Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures

Transcription

1 Ruelle Operator or Continuous Potentials and DLR-Gibbs Measures arxiv: v4 [math.ds] 22 Apr 2018 Leandro Cioletti Departamento de Matemática - UnB , Brasília, Brazil cioletti@mat.unb.br April 24, 2018 Abstract Artur Lopes Departamento de Matemática - UFRGS , Porto Alegre, Brazil arturoscar.lopes@gmail.com In this work we study the Ruelle Operator associated to continuous potentials on general compact spaces. We present a new criterion or uniqueness o the eigenmeasures associated to the spectral radius o this operator acting on the space o all real-valued continuous unctions deined on some suitable compact metric space. This uniqueness result is obtained by showing that set o eigenmeasures or any continuous potential is a subset o the DLR-Gibbs measures or a suitable quasilocal speciication. In particular, we proved that the phase transition in the DLR sense can be obtained by showing the existence o more than one eigenprobability or the dual o the Ruelle operator. On the context o compact metric state spaces, we obtain the uniorm convergence o n 1 log L n (1)(x), when n, to the topological pressure P (), or any continuous potential. We also consider bounded extensions o the Ruelle operator to the Lebesgue space o integrable unctions with respect to the eigenmeasures. We give very general suicient conditions or the existence o integrable and almost everywhere positive eigenunctions associated to the spectral radius o the Ruelle operator. Techniques o super and sub solutions to the eigenvalue problem is developed and employed to construct (semi-explicitly) eigenunctions or a very large class o continuous potentials having low regularity. Keywords: Thermodynamic Formalism, Ruelle operator, Eigenunctions, Equilibrium states, DLR-Gibbs Measures, Strongly non-null Condition. MSC2010: 37D35, 28Dxx, 37C30. The authors are supported by CNPq-Brazil. 1

2 1 Introduction The classical Ruelle operator needs no introduction and nowadays is a key concept in the Thermodynamic Formalism. It was irst deined in ininite dimensions in 1968 by David Ruelle in a seminal paper [22] and since then has attracted the attention o the Dynamical System community. Remarkable applications o this operator to Hyperbolic dynamical systems and Statistical Mechanics were presented by Ruelle, Sinai and Bowen, see [7, 22, 24]. The Ruelle operator was generalized in several directions and its generalizations are commonly called transer operators. Nowadays transer operators are present in several applications in pure and applied mathematics and is ruitul area o active development, see [2] or comprehensive overview o the works beore the two thousands. The classical theory o Ruelle operator with the dynamics given by the ull shit σ : is ormulated in the symbolic space M N, where M = {1,..., n} with the operator acting on the space o all real-valued γ-hölder continuous unctions deined on, which is denoted here by C γ (). On its classical orm, given a continuous unction : R, the Ruelle operator L is such that L (ϕ) = ψ, where the unction ψ or any x is given by ψ(x) = e (y) ϕ(y). y : σ(y)=x A central problem is this setting is to determine the solutions o the ollowing variational problem proposed by Ruelle 1967, [21] and Walters 1975, [27]: sup µ P σ(,f ) {h(µ) + dµ}, (1) where h(µ) is the Kolmogorov-Sinai entropy o µ and P σ (, F ) is the set o all σ- invariant Borel probability measures over and F is the sigma algebra generated by the cylinder sets. A probability measure which attains this supremum is called an equilibrium state or the potential. It is well known that the Ruelle operator L is quite useul or getting equilibrium states (see [2] and [20]). We remark that the simple existence o the solution to the variational problem can be easily obtained through abstract theorems o convex analysis, but as we said the Ruelle operator approach to this problem give us much more inormation about the solution as uniqueness, or instance. One can also get dierentiability properties with variation o the potential using this approach (see [5]). But that comes with a cost which is the determination o certain spectral properties o the Ruelle operator. For the symbolic space we mentioned above all these problems were overcome around the eighties and nineties and in that time much more general potentials were considered, see [6, 28, 29, 30] or their precise deinitions. These new space o potentials are denoted by W (, σ) and B(, σ), where σ in the notation reers to the dynamics and is particularized here to the let shit map. Potentials on W (, σ) are said to satisy the Walters condition and those in B(, σ) Bowen s condition. We shall remark that C γ () W (, σ) B(, σ), but the spectral analysis o the Ruelle operator is more eicient in C γ () because o the spectral gap property o the Ruelle 2

3 operator acting in this space. Although this operator has no spectral gap in W (, σ) and B(, σ) the Ruelle operator approach is still useul to attack the above variational problem and provides rich connections among symbolic dynamics, Statistical Mechanics and probability theory. The above mentioned works o Peter Walters are not restricted to dynamics given by the shit mapping and spaces like ininite cartesian products o inite sets. Basically he considered expanding mappings T : on compact spaces satisying the property that the number o preimages o each point is inite. So symbolic spaces like = (S 1 ) N with the shit acting on it does not it his theory. The problem is related to the act that the number o preimages under the shit map is not countable. Let us recall that several amous models o Statistical Mechanics are deined over the alphabet S n 1, the unit sphere o R n (which is uncountable or n 2). For example, n = 0 give us the Sel- Avoiding Walking (SAW), n = 1 is the Ising model, n = 2 (the irst uncountable example on this list) the so-called XY model, or n = 3 we have the Heisenberg model and or n = 4 the toy model or the Higgs sector o the Standard Model, see [1, 3, 14, 15, 16, 17, 25] or more details. In [3] the authors used the idea o an a priori measure p : B(S 1 ) [0, 1] to circumvent the problem o uncountable alphabets and developed the theory o the Ruelle operator or Hölder potentials on (S 1 ) N with the dynamics given by the let shit map. The scheme developed to handle the (S 1 ) N case, works similarly or Hölder potentials when one replaces the unit circle S 1 by a more general compact metric space M, but in the general case we have to be careul about the choice o the a priori measure p : B(M) [0, 1], see [18] or details. In this more general setting the operator is deined as L (ϕ) (x) = e (a x) ϕ(a x) dp(a), M where ax := (a, x 1, x 2,...). A ull support condition is imposed on the a priori measure in [18] but, this is not a strong restriction, since in the majority o the applications there is a natural choice or the a priori measure and it always satisies this ull support condition. For instance, in the classical Ruelle operator L the metric space is some inite set as M = {1, 2,.., n} and one normally consider the normalized counting measure as the a priori measure p on M. When M is a general compact group one can consider the normalized Haar Measure. For example, i M = S 1, then one can consider the Lebesgue measure dx on S 1 as a natural choice or the a priori probability measure, see [3]. Another progress towards considering more general potentials deined, on ininite cartesian products o a general metric compact spaces, was obtained by one o the authors in [11]. In this work a version o the Ruelle-Perron-Fröbenius theorem is obtained or what the authors called weak and strong Walters conditions, which are natural generalizations o the classical Walters condition. Some results mentioned above have its counterpart in case M is a countable ininite set but not compact when regarded as topological space. The Thermodynamic Formalism or such alphabets are motivated in parts by application to non-uniormly hyperbolic dynamical systems [23] and reerences therein. 3

