conformal measure.

Theorem A.Let f :M→M be a local homeomorphism with Lipschitz continuous inverse and φ:M→Rbe a Hölder continuous potential satisfying(H1), (H2)and (P). Then the Ruelle–Perron–Frobenius has a spectral gap property in the space of Hölder continuous observables, there exists a unique equilibrium stateμforf with respect toφand the densitydμ/dνis Hölder continuous.

Let us mention that the previous result holds for more general compact invariant subsetsK⊂M(with the induced topology) also under the assumption that every point has constant number of preimages inKand at least one preimage in the expanding region, as considered in[41]. Since we will be interested in further extensions to differentiable dynamics as discussed below we will not prove or use this fact here. Let us give two important consequences of the previous result.

(5)

Corollary 1.The equilibrium stateμhas exponential decay of correlations for Hölder continuous observables:there exists some constant0< τ <1such that for allϕ∈L¹(ν), ψ∈C^α(M)there existsK(ϕ, ψ ) >0satisfying

M

ϕ◦fⁿ ψ dμ−

M

ϕ dμ

M

ψ dμ

K(ϕ, ψ )·τⁿ, for everyn1.

As a byproduct of the previous theorem we also obtain a Central Limit Theorem.

Corollary 2.Letϕbe a Hölder continuous function and set σ_ϕ²:=

v²dμ+2

∞ j=1

v· v◦f^j

dμ, wherev=ϕ−

ϕ dμ.

Thenσ_ϕ<∞andσ_ϕ=0iffϕ=u◦f−ufor someu∈L¹(μ). Furthermore, ifσ_ϕ>0then the following convergence on distribution

μ

x∈M: 1

√n

n−1

j=0

ϕ

f^j(x)

−

ϕ dμ

∈A

→ 1

σϕ

√2π

A

e⁻

t2 2σ2

ϕ dt,

holds asn→ ∞for every intervalA⊂R.

The stability of the equilibrium state under deterministic perturbations is more subtle. In fact, the Ruelle–Perron–

Frobenius operatorLf,φacting on the space of Hölder continuous observables is continuous on the potentialφbut in general itmay not vary continuouslywith the underlying dynamicsf, as shown in Example4.14. Nevertheless we could obtain further that the Hölder continuous densities of the equilibrium states with respect to the conformal measures vary continuously with the dynamics in theC⁰-topology and that the topological pressure varies continuously, which gives a nontrivial extension of the weak^∗stability results in[41].

Theorem B.LetFbe a family of local homeomorphisms with Lipschitz inverse and letW be some family of Hölder continuous potentials satisfying(H1), (H2)and (P)with uniform constants. Then the topological pressure function F×W(f, φ)→P_top(f, φ)is continuous. Moreover, the invariant density function

F×W→C^α(M,R), (f, φ)→dμ_f,φ

dνf,φ

is continuous wheneverC^α(M,R)is endowed with theC⁰-topology.

2.3.1. Stronger stability results

Now we pay attention to the stability of the equilibrium states under both deterministic and an arbitrary random perturbations. To obtain stronger statistical stability results we will admit that the dynamics isC^r-differentiable (r1) and give a detailed study of the spectral properties for the Ruelle–Perron–Frobenius operator acting on the space C^r(M,R). Associated toφ∈C^r(M,R)consider the condition:

(P) supφ−infφ < εφand maxsr D^sφ 0< ε_φ

for some ε_φ >0 expressed precisely in Eq. (4.2) and depending on L, σ, q, deg(f ), ε_φ and r. This is an open condition on the set of potentials, satisfied by constant potentials, and a natural generalization of condition (P) to the differentiable setting.

Theorem C.Given an integer r1, let F^r be a family ofC^r-local diffeomorphisms and let W^r be a family of C^r-potentials satisfying (H1), (H2) and (P) with uniform constants. Then the topological pressure F^r ×W^r (f, φ)→P_top(f, φ)and the invariant density

(6)

F^r×W^r→C^r(M,R), (f, φ)→ dμ_f,φ

dν_f,φ

vary continuously in theC^r-topology. Moreover, the conformal measure function F^r×W^r→M(M),

(f, φ)→ν_f,φ

is continuous in the weak^∗topology. In consequence, the equilibrium measureμ_f,φvaries continuously in the weak^∗ topology.