4 When M = S 1, or example, the let shi mapping losses the important dynamical property which is the uniormly expansivity. This is intimately connected to the existence and uniormly convergence o 1 lim n log L n (1)(x) to the topological pressure and also in studying dierentiability and analyticity o the Thermodynamic quantities. Some progress regarding these problems are obtained, in [11]. The goal o the present paper is to study the Ruelle operator L associated to a general continuous potential deined over an ininite cartesian product o a general compact metric space. One o the main results here is the extension o the results mentioned above to potentials satisying a Bowen s like condition (see Theorem 4). We also construct a Perturbation theory or the Ruelle operator in the sense o C() perturbations and present a constructive approach to solve the classical variational problem, or continuous potentials. In Statistical Mechanics (and also Thermodynamic Formalism) the problem o existence and multiplicity o DLR Gibbs Measures play a very important role (see [23]). An analysis regarding the uniqueness in the Thermodynamic Formalism setting was done in [10], in case where the state space is inite or a large class o continuous potentials. We shall mention that the equivalence and dierences between the concepts o Gibbs measures in Statistical Mechanics and Dynamical System were investigated, or inite state space in the lattice Z, in [12] and more recently a rather complete description was obtained in [4]. The study o the multiplicity o the DLR Gibbs Measures is an important problem in Statistical Mechanics and Thermodynamic Formalism because o the Dobrushin interpretation o it as phase transitions. There is no universal deinition o what a phase transition is but nowadays it is understood as either the existence o more than one DLR state, more than one eigenprobability or the dual o the Ruelle operator or, more than one equilibrium state and so on (see [9, 13] or more details). These concepts and their connections will be careully described here and this is helpul or understanding when there exist or not phase transitions. This paper is structured as ollows. In section 2 the Ruelle Operator acting on C(), where is a ininite cartesian product o an arbitrary compact metric space, is introduced. We recall the standard strategy used to obtain the maximal eigenvalue λ (the maximality is discussed in urther section) o L as well as the probability measures ν satisying L (ν ) = λ ν. In section 3 we obtain an intrinsic ormulae or the asymptotic pressure or arbitrary continuous potential. This result is similar to the one obtained by Peter Walters in [29] but use dierent approach since the entropy arguments used there can not be directly applied or the case o general compact metric space M as we are considering here. In this section we also prove that the classical strategy, above mentioned to construct ν, always provide us eigenprobabilities associated to the maximal eigenvalue λ. 4

5 Section 4 we prove that the eigenmeasures o the dual o the Ruelle operator is contained in the set o DLR-Gibbs measures, or any continuous potential. In particular, we explain how to construct a quasilocal speciication rom the continuous potential and the Ruelle operator. We also show that these DLR-Gibbs Measures that are shit-invariant on the tail sigma-algebra are contained in the set o the eigenmeasures. So letting clear what kind o results one can immediately import rom the DLR-Gibbs Measures to the Thermodynamic Formalism. In sections 5 and 6 we prove the quasilocality property o the speciications introduced in this work, and the strongly non-null condition is analyzed when the state space M is inite, respectively. In Section 6 is presented an example where the speciications constructed here rom a continuous potential is not strongly non-null. This example in inite state space is rather surprising. In section 7 we obtain a kind o Bowen criteria or the uniqueness o the eigenprobabilities applicable on the setting o general compact metric state space. This theorem is non trivial generalization o the classical ones or Hölder, Walters and Bowen potentials on the setting o inite state space. Section 8 the extension o L to the Lebesgue space L 1 (ν ) is considered. We show that the operator norm is given by λ thus proving the maximality o the eigenvalue λ obtained in the Section 2. We also point out that the classical duality relation L (ϕ) dν = λ ϕdν extends naturally or test unctions ϕ L 1 (ν ). In Section 9 we prove continuity results or sequences o Ruelle operators in the uniorm operator norm. In Section 10 we give very general conditions or the existence o the eigenunctions in the L 1 sense or potentials having less regularity than Hölder, Walters and Bowen, or example. Under mild hypothesis on the potential we prove that lim sup h n (here n is a sequence o potentials converging uniormly to ) is a non trivial Lebesgue integrable (with respect to any eigenprobability) eigenunction associated to the maximal eigenvalue. Another remarkable result concerning to the eigenunctions is the proo that lim sup L n (1)/λn is non trivial eigenunction o L under airly general conditions. In Section 11 we present applications o our previous results. The irst application concerns to cluster points o the sequence o eigenprobabilities (µ n ) n N, where n is a suitable truncation o. We prove that such cluster points belongs to the set o eigenprobabilities or. In this application n and are assumed to have Hölder regularity. 2 Preliminaries Here and subsequently (M, d) denotes a compact metric space endowed with a Borel probability measure µ which is assumed to be ully supported in M. Let denote the ininite cartesian product M N and F be the σ-algebra generated by its cylinder sets. We will consider the dynamics on given by the let shit map σ : which is deined, as usual, by σ(x 1, x 2,...) = (x 2, x 3,...). We use the notation C() or the space o all real continuous unctions on. When convenient we call an element C() a potential and unless stated otherwise all the potentials are assumed to be a general 5

6 continuous unction. The Ruelle operator associated to the potential is a mapping L : C() C() that sends ϕ L (ϕ) which is deined or each x by the ollowing expression L (ϕ)(x) = exp((ax))ϕ(ax) dp(a), where ax := (a, x 1, x 2,...). (2) M Due to compactness o, in the product topology and the Riesz-Markov theorem we have that C () is isomorphic to M s (, F ), the space o all signed Radon measures. Thereore we can deine L, the dual o the Ruelle operator, as the unique continuous map rom M s (, F ) to itsel satisying or each γ M s (, F ) the ollowing identity L (ϕ) dγ = ϕ d[l γ] ϕ C(). (3) From the positivity o L ollows that the map γ L (γ)/l (γ)(1) sends the space o all Borel probability measures P(, F ) to itsel. Since P(, F ) is convex set and compact in the weak topology (which is Hausdor in this case) and the mapping γ L (γ)/l (γ)(1) is continuous the Schauder-Tychono theorem ensures the existence o at least one Borel probability measure ν such that L (ν) = L (ν)(1) ν. Notice that this eigenvalue λ L (ν)(1) is positive but strictly speaking it could depend on the choice o the ixed point when it is not unique, however any case such eigenvalues trivially satisies exp( ) λ exp( ) so we can always work with { } λ = sup L ν P(, F ) and ν is ix point or (ν)(1) : γ L (γ)/l (γ)(1). (4) O course, rom the compactness o P(, F ) and continuity o L attained and thereore the set deined below is not empty. the supremum is Deinition 1 (G ()). Let be a continuous potential and λ given by (4). We deine G () = {ν P(, F ) : L ν = λ ν}. To study the eigenunctions o L, where is a general continuous potential, we will need the RPF theorem or the Hölder class. This theorem is stated as ollows, see [3] and [18] or the proo. We consider the metric d on given by d (x, y) = n=1 2 n d(x n, y n ) and or any ixed 0 < γ 1 we denote by C γ () the space o all γ-hölder continuous unctions, i.e, the set o all unctions ϕ : R satisying Hol γ (ϕ) = ϕ(x) ϕ(y) sup x,y :x y d (x, y) γ < +. Theorem 1 (Ruelle-Perron-Fröbenius: Hölder potentials [3, 18] ). Let (M, d) be a compact metric space, µ a Borel probability measure o ull support on M and be a potential in C γ (), where 0 < γ < 1. Then L : C γ () C γ () have a simple positive eigenvalue o maximal modulus λ and there are a strictly positive unction h satisying L (h ) = λ h and a Borel probability measure ν or which L (ν ) = λ ν L (ν )(1) = λ. and 6

7 3 The Pressure o Continuous Potentials Next proposition it is an extension o Corollary 1.3 in [19]. Here M is allowed to be any general compact metric space and it is worth to mention that the techniques employed in our proo are much simpler. Proposition 1. Let C() be a potential and λ given by (4). Then, or any x we have 1 lim n log L n (1)(σ n x) = log λ. Proo. Let ν G () a ixed eigenprobability. Without loss o generality we can assume that diam(m) = 1. By the deinition o d or any pair z, w such that z i = w i, i = 1,..., N we have that d (z, w) 2 N. From uniorm continuity o given ε > 0, there is N 0 N, so that (z) (w) < ε/2, whenever d (z, w) < 2 N 0. I n > 2N 0 and a := (a 1,..., a n ) we claim that or any x, y we have Indeed, or any n 2N 0, we have S n ()(ax) S n ()(ay) (n N 0 ) ε N 0. (5) S n ()(ax) S n ()(ay) = n N 0 j=1 N 0 j=1 n (σ j (a 1,..., a n, x) j=1 n (σ j (a 1,..., a n, y) j=1 (σ j (a 1,..., a n, x) (σ j (a 1,..., a n, y) + (σ j (a n N0,..., a n, x) (σ j (a n N0,..., a n, y) (n N 0 ) ε 2 + 2N 0. The last inequality comes rom the uniorm continuity or the irst terms and rom the uniorm norm o or the second ones. We recall that or any probability space (E, E, P ), ϕ and ψ bounded real E -measurable unctions the ollowing inequality holds log e ϕ(ω) dp (ω) log e ψ(ω) dp (ω) ϕ ψ. (6) E From the deinition o the Ruelle operator, or any n N, we have n L n (1)(σ n x) = exp S n ()(aσ n x) dp(a i ) M n E and rom (5) and (6) with ϕ(a) = exp(s n ()(aσ n x)) and ψ(a) = exp(s n ()(ay)) we get or n max{2n 0, 4ε 1 N 0 } the ollowing inequality 1 n log(l n (1)(σ n x)) log(l n (1)(y)) 1 n ((n N 0) ε N 0 n i=1 ε. 7

8 By using Fubini s theorem, sum and subtract exp(s n ()(ay)), the identity (3) iteratively and the last inequality or n max{2n 0, 4ε 1 N 0 } we obtain L n (1)(σ n x) = exp(s n ()(aσ n x)) M n n = exp(s n ()(aσ n x)) dν(y) M n exp((n N 0 ) ε N 0 ) exp(nε) L n (1)(y) dν(y) = exp(nε)λ n. M n i=1 n dp(a i ) i=1 dp(a i ) exp(s n ()(ay)) dν (y) Similarly we obtain the lower bound L n(1)(σn x) exp( nε)λ n ollows. n dp(a i ) i=1 so the proposition Corollary 1. Let be a continuous potential. I ν and ˆν are ixed points or the map γ L (γ)/l (γ)(1) then L (ν)(1) = L (ˆν)(1) = λ. Proo. For any x 0 by repeating the same steps o the proo o the above previous proposition one shows that log(l (ν)(1)) log(λ 1 (ν)) = lim log L n n (1)(x 0) = log(λ (ˆν)) = log(l (ˆν)(1)). Deinition 2 (The Pressure Functional). The unction p : C() R given by p() = log λ is called pressure unctional. In Thermodynamic Formalism what is usually called pressure unctional is a unction P : C() R given by P () sup {h(µ) + dµ}. µ P σ(,f ) Ater developing some perturbation theory we will show latter that both deinitions o the Pressure unctional are equivalent or any continuous potential, i.e., P = p. Since is compact and the space o all γ-hölder continuous unction C γ () is an algebra o unctions that separate points and contain the constant unctions, we can apply the Stone-Weierstrass theorem to conclude that the closure o C γ () in the uniorm topology is C(). Thereore or any arbitrary continuous potential there is a sequence ( n ) n N o Hölder continuous potentials such that n 0, when n. For such uniorm convergent sequences we will see that p( n ) converges to p(). In act, a much stronger result can be stated. The pressure unctional is Lipschitz continuous unction on the space C(). Proposition 2. I, g : R are two arbitrary continuous potentials then p() p(g) g. 8