Finally we will describe our results on the stability of the spectra of the Ruelle–Perron–Frobenius operator under random perturbations. Givenr∈N, and familiesF^r of local diffeomorphisms andW^r ofC^r-observables satisfying (H1), (H2) and (P) with uniform constants, arandom perturbationoff ∈F is a familyθ_ε, 0< ε1 of probability measures inF^r ×W^r such that there exists a familyV_ε(f, φ), 0< ε1 of neighborhoods of (f, φ), depending monotonically onεand satisfying

suppθ_ε⊂V_ε(f, φ) and

0<ε1

V_ε(f, φ)= (f, φ)

.

This dynamics can be codified by considering the skew product map F :F^N×M→F^N×M,

(f , x)→

σ ( f ), f₁(x)

wheref=(f1, f2, . . .)andσ:F^N→F^Nis the shift to the left. Associated to this random dynamical system consider theintegrated Ruelle–Perron–Frobenius operatorLεgiven by

Lεϕ(x)=

(Lf,φϕ)(x) dθ_ε(f ). (2.1)

We say that (f, φ)has C^r-spectral stabilityunder the random perturbation if the operatorLε in the Banach space C^r(M,R)has the spectral gap property and the leading eigenvalueλ_εand associated eigenfunctionh_εvary continuously withεand accumulate, asε→0, respectively on the leading eigenvalue and eigenfunction of the unperturbed operator. We prove the following spectral stability under random perturbations.

Theorem D.Let(θ_ε)_εbe any random perturbation of(f, φ)∈F^r×W^r. Then(f, φ)hasC^r-spectral stability under the random perturbation(θ_ε)_ε.

Some comments are in order. Weaker stochastic stability results were previously obtained in[41]under a non- degeneracy assumption. Namely, assuming that allf ∈Fare non-singular with respect to a fixed conformal measure it follows that there are stationary measures μ^ε absolutely continuous with respect to the conformal measureνand that converge to the equilibrium state μ in the weak^∗ topology as the noise levelεtends to zero. Here we obtain spectral stability under arbitrary random perturbations.

3. Preliminaries

In this section we provide some preparatory results needed for the proof of the main results. Namely, we study the combinatorics of the orbits, hyperbolic times and some pressure estimates.

3.1. Combinatorial estimates for orbits

Here we give a description of the orbits of points according to the visit to the possibly not expanding region A using an auxiliary partitionPbuilt using (H2).

(7)

Lemma 3.1.There exists a partitionP ofM of domains of injectivity forf with cardinality at mostU and such that

{U∈U: U∩A= ∅} =

{P ∈P: P∩A= ∅}. In particular there are at mostq <deg(f )elements ofPthat coverA.

Proof. Pick an enumeration{U_i}of the open coveringU given by (H2) in such a way that the regionAis covered by the firstq elements ofU. Consider the partitionP given byP₁=U₁and, recursively,P_i₊₁=U_j₊₁\(_i

j=1P_j) fori=1. . .#U−1. It is clear thatPU. Moreover,f|Piis injective for every nonemptyP_i since by construction P_i ⊂U_i and

{U∈U: U∩A= ∅} = q j=1

U_j= q j=1

P_j=

{P ∈P: P∩A= ∅}.

Since the last statement in the lemma is immediate from the construction this finishes the proof of the lemma. 2 Since the regionAis contained inq elements of the partitionP we can assume without any loss of generality that Ais contained in the firstqelements ofP. For allxwe can associate anitineraryi(x)∈(i₀, . . . , i_n₋₁)∈ {1, . . . ,#P}ⁿ byi_j=if and only iff^j(x)∈P. Givenγ∈(0,1)andn1, let us consider also the setI (γ , n)of all itineraries (i₀, . . . , i_n₋₁)so that #{0j n−1:i_jq}> γ n.