9 Proo. The proo is an immediate consequence o the Proposition 1 and the inequality (6). Corollary 2. Let ( n ) n N be a sequence o continuous potentials such that n uniormly, then p( n ) p(). In particular, λ n λ. 4 DLR-Gibbs Measures and Eigenmeasures In this section we discuss the concept o speciications in the Thermodynamic Formalism setting. Some o its elementary properties or inite state space is discussed in details within this ramework in the reerence [10]. For each n N, we deine the projection on the n-th coordinate π n : M by π n (x) = x n. We use the notation F n to denote the sigma-algebra generated by the projections π 1,..., π n. On the other hand, the notation σ n (F ) stands or the sigmaalgebra generated by the collection o projections {π k : k n + 1}. Let C() a potential and or each n N, x and E F consider the mapping K n : F [0, 1] given by K n (E, x) L n (1 E)(σ n (x)) L n (1)(σn (x)). (7) For any ixed x ollows rom the monotone convergence theorem that the map F E K n (E, x) is a probability measure. For any ixed measurable set E F ollows rom the Fubini theorem that the map x K n (E, x) is σ n (F )-measurable. So K n is a probability Kernel rom σ n (F ) to F. Notice or any ϕ C() that K n (ϕ, x) is naturally deined because the rhs o (7). It is easy to see (using rhs o (7)) that they are proper kernels, meaning that or any bounded σ n (F )-measurable unction ϕ we have K n (ϕ, x) = ϕ(σ n (x)). The above probability kernels have the ollowing important property. For any ixed continuous unction ϕ the map x K n (ϕ, x) is continuous as consequence o Lebesgue dominated convergence theorem. We reer to this saying that (K n ) n N has the Feller property. Deinition 3. A Gibbsian speciication with parameter set N in the translation invariant setting is an abstract amily o probability Kernels K n : (F, ) [0, 1], n N such that a) x K n (E, x) is σ n F -measurable unction or any E F ; b) F E K n (F, x) is a probability measure or any x ; c) or any n, r N and any bounded F -measurable unction ϕ : R we have the compatibility condition, i.e., K n+r (ϕ, x) = K n (ϕ, )dk n+r (, x) K n+r (K n (ϕ, ), x). 9

10 Remark 1. The classical deinition o a speciication as given in [13] requires even in our setting a larger amily o probability kernels. To be more precise we have to deine a probability kernel or any inite subset Λ N and the kernels K Λ have to satisy a), b) and a generalization o c). In translation invariant setting on the lattice N the ormalism can be simpliied and one needs only to consider the amily K n, n N, as deined above. Strictly speaking to be able to use the results in [13] one has irst to extend our speciications to any set Λ = {n 1,..., n r }, but this can be consistently done by putting K Λ K nr. This simpliied deinition adopted here is urther justiied by the act that the DLR-Gibbs measures, compatible with a speciication with parameter set N, are completely determined by the kernels indexed in any coinal collection o subsets o N. So here we are taking advantage o this result to deine our kernels only on the coinal collection o subsets o N o the orm {1,..., n} with n N. Thereore when we write K n we are really thinking, in terms o the general deinition o speciications, about K {1,...,n}. The only speciications needed here are the ones described by (7), which is deined in terms o any continuous potential. Notice that in the translation invariant setting the construction in (7), or the lattice N, extends the usual construction made in terms o regular interactions. But in any case (7) give us particular constructions o quasilocal speciications which allow us to use some o the results rom [13]. We reer the reader to [10] and [29] or results about speciications on the Ruelle operator when the dynamics have inite pre-images property. Using the same ideas employed in the proo o Theorem 23 in [10] one can prove or any r, n N, x and ϕ C() the ollowing identity ( ) L n L n+r (ϕ)(σ n+r (x)) = L n+r (ϕ)( σ n ( ) ) (σ n+r (x)). (8) L n(1)(σn ( ) ) The above identity immediately implies or the Kernels deined by (7) that K n+r (, x) = K n (, )dk n+r (, x) K n+r (K n (, ), x). (9) As we mentioned beore, we reer to the above set o identities as compatibility conditions or the amily o probability kernels (K n ) n N or simply DLR equations. Similar kernels are also considered in [29] but here we are working with a dynamical system that may have uncountable many elements in the preimage o any point. Deinition 4. We say that µ P(, F ) is a DLR-Gibbs measure (or, just DLR) or the continuous potential i or any n and any continuous unction ϕ : R we have or µ-almost all x that E µ [ϕ σ n (F )](x) = ϕ(y) dk n (y, x). The set o all DLR-Gibbs measures or is denoted by G DLR (). 10

11 One very important and elementary result on DLR-Gibbs measure is the equivalence between the two conditions below: a) µ G DLR (); b) or any n N and E F we have that µ(e) = K n(e, ) dµ. As usual, to prove that µ G DLR () is not empty one uses the ollowing result. Lemma 1. For any C() the closure o the convex hull o the weak limits o the orm K n (, x), where x runs over all elements o is equal to G (). Proo. I is Hölder continuous potential then we can proceed as in [10] with the suitable adaptations to consider general compact state space M. For the general case we use Theorem 2 below and then the result ollows rom Corollary 7.30 o [13]. Lemma 2. Let C() be a potential and (K n ) n N the speciication deined by (7). Then we have G () G DLR (). Proo. Let ν be such that L ν = λ ν and ϕ a bounded F -measurable unction. Notice that the quotient appearing in the irst integral below is σ n (F )-measurable. Thereore or any bounded F -measurable ψ the ollowing equality holds. (ϕ σ n )(x) L n (ψ) (σn (x)) L n (1) (σn (x)) d ν(x) = = L n (ψ (ϕ σn )) (σ n (x)) L n (1) d ν(x) (σn (x)) [ ] 1 L n λ L n (ψ (ϕ σ n )) (σ n ( )) n L n (1) (x) d ν(x). (σn ( )) By using the equation (8) we see that rhs above is equals to 1 λ L n n (ψ (ϕ σ n ))(x) d ν(x) = ψ(x) (ϕ σ n )(x) d ν(x). Since ϕ is an arbitrary F -measurable unction we can conclude that ν[e σ n F ](y) = L n (I E) (σ n (y)) L n (1) (σn (y)) ν a.s. so the equation (7) implies that ν G DLR (). The next lemma establishes the reverse inclusion between the set G DLR () and the set G () o eigenprobabilities or the dual o the Ruelle operator, under some additional assumptions. Its proo is much more involved than previous one and beore proceed we recall some classical results about Martingales and Speciication Theory which we will be used in the sequel. 11

12 Theorem A (Backward Martingale Convergence Theorem). Consider the ollowing sequence o σ-algebras F σf... n N σ n F and a bounded F -measurable unction ϕ : R. Then, or any µ P(, F ) we have µ[ϕ σ n F ] µ[ϕ j=1 σ j F ], a.s. and in L 1 (, F, µ). Theorem B. Let (K n ) n N be the speciication given in (7). Then the ollowing conclusion holds. A probability measure µ G DLR () is extreme in G DLR (), i and only i, µ is trivial on n N σ n F. As consequence i µ is extreme in G DLR (), then every n N σ n F -measurable unction is constant µ a.s.. We give a proo o the above result in our setting in the appendix. From now on we eventually reer to n N σ n F as the tail sigma-algebra. Now we are ready to prove one o the main results o this section. Lemma 3. Let C() be a potential and (K n ) n N deined as in (7). I µ G DLR () is such that µ(a) = µ(σ 1 (A)) or all A n N σ n F then µ G () = {ν P(, F ) : L ν = λ ν}. Proo. Let µ G DLR () be an extreme element, ϕ C() and λ the eigenvalue o L. From the elementary properties o the conditional expectation and deinition o G DLR (), we have λ ϕ dµ = λ E µ [ϕ σ n+1 F ] dµ = λ The integrand on rhs can be rewritten as ollows λ L n+1 (ϕ) (σ n+1 (x)) L n+1 (1) (σ n+1 (x)) = L n (L ϕ)(σ n (σx)) L n(1)(σn (σx)) L n+1 (ϕ) σ n+1 L n+1 dµ. (1) σn+1 L n λ (1)(σn+1 (x)) L n+1 (1)(σ n+1 (x)). From the above equation and deinition o G DLR () we have µ a.s. the ollowing identity λ E µ [ϕ σ n+1 F ] = E µ [L ϕ σ n F ] σ λ L n (1) σn+1 L n+1 (1) σ. n+1 By using the Backward Martingale Convergence Theorem and µ-triviality o n N σ n F - measurable unctions (see Corollary 5 in the Appendix) and shit invariance o µ on the tail sigma-algebra, we can ensure that there exist the limit below and it is µ a.s. constant lim λ L n (1)(σn+1 (x)) L n+1 (1)(σ n+1 (x)) ϱ. Thereore we have the ollowing equality µ a.s. λ E µ [ϕ n N σ n F ] = ϱ E µ [L ϕ n N σ n F ] σ 12

13 Since µ is extreme in G DLR () it ollows that the r.v. E µ [L ϕ n N σ n F ] is constant µ a.s. and one version o this conditional expectation is given by the constant unction L (ϕ) dµ. Using this inormation and taking expectations on both sides above we get λ ϕ dµ = ϱ E µ [L ϕ n N σ n F ] σ dµ = ϱ L (ϕ) dµ, where in the last equality we use that the restriction o µ to the tail sigma-algebra is shit invariant. Thereore L (µ) = (λ /ϱ )µ. At this point we already proved that µ is an eigenprobability, but more stronger result can be shown, which is, ϱ = 1 and so µ is eigenprobability associated to λ, the maximal eigenvalue. Indeed, by proceeding as in the proo o Proposition 1 we can see that L n (1)(σ n x) exp(nε) L n (1)(y) dµ(y) = exp(nε) λn. ϱ n Thereore 1 lim n log L n (1)(σ n x) ε + log λ log ϱ Similar lower bounds can be shown, so the Proposition 1 implies that log ϱ = 0 which implies that ϱ = 1 and the lemma ollows. 5 Quasilocality o K n Deinition 5 (Local Function). A unction : R is said to be local i is F n - measurable or some n N. For each n N, we denote by L n the space o all bounded F n -measurable local unctions and L = n N L n the set o all bounded local unctions. Remark 2. Note that i M is inite, then any local unction is continuous. Indeed, any element L n is a unction depending only on the irst n coordinates. Deinition 6 (Quasilocal Function). A unction : R is said to be a quasilocal unction i there is a sequence ( n ) n N in L such that n 0, when n. Here is the sup-norm. We write L to denote the space o all bounded quasilocal unctions. Deinition 7. We say that the speciication (K n ) n N is quasilocal, i or each n N and ϕ L the mapping x K n (ϕ, x) is quasilocal. Proposition 3. For each C() there is a sequence ( n ) n N such that n is continuous, local (more precisely F n -measurable) and n 0, when n. Proo. We ix a point in M, or sake o simplicity, this point will be denoted by 0. For each n N we deine n : R by n (x) = (x 1,..., x n, 0, 0,...). Since is compact the unction is uniormly continuous, so or any given ε > 0 there is δ > 0 such that 13