Lemma 3.2.Givenε >0there existsγ₀∈(0,1)such that c_γ:=lim sup

n→∞

1

nlog #I (γ , n) <logq+ε for everyγ∈(γ₀,1).

Proof. See[41, Lemma 3.1]. 2

We are in a position to state our precise condition on the constantLin assumption (H1) and the constantcin the definition of hyperbolic times. First note that if supφ−infφ <log deg(f )−logq as in Theorem2.1then it follows from Lemma3.2that one may findγ <1 such thatc_γ <log deg(f )−supφ+infφ. We assume thatLis close enough to 1, andc >0 and 0< ε_φ<log deg(f )−logq are so that

σ⁻⁽¹⁻^{γ )}L^γ < e⁻^2c<1 (3.1)

and e^ε^φ·

(deg(f )−q)σ⁻^α+qL^α[1+(L−1)^α] deg(f )

<1. (3.2)

The first condition is to guarantee the existence of infinitely many hyperbolic times with respect to the reference measure in the proposition below. The second technical condition roughly means that f has some average backward contraction and will be used to obtain the invariance of a cone of functions under the Ruelle–Perron–Frobenius operator in Proposition4.1.

3.2. Ruelle–Perron–Frobenius operators and conformal measures

Recall that the Ruelle–Perron–Frobenius transfer operatorLφ:C⁰(M,R)→C⁰(M,R)associated tof :M→M andφ:M→Ris the linear operator defined on the spaceC⁰(M,R)of continuous functionsϕ:M→Rby

Lφϕ(x)=

f (y)=x

e^{φ (y)}ϕ(y).

In fact,Lφϕ is continuous sincef is a local homeomorphism andϕis continuous. Moreover, it is not hard to check thatLφis a bounded operator, relative to the norm of uniform convergence inC⁰(M,R)and Lφ deg(f )e^sup^|^φ^|. Consider also the dual operator L^∗_φ:M(M)→M(M) acting on the spaceM(M) of Borel measures in M by

(8)

ϕ d(L^∗_φη)=

(Lφϕ) dηfor everyϕ∈C⁰(M,R). Letr(Lφ)be the spectral radius ofLφ. In our context conformal measures associated to the spectral radius always exist as stated in the next proposition, whose proof can be found in the proofs of Theorem B and Theorem 4.1 in[41].

Proposition 3.3.If f is topologically exact and satisfies(H1), (H2)and φ satisfies supφ−infφ <log deg(f )− logq then there exists an expanding conformal measure such thatL^∗_φν=λν andsupp(ν)=H, whereλ=r(Lφ) deg(f )e^inf^φ. Moreover,ν is a non-lacunary Gibbs measure and has a Jacobian with respect tof given byJνf = λe⁻^φ.

Just for completeness let us mention that one key ingredient is that our assumptions guarantee we obtain volume expansion with respect to the conformal measure, that is, J_νf (x)deg(f )e^inf^φ⁻^supφ> e^c^γ for all x∈M. This is enough to guarantee thatν-almost every point spends at most a fractionγ of time inside the domainAwheref may fail to be expanding. Notice also thatλ=

Lφ1dν.

Finally, we collect the main estimates concerning the pressure of the invariant setsH andH^c, which play a key role in the construction of equilibrium states.

Proposition 3.4. P_top(f, φ)=P_H(f, φ)=logλ > P_H^c(f, φ), where λ denotes the spectral radius of the Ruelle–

Perron–Frobenius Lφ acting on the space of continuous observables. In consequently, any equilibrium state is an expanding measure.

Proof. See Proposition 6.1, Lemma 6.4 and Lemma 6.5 in[41]. 2 3.3. Regularity of the observables

Here we study a relation between Hölder and locally Hölder continuous functions. We say that ϕ:M→Ris (C, α)-Hölder continuous in balls of radiusδif

ϕ(x)−ϕ(y)Cd(x, y)^α

for everyy∈B(x, δ)andx∈M. Our first auxiliary lemma for the regularity of observables is as follows.