14 or every pair x, y satisying d (x, y) < δ we have (x) (y) < ε. Notice that or any x we have d ( x, (x 1,..., x n, 0, 0,...)) = j=n j d(x j, 0) diam() I n is such that diam()2 n < δ, then or all x we have j=n+1 (x) n (x) = (x) (x 1,..., x n, 0, 0,...) < ε. 1 2 = diam(). j 2 n Since ε > 0 is arbitrary and the last inequality is uniorm in x we have n 0, when n. For each n N we clearly have that n is continuous, bounded and depends only on the irst n coordinates and thus an element o L n. Theorem 2. For any ixed potential C() and n N, we have that the Kernel K n as deined in (7) is quasilocal. Proo. Let ϕ a quasilocal unction and (ϕ n ) n N a sequence o bounded local unctions converging uniormly to ϕ. From the deinition o K n and the Ruelle operator we have K n (ϕ, x) is equal to M n exp(s n ()(a 1,..., a n, x n+1, x n+2,...))ϕ(a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j) M n exp((a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j), where S n ()(x) (x) + (σx) (σ n 1 x). For each m N deine ψ m : R, where ψ m (x), or any x, is given the ollowing expression exp(s M n n ( m )(a 1,..., a n, x n+1, x n+2,...))ϕ m (a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j) exp(s M n n ( m )(a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a, j) Since S n ( m ) and ϕ m are local, it ollows that ψ m is local. Clearly ψ m is bounded. We claim that or each ixed n N we have ψ m K n (ϕ, ) 0, when m. Since n is ixed, we can introduce a more convenient notation W (g)(a, x) = exp(s n (g)(a 1,..., a n, x n+1, x n+2,...)). By using that m 0, when m, we get or each ixed (a 1,..., a n ) E n that lim sup W ( m )(a, x) W ()(a, x) = 0. (10) m x Using the W notation ψ m has simpler expression ψ m (x) = M n W ( m )(a, x)ϕ m (a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j) M n W ( m )(a, x) n j=1 dp(a j) 14

15 It is easy to see that exp( n ) W ( m )(a, x) exp(n ) and ϕ m 1 + ϕ or m large enough. From these estimates and the Dominated Convergence Theorem we have have lim ψ m(x) = lim m = W ( M n m )(a, x)ϕ m (a 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j) m W ( M n m )(a, x) n j=1 dp(a j) W ()(a, x)ϕ(a M n 1,..., a n, x n+1, x n+2,...)) n j=1 dp(a j) W ()(a, x) M n n j=1 dp(a j) = K n (ϕ, x). Now we prove that the convergence ψ m (x) K n (ϕ, x) 0, when m is uniorm in x. Indeed, by using that ϕ m ϕ 0, when m and (10) we have or any ixed a 1,..., a n that lim sup W ( m )(a, x)ϕ m (a 1,..., a n, x n+1, x n+2,...)) m x W ()(a, x)ϕ(a 1,..., a n, x n+1, x n+2,...)) = 0. (11) The above limit implies that the numerator o ψ m converges uniormly to the numerator o K n (ϕ, ). The expression 10 guarantees that the denominator o ψ m converges uniormly to the denominator o K n (ϕ, ), thus proving the theorem. 6 Strongly Non-null Speciications Let us assume that the state space M is a inite set o the orm M = {1, 2,..., d}. We recall that in the general theory o speciications, we say that a speciication γ = (γ Λ ) Λ S with parameter set S on (M S, F ) is strongly non-null i there exist a constant c such that or all i S and ω we have γ {i} (ω i ) c > 0, see [13, 12] or more details on speciications. For our particular speciication (K n ) n N the above condition is simply K i (C i (a), ) L i (1 C i (a))(σ i ( )) L i (1)(σi ( )) > c > 0, where a {1, 2,..., d} and C i (a) = {ω : ω i = a} is cylinder set. Speciications Associated to Hölder Potentials are Strongly Non-null Let {1, 2,..., d} N, : R be a normalized Hölder potential, i.e., L (1)(x) 1, x, and (K n ) n N the speciication (7) deined by. We ix a {1, 2,..., d} and x = (x 1, x 2,...). From the deinition o the speciication (K n ) n N it ollows that K i (C i (a), (x 1, x 2, x 3,...)) = L i (1 C i (a))(σ i (x 1, x 2,...)) L i (1)(σi (x 1, x 2,...)) = L i (1 Ci (a))(σ i (x 1, x 2,...)) 15

16 = L i 1 (1)(a, x i+1, x i+2,...)e (a,x i+1,x i+2,...). Since e (a,x i+1,x i+2,...) is uniormly bounded away rom zero and L i 1 (1)(a, x i, x i+1,...) = 1, then the speciication (K n ) n N associated to satisies the strongly non-null condition. Now we consider the case o a general Hölder potential. Let h > 0 the main eigenunction o the Ruelle operator L, associated to the main eigenvalue λ. We assume that the equilibrium state µ or satisies h dµ = 1. We denote by the associated normalized potential, i.e., = + log h log h σ log λ. Now, or any ixed a {1, 2,..., d} and x = (x 1, x 2,...) we get K i (C i (a), (x 1, x 2, x 3,...)) = L i (1 C i (a))(σ i (x 1, x 2,...)) L i (1)(σi (x 1, x 2,...)) = L i 1 (h 1 )(a, x i+1, x i+2,...)h (a, x i+1, x i+2,...)λ i L i(1)(x. i+1, x i+2,...) It is known that h is uniormly bounded away rom zero and lim (h 1 )(a, x i+1, x i+2,...) = h 1 dµ > 0. Moreover, we have that L i 1 i λ i L i (1) h 0, thus proving that or any choice o a {1, 2,..., d} that K i (C i (a), x) is uniormly bounded away rom zero, in both x and i N. Breaking the Strongly Non-null Condition - Double Hobauer We now present an example o a continuous potential, called Double Hobauer, or which the associated speciication does not satisy the strongly non-null condition. The Double Hobauer potential is deined on the symbolic space = {0, 1} N as ollows: or each n 1 we consider the ollowing cylinder subsets o L n = 000 }{{... 0} 1 and R n = 111 }{{... 1} 0, or all n 1. (12) n n Note that these cylinders are disjoint and ( n 1 L n ) ( n 1 R n ) = \ {0, 1 }. We ix two real numbers γ > 1 and δ > 1, satisying δ < γ and the Double Hobauer potential is given by the ollowing expression: γ log n, i x L n 1 n, or some n 2; δ log n, i x R n 1 n, or some n 2; (x) = log ζ(γ), i x L 1 ; log ζ(δ), i x R 1 ; 0, i x {1, 0 }, 16