Lemma 3.5.Given1ζ 2and δ >0, ifϕ:M→Ris(C, α)-Hölder continuous in balls of radiusδ then it is (C(1+r^α), α)-Hölder continuous in balls of radius(1+r)δζ δ, with0< r1.

Proof. SinceMis connected then giveny, z∈Mso thatd(y, z) < (1+r)δby considering a geodesic arc connecting y andzinMthere existswso thatd(z, w)=δandd(w, y) < rd(z, w) < δ. Therefore

ϕ(z)−ϕ(y)ϕ(z)−ϕ(w)+ϕ(w)−ϕ(y)Cd(z, w)^α+Cd(w, y)^α

C

1+r^α

d(z, w)^αC 1+r^α

d(z, y)^α, which proves the lemma. 2

The next lemma asserts that every locally Hölder continuous observable is indeed Hölder continuous. Moreover, we give an estimate for the Hölder constant.

Lemma 3.6.LetN be a compact and connected metric space. Givenδ >0there existsm1 (depending only onδ) such that the following holds:ifϕ:N→Ris(C, α)-Hölder continuous in balls of radiusδthen it is(Cm, α)-Hölder continuous.

Proof. Fixδ >0 and letB= {B(x_i, δ/3)}i=1...s be a finite covering ofN. We can assume, without loss of generality, thatx_j∈B(x_j₊₁, δ)for everyj=1, . . . , s−1. Our hypothesis guarantee that ifx, w∈N withd(x, w) < δwe have

|ϕ(x)−ϕ(w)|Cd(x, w)^α. Hence, ifd(x, w)δthen it is not hard to use the triangular inequality to get ϕ(x)−ϕ(w)(s+2)Cδ^αC(s+2) d(x, w)^α.

Thus it is enough to takems+2 in the lemma. 2

(9)

3.4. Positive operators and cones

In this subsection we shall recall some results concerning the theory of projective metrics on cones and positive operators due to G. Birkhoff. Despite the great generality of this theory we shall concentrate on cones and positive operators on Banach spaces. We refer the reader to[27,3]for detailed presentations.

LetBbe a Banach space. A subsetΛ⊂B− {0}is aconeifr·v∈Λfor allv∈Λandr∈R⁺. The coneΛis closedifΛ=Λ∪ {0}, andΛisconvexifv+w∈Λfor allv, w∈Λ. Notice that a convex coneΛwithΛ∩(−Λ)= ∅ determines apartial orderingonBgiven by:

wv iff v−w∈Λ∪ {0}.

In the sequel, our conesΛare assumed to be closed, convex andΛ∩(−Λ)= ∅. Given a coneΛand two vectors v, w∈Λ, we defineΘ(v, w)=Θ_Λ(v, w)by

Θ(v, w)=logB_Λ(v, w) A_Λ(v, w),

whereAΛ(v, w)=sup{r∈R⁺: r·vw}andBΛ(v, w)=inf{r∈R⁺: wr·v}. The (pseudo-)metricΘis called theprojective metricofΛ(orΛ-metricfor brevity). Defining the equivalence relationv∼wiffw=r·vfor some r∈R⁺, then Θ induces a metric on the quotientΛ/∼. The following key result is due to Birkhoff, which can be found e.g. in[42, Proposition 2.3].

Theorem 3.7.LetΛ_ibe a closed convex cone(withΛ_i∩(−Λ_i)= ∅)in a Banach spaceBi, fori=1,2. IfL:B1→ B2is a linear operator such thatL(Λ₁)⊂Λ₂and=diamΘΛ2(LΛ₁) <∞then

Θ_Λ₂(Lv,Lw)

1−e⁻

·Θ_Λ₁(v, w), for anyv, w∈Λ1.

In consequence of the previous theorem, if the diameter of the coneL(Λ₁)is finite inΛ₂thenLis a contraction in the projective metric which enables us to prove that it admits a unique fixed point.

3.5. Combinatorial lemma on preimages matching

Here we establish an auxiliary lemma to bound for the distance of preimages associated to different functions inF which will play a key role in the proof of the stability results. LetV_ε(f )⊂Fbe an open neighborhood off ∈F.