17 where ζ(s) = n 1 1/ns. Note that this is a continuous potential and below we have a sketch o its graph Figure 1: The Double Hobauer Potential represented in the closed interval [0, 1]. We reer the reader to [9] or the analysis o phase transition in this model and some acts about this model that will be used in the sequel. The irst important act we use next is that the Ruelle operator associated to this potential has a non-negative eigenunction h which is continuous in almost all points. An explicit description o this eigenunction is given in section 4 in [9]. Nevertheless we can deine a normalized potential = log J = + log h log(h σ) log λ, which is only discontinuous at a inite number o points. We observe that exp( ) = J is continuous, although = log J can attain the value. The continuous unction J, can take the values 0 and 1 on a inite number o points, see item ) on the beginning o the proo o Proposition 29 in [9]. Thereore L = L log J deines a positive bounded linear operator acting on C(). Let us proceed to analyze, irst, the strongly non-null condition or (which is not continuous). In order to prove that the strongly non-null condition is broken we take a = 0 and x = 1. In the same way as beore we get K i (C i (0), (1, 1, 1,...)) = L i 1 (1)(0, 1, 1,...) exp( (0, 1, 1,...)). From items e) and ) on the explicit computation o exp( ) appearing in Proposition 29 o [9], we get that exp( (0, 1, 1,...)) = 0. From the computation on page 27 in [9] ollows that L ī lim (1 C i (0))(σ i (1 )) i L ī (1)(σi (1 )) = 0. 17

18 Thus showing that the speciication associated to the potential = log J does not satisy the strongly non-null condition. Now we turn attention to the Double Hobauer potential. Beore to present computations or we will need to some expressions evolving its normalization. As we observed above exp( ) = J is continuous mapping so the duality relation is well deined and we have ϕ d[l ν] = L ϕ dν, ϕ C(). This allow us to deine or each n 1 and y the ollowing probability measure on µ y n = [ (L ) ] n (δ σ n (y)). For a ixed y any cluster point, with respect to the weak topology, o the sequence (µ y n) n N, is called a thermodynamic limit with boundary condition y (see Section 8 in [9]). Let µ be any thermodynamic limit o the sequence (µ 01 n ) n N, where the boundary condition 01 (0, 1, 1,...). It was shown in [9] that (a true limit on i not a subsequence) lim L log i J(1 C1 (0))(0, 1, 1,...) = 1 C1 (0) d µ. i We now back to the original potential (non-normalized Hobauer), but considering urther restrictions γ, δ > 2. In this case it is known that the eigenvalue λ = 1. By using this act, ixing a = 0 and proceeding in the same way as beore, or any x, we get that K i (C i (0), (x 1, x 2, x 3,...)) = L i (1 C i (0))(σ i (x 1, x 2,...)) L i (1)(σi (x 1, x 2,...)) = L i 1 log J (h 1 )(0, x i+1, x i+2,...)h (0, x i+1, x i+2,...) L i(1)(x. i+1, x i+2,...) From now on x = (1, 1, 1,...). It was shown in [9] that h (0, 1, 1,...) is inite and nonzero. In Section 4 o [9] it is shown that h does not vanish. Notice that the cylinder C 1 (0) is a countable union o cylinders sets o the orm L k (ollowing the notation (12)) where k 1. Since µ(c 1 (0)) > 0, there exist n N such that µ(l n ) > 0. This means that or some sequence (i k ) k N we have lim k L i k log J (1 L n )(0, 1, 1,...) > 0. From the properties o h deduced in [9] ollows that 0 < c in{h (x) : x L n } < +. For any pair o unctions ϕ, ψ C() satisying ϕ ψ, we have rom the positivity o the Ruelle operator, that L i 1 i 1 log J (ϕ)(0, 1, 1,...) Llog J (ψ)(0, 1, 1,...). By taking ϕ h 1 and ψ c 1 1 Ln we immediately have lim L i 1 i log J (h 1 )(0, 1, 1,...) lim i L i 1 log J (1 L n c 1 )(0, 1, 1,...) > 0. 18

19 On the other hand, lim i L i (1)(1, 1, 1,...) =. By using the estimates obtained above we can inally conclude that L i lim (1 C i (0))(σ i (1 )) i L i(1)(σi (1 )) showing that the speciication (K n ) n N associated to the Double Hobauer potential does not satisy the strongly non-null condition. = 0, 7 Uniqueness Theorem or Eigenprobabilities Theorem 3. Let be continuous potential and (K n ) n N be the speciication deined as in (7). Suppose that there is constant c > 0 such that or every cylinder set F F there is n N such that K n (F, x) ck n (F, y) or all x, y. Then, the set G () has only one element. Proo. Because o Lemma 2 it is enough to show that G DLR () is a singleton. Suppose that G DLR () contains two distinct elements µ and ν. Then the convex combination (1/2)(µ+ν) G ()\ex(g ()), where ex(g ()) denotes the set o extremes measures o G (). Thereore it is suicient to show that G () ex(g ()). Let µ G (), E 0 j N σ j (F ) and suppose that µ(e 0 ) > 0. The existence o such set is ensured by the Theorem 7.7 item (c) in [13], which says that any element µ G DLR () is uniquely determined by its restriction to the tail σ-algebra j N σ j (F ) (see Corollary 5 in the Appendix). Since µ(e 0 ) > 0 the probability measure ν µ( E 0 ) G DLR (), see Theorem 7.7 (b) in [13] (or, see Corollaries 9 and 4 in the Appendix). Let us prove that or all E F we have ν(e) cµ(e). Fix a cylinder set F F then or n big enough ollows rom the characterization o the DLR-Gibbs measures and rom the hypothesis that [ ] ν(f ) = K n (F, x) dν(x) = K n (F, x) dν(x) dµ(y) [ ] c K n (F, y) dν(x) dµ(y) [ ] = c K n (F, y) dµ(y) dν(x) = cµ(f ). Using the monotone class theorem we may conclude that or all E F we have ν(e) cµ(e). In particular, 0 = ν( \ E 0 ) cµ( \ E 0 ) thereore µ(e 0 ) = 1. Consequently µ is trivial on j N σ j (F ). Hence another application o Theorem 7.7 (a) o [13] (or, see Corollary 4) ensures that µ is extreme. 19