Lemma 3.8. Givenn1,f , g∈F^N and x, y∈M there exists bijection between the sets of preimages {z∈M: fⁿ(z)=x}and{z∈M:gⁿ(z)=y}. Moreover, for everyn∈Nthere existsε(n) >0such that for every0< εε(n) the distance between pairedn-preimages is such that ifd(x, y) < εandg∈V_ε( f )then

d

x_i⁽ⁿ⁾, y_i⁽ⁿ⁾

Lⁿd(x, y)+

n

j=1

Lⁿ⁻^j⁺¹ f_j−g_j _α,

for everyi=1. . .deg(f )ⁿ.

Proof. LetUˆ be a finite open cover by balls obtained using domains of invertibility forf and let 2δˆbe the Lebesgue number of the covering Uˆ. If ε >0 is small enough the constant 2δˆ can be taken uniform for everyf˜∈V_ε(f ).

Letx, y∈M satisfy d(x, y) < εand takef =(f_i)_i_∈N andg=(g_i)_i_∈N withg∈V_ε( f ). We will prove the result recursively.

First notice that the sets{z∈M: f1(z)=x}and{z∈M: g1(z)=y}have the same cardinality deg(f )and thus there exists a one-to-one correspondence. Moreover, reducingε >0 if necessary, we obtain that the paired enumerations{x_i}and{y_i}of such elements verify

(10)

d

g₁(y_i), g₁(x_i)

=d

y, g₁(x_i)

d(y, x)+d

x, g₁(x_i)

=d(y, x)+d

f1(xi), g1(xi)

d(y, x)+ f1−g1 α<δ.ˆ

Sinceg₁∈Fthen it satisfies (H1), (H2) and sod(x_i, y_i)L[d(y, x)+ f₁−g₁ _α]for everyi. The same argument as above applied to the pairsx_i=f₂(x_j⁽²⁾)andy_i=g₂(y_j⁽²⁾)proves thatd(g₂(y_j⁽²⁾), g₂(x_j⁽²⁾))d(x_i, y_i)+ f₂−g₂ _α and, consequently,

d

y_j⁽²⁾, x_j⁽²⁾

L

d(x_i, y_i)+ f2−g2 α

L²

d(x, y)+ f₁−g₁ _α

+L f₂−g₂ _α

=L²d(x, y)+L² f₁−g₁ _α+L f₂−g₂ _α,

which can be taken also smaller thanδˆprovided that we reduceε. Using the same reasoning recursively, ifd(x, y) <

ε(n)small so that the corresponding paired enumerations of preimages(x_i^(k))and(y_i^(k)),i=1. . .deg(f )^kin the sets {z∈M: f^k(z)=x}and{z∈M: g^k(z)=y}areδ-close for every 1ˆ kn−1. Moreover, applying the previous reasoning it follows that

d

x_i⁽ⁿ⁾, y_i⁽ⁿ⁾

Lⁿd(x, y)+

n

j=1

Lⁿ⁻^j⁺¹ f_j−g_j _α

as claimed. This finishes the proof of the lemma. 2

We also get a simple expression for the distance ofn-preimages associated to the same close functions inFand the same base point inM.

Corollary 3.9.Givenn∈Nthere existsε(n) >0such that for anyf₁, f₂∈Fwith f₁−f₂ _α< ε(n)the following property holds:givenx∈Mand paired preimages(x_1i⁽ⁿ⁾)and(x_2i⁽ⁿ⁾)byf₁ⁿandf₂ⁿ, respectively, then

d

x_1i⁽ⁿ⁾, x_2i⁽ⁿ⁾

nLⁿ f₁−f₂ _α for alli.

Proof. This is a direct consequence of the previous lemma, by consideringx=y and the sequences of functions f=(f1, f1, f1, . . .)andg=(f2, f2, f2, . . .)inVε(f )^Nwithεsmall. 2

We finish this section by proving that paired preimages associated to any close points have similar behavior with respect to the regionA. More precisely,

Lemma 3.10.Letf satisfy assumptions(H1)and(H2). Then there existsδ >0so that for every ballBof radiusδ has at mostq <deg(f )connected components inf⁻¹(B)that intersectA. In particular, ifd(x, y) < δthen there are at mostqpairs of paired preimages byf associated toxandythat belong toA.

Proof. Assume that δ0 >0 is small so that every inverse branch is well defined in a ball of radius δ0. Since ({f⁻¹(x)} ∩A)q then for everyx∈Mthere exists 0< δ_x< δ₀so thatf⁻¹(B(x, δ_x))has at mostq connected components that intersectA. By compactness ofM pick a finite subcoverB= {B(x_i, δ_i)}i∈I, set 2δto be Lebesgue number ofBand assume, without loss of generality, thatδ < δ₀. Therefore, by construction, given any ballBof radius δit follows thatB⊂B(x_i, δ_i)for somei∈I. In consequence, the number of connected components satisfy

c.c.

f⁻¹(B)∩A

c.c.

f⁻¹

B(x_i, δ_i)

∩A

q.

This finishes the proof of the lemma. 2

(11)

4. Ruelle–Perron–Frobenius operator inC^α(M,RRR): Spectral gap and statistical consequences

In this section we prove that the action of the transfer operator in the space of Hölder continuous observables has the spectral gap property. In consequence, we provide an alternative proof for the existence and uniqueness of equilibrium states as well as further statistical properties: exponential decay of correlations and central limit theorem.

We also get that the densities of the unique equilibrium state with respect to the conformal measures are Hölder and vary continuously in a uniform way with the dynamical system. Finally, the topological pressure also varies continuously in this non-uniformly expanding setting.

4.1. Invariant cones for the transfer operator inC^α(M,R)

To prove that the Ruelle–Perron–Frobenius operator has a spectral gap in the space of Hölder continuous observables one first introduce some notations. Recall that the Hölder constant ofϕ∈C^α(M,R)is

|ϕ|α=sup

x=y

|ϕ(x)−ϕ(y)| d(x, y)^α

and set|ϕ|α,δas the least constantC >0 such that|ϕ(x)−ϕ(y)|Cd(x, y)^αfor all pointsx, ysuch thatd(x, y) < δ.

Now, consider the cone of locally Hölder continuous observables Λ_κ,δ=

ϕ∈C⁰(M,R):ϕ >0 and|ϕ|α,δ

infϕ κ

.

Throughout, letδ >0 be fixed and given by Lemma3.10. Fix alsomgiven by Lemma3.5associated to balls of radiusδ. We are now in a position to state the precise condition on the constantsε_φandε_φon (P) and (P) respectively.

Then taking into account (3.2) we assume:

e^ε^φ·

+εφ2mL^αdiam(M)^α<1 (4.1)

and

1+ε_φ

·e^ε^φ·

<1. (4.2)

Notice that having (3.2) it is possible to considerε_φ satisfying the later condition. Our main result in this section is as follows.

Theorem 4.1.Assume thatf satisfies(H1), (H2)and thatφsatisfies(P). Then there existδ >0and0<λ <ˆ 1such thatLφ(Λ_κ,δ)⊂Λ_λκ,δ_ˆ for every large positive constantκ.

Proof. Take κ >0 and let ϕ∈Λ_κ,δ be given. Moreover, given x∈M we consider the set (x_j)_j₌_1..._{deg(f )} of the preimages by f of the point x, that is f (x_j)=x, and letK= |ϕ|α,δ be the α-Hölder constant of ϕ on balls of radiusδ. We will prove that there exists 0<λ <ˆ 1 such thatLφϕ∈Λ_λκ,δ_ˆ provided thatκis large enough. Indeed, if d(x, w) < δthen

|Lφϕ(x)−Lφϕ(w)| infz∈M{Lφϕ(z)}d(x, w)^α

deg(f )

j=1 |ϕ(x_j)e^{φ (x}^j⁾−ϕ(w_j)e^{φ (w}^j⁾| deg(f )·e^infφinfϕ·d(x, w)^α

deg(f )

j=1 |e^{φ (x}^j⁾(ϕ(xj)−ϕ(wj))|

deg(f )·e^infφinfϕ·d(x, w)^α (4.3)

+

deg(f )

j=1 |ϕ(w_j)(e^{φ (x}^j⁾−e^{φ (w}^j⁾)|

deg(f )·e^inf^φinfϕ·d(x, w)^α . (4.4)

(12)

We subdivide the sum in (4.3) according to the possible backward contraction of the inverse branches off. Using Lemma3.5we obtain thatϕ isK-Hölder on balls of radius˜ LδwithK˜ =K(1+(L−1)^α). SinceKκinfgand supφ−infφ < ε_φwe get that (4.3) is bounded from above by

e^ε^φ(deg(f )−q)σ⁻^α+qL^α[1+(L−1)^α]

deg(f ) κ.

For estimating (4.4) we first note that sinceϕis Hölder continuous then supϕinfϕ+m|ϕ|α,δdiam(M)^α. Therefore, usingϕ∈Λ_κ,δwe get

(4.4)supϕ· |e^φ|α·L^α

infϕ·e^inf^φ infϕ+m ϕ _α,δdiam(M)^α infϕ

|e^φ|α

e^inf^φL^α |e^φ|α

e^infφL^α

1+mκdiam(M)^α |e^φ|α

e^infφ2mL^ακdiam(M)^α.

Using (P) we have|e^φ|α< ε_φe^infφand our previous choice ofε_φyields that Lφϕ _α,δλκˆ inf(Lφϕ). This completes the proof of the theorem. 2

Observe that assumption (P) can be rewritten as sup(φ)−inf(φ) < εφande^φ∈Λε_φ. To prove that the cone has finite diameter inΛκ,δwe compute an explicit expression for the projective metric.

Lemma 4.2.TheΛ_κ,δ-cone metricΘ_κis given byΘ_κ(ϕ, ψ )=log^B_A^κ^(ϕ,ψ)

κ(ϕ,ψ), where Aκ(ϕ, ψ )= inf

0<d(x,y)<δ, z∈M

κ|x−y|^αψ (z)−(ψ (x)−ψ (y)) κ|x−y|^αϕ(z)−(ϕ(x)−ϕ(y)) , and

B_κ(ϕ, ψ )= sup

0<d(x,y)<δ, z∈M

κ|x−y|^αψ (z)−(ψ (x)−ψ (y)) κ|x−y|^αϕ(z)−(ϕ(x)−ϕ(y)) .

Proof. By definition,Aϕψif and only ifψ (x)−Aϕ(x)0 for everyx∈Mand ψ−Aϕ _α,δκinf(ψ−Aϕ).

In particular one gets Amin

infx

ψ (x)

ϕ(x), inf

0<d(x,y)<δ, z∈M

κ|x−y|^αψ (z)−(ψ (x)−ψ (y)) κ|x−y|^αϕ(z)−(ϕ(x)−ϕ(y))

.

We will prove that minimum in the right-hand side is always attained by the second term. Pick x₀∈M such that infxψ(x)

ϕ(x) =^ψ(x_ϕ(x₀⁰₎⁾. Then it is immediate that

xlim→x0

κ|x−x₀|^αψ (x₀)−(ψ (x)−ψ (x₀))

κ|x−x₀|^αϕ(x₀)−(ϕ(x)−ϕ(x₀)) ψ (x₀) ϕ(x₀), which guarantees that

A_κ(ϕ, ψ )= inf

0<d(x,y)<δ, z∈M

κ|x−y|^αψ (z)−(ψ (x)−ψ (y)) κ|x−y|^αϕ(z)−(ϕ(x)−ϕ(y)) . Similar computations lead to the expression forBκ(ϕ, ψ ). 2

Proposition 4.3.Given0<λ <ˆ 1, the coneΛ_λκ,δ_ˆ has finiteΛ_κ,δ-diameter.

Proof. For allϕ∈Λ_λκ,δ_ˆ by definition we have|ϕ|α,δλκˆ infϕand, consequently, supϕ[1+mˆλκ(diamM)^α]infϕ.

So, using the previous expression for the projective metric, givenϕ, ψ∈Λ_λκ,δ_ˆ one can easily check that

(13)

Θ_κ(ϕ, ψ )log

κ·supϕ+ ˆλκinfϕ

κ·infϕ− ˆλκinfϕ ·κ·supψ+ ˆλκinfψ κ·infψ− ˆλκinfψ

log

κ(1+mˆλκ(diamM)^α)(1+ ˆλ)infϕ κ·(1− ˆλ)·infϕ

+log

κ(1+mˆλκ(diamM)^α)(1+ ˆλ)infψ κ·(1− ˆλ)·infψ

2 log 1+ ˆλ

1− ˆλ

+2 log

1+mcdiam(M)^α

for some positive constantc. This implies the finiteΘκ-diameter ofΛ_λκ,δ_ˆ and finishes the proof of the lemma. 2 4.2. Consequences of the spectral gap inC^α(M,R)

Now we shall deduce the existence of equilibrium states and some of their ergodic properties.

4.2.1. Existence of equilibrium states

Using the spectral gap property in the space of Hölder continuous observables we get the existence of a unique Hölder continuous invariant densityh.

Proposition 4.4.There exists a unique densityh∈C^α(M,R)such thatLφh=λh. In particular,μ=hν an equilib- rium state forf with respect toφ. Finally, the densitydμ/dν is bounded away from zero and infinity and Hölder continuous.

Proof. Consider the normalized operator L˜φ=λ⁻¹Lφ, whereλ is the spectral radius ofL and writeΛ⁺ for the cone of strictly positive continuous functions onM. SinceΛκ,δ⊂Λ⁺then the projective metrics satisfyΘ⁺(ϕ, ψ ) Θ_κ(ϕ, ψ )for anyϕ, ψ∈Λ_κ,δ, where

Θ⁺(ϕ, ψ )=log

sup_x_∈_M{ϕ(x)/ψ (x)} infy∈M{ϕ(y)/ψ (y)}

.

By the previous proposition,L˜φ(Λ_κ,δ)has finite diameter inΛ_κ,δfor any sufficiently largeκ. Therefore, as discussed at the end of Subsection 3.4, L˜φ is a contraction in the Θ_κ-metric and there exists 0< τ <1 such that for any ϕ, ψ∈Λ_κ,δandn, k1

Θ⁺L˜ⁿ_φ⁺^k(ϕ),L˜ⁿφ(ψ ) Θκ

L˜ⁿ_φ⁺^k(ϕ),L˜ⁿφ(ψ )

τⁿ, (4.5)

where is the Θκ-diameter of the cone Λ_λκ,δ_ˆ . This proves that (L˜ⁿ_φϕ)n is a Cauchy sequence in the projective metric. For the reference measureν we have thatL˜φϕ dν=

ϕ dν, ∀ϕ∈C⁰(M,R). Given ϕ∈Λ_κ,δ with ϕ dν=1 it is clear that supϕ1 and infϕ1. Together with the remark that any ϕ∈Λ_κ,δ satisfies supϕ [1+mκdiam(M)^α]infϕthis shows that

1

R1infϕ1supϕR₁ (4.6)

whereR₁=1+mκdiam(M)^α. Writeϕ_n= ˜Lⁿ_φ(ϕ). First notice that(ϕ_n)_nis an equi-Hölder sequence since|ϕ_n(x)− ϕ_n(y)|κinfϕd(x, y)^ακd(x, y)^αfor alld(x, y) < δand alln, which proves that allϕ_nareκm-Hölder continuous.

From the previous discussion we know that

ϕ_ndν=1 for everynand, consequently, the sequenceϕ_n is uniformly bounded from above and below. In fact, observe first that

ϕ_kdν=

ϕ_ldν=1 implies inf^ϕ_ϕ^k

l 1sup^ϕ_ϕ^k

l. Therefore, from (4.5) we get

e⁻^τⁿ<sup_x_∈_M{ϕ_k(x)/ϕ_l(x)}

infy∈M{ϕ_k(y)/ϕ_l(y)} =e^θ⁺^(ϕ^k^,ϕ^l⁾< e^τⁿ for everykandln. In consequence,