20 A consequence o this theorem is the ollowing generalization to uncountable alphabets o the amous Bowen s condition, see [29]. Theorem 4. Let be a continuous potential satisying D sup n N sup x,y ; x i =y i,i=1,...,n S n ()(x) S n ()(y) < then the set G () = {ν P(, F ) : L ν = λ ν} is a singleton. Proo. Let D be the constant deined on the above theorem and C a cylinder such that its basis is contained in the set {1,..., p}, i.e., or every n p we have 1 C (x 1... x n σ n (z)) = 1 C (x 1... x n σ n (y)) or all y, z. We claim that or any choice o y, z and or all n p, we have e 2D K n (C, z) K n (C, y) e 2D K n (C, z). By deinition o D we have, uniormly in n N, x, y, z, the ollowing inequality D S n ()(x 1... x n σ n (z)) S n ()(x 1... x n σ n (y)) D which immediately imply the inequalities exp( D) exp( S n ()(x 1... x n σ n (z))) exp( S n ()(x 1... x n σ n (y))) and exp( S n ()(x 1... x n σ n (y))) exp(d) exp( S n ()(x 1... x n σ n (z))). Using theses two previous inequalities we get that e D L n(1)(σn (z)) L n(1)(σn (y)) e D L n(1)(σn (z)) and also K n (C, y) = L n (1 C)(σ n (y)) L n (1)(σn (y)) L n (1 C e D )(σ n (z)) e D L n (1)(σn (z)) = e2d K n (C, z). Analogously we obtain e 2D K n (C, z) K n (C, y) and so the claim is proved. Let µ and ν be distinct extreme measures in G DLR (). Since we are assuming that M is compact ollow rom Theorem 7.12 o [13] that there exist y, z such that both measures µ and ν are thermodynamic limits o K n (, y) and K n (, z), respectively, when n. Given an open cylinder set C such that its basis is contained in the set {1,..., p} there is an increasing sequence o closed cylinders C 1 C 2... such that or all k N the basis o C k is contained in the set {1,..., p}, and k N C k = C. By Urysohn s lemma or each k N there is a continuous unction ϕ k : [0, 1] such that 1 Ck ϕ k 1 C and ϕ k 1 C pointwise. Since C and (C k ) k N have their basis contained in {1,..., p}, then the unction ϕ k can be chosen as a continuous unction depending only on its irst p coordinates. By using the claim and a standard approximation arguments we get, or any ixed k, the inequality K n (ϕ k, y) e 2D K n (ϕ k, z) or all n p. By taking the limits, when n goes to ininity and next when k goes to ininity we get µ(c) e 2D ν(c). Clearly the collection D = {E F : µ(e) e 2Dβ ν(e)} is a monotone class. Since it contains the open cylinder sets, which is stable under intersections, we have that D = F. Thereore µ e 2Dβ ν, in particular µ ν. This contradicts the act that two distinct extreme DLR-Gibbs measures are mutually singular, thereore G DLR () is a singleton and by Lemma 2 we are done. This result generalize two conditions or uniqueness presented in two recent works by the authors when general compact state space M is considered, see [11] and [18]. In act, 20

21 the above theorem generalizes the Hölder, Walters (weak and stronger as introduced in [11]) and Bowen conditions because it can be applied or potentials deined on = M N, where the state space M is any general compact metric space. 8 The Extension o the Ruelle Operator to the Lebesgue Space L 1 (, F, ν ) Let ν be the Borel probability measure obtained in previous sections and any ixed continuous potential. In this section we show how to construct a bounded linear extension o the operator L : C() C() acting on L 1 (, F, ν ), by abusing notation also called L, and prove the existence o an almost surely non-negative eigenunction ϕ L 1 (, F, ν ) associated to the eigenvalue λ constructed in the previous section. Proposition 4. Fix a continuous potential and let λ and ν be the eigenvalue and eigenmeasure o L /L (1), respectively. Then the Ruelle operator L : C() C() can be uniquely extended to a bounded linear operator L : L 1 (, F, ν ) L 1 (, F, ν ) having its operator norm given by L L 1 (,F,ν ) = λ. Proo. I ϕ C() then ϕ ± max{0, ±ϕ} C(). Thereore ollows rom the positivity o the Ruelle operator and (3) that L (ϕ) L 1 = L (ϕ + ϕ ) dν L (ϕ + ) + L (ϕ ) dν = L (ϕ + ) + L (ϕ ) dν = (ϕ + + ϕ ) d(l ν ) = λ (ϕ + + ϕ ) dν = λ ϕ dν = λ ϕ L 1. Since is a compact Hausdor space we have C(, R) L1 (,F,ν ) = L 1 (, F, ν ), thereore L admits a unique continuous extension to L 1 (, F, ν ). By taking ϕ 1 it is easy to see that L L 1 ( F,ν ) = λ. Proposition 5. For any ixed potential C() we have that { L 1 ( F, ν ) = Ξ() ϕ L 1 (, F, ν ) : L (ϕ) dν = λ ϕdν }. Proo. From (3) it ollows that C() Ξ(). Let {ϕ n } n N be a sequence in Ξ() such that ϕ n ϕ in L 1 (, F, ν ). Then ϕ n dν ϕ dν ϕ n ϕ dν 0 21

22 and using the boundedness o L, we can also conclude that L (ϕ n ) dν L (ϕ) dν L (ϕ n ϕ) dν λ ϕ n ϕ L 1 0. By using the above convergences and the triangular inequality we can see that Ξ() is closed subset o L 1 (, F, ν ). Indeed, L (ϕ) dν λ ϕ dν L (ϕ) dν L (ϕ n ) dν + L (ϕ n ) dν λ ϕ dν and the rhs goes to zero when n thereore ϕ Ξ(). Since C(, R) Ξ() and Ξ() is closed in L 1 (, F, ν ) we have that L 1 (, F, ν ) = C(, R) L1 (,F,ν ) Ξ() L 1 (,F,ν ) = Ξ() L 1 (, F, ν ). 9 Strong Convergence o Ruelle Operators Proposition 6. For any ixed potential C() there is a sequence ( n ) n N contained in C γ () such that n 0. Moreover, or any eigenmeasure ν associated to the eigenvalue λ we have that L n has a unique continuous extension to an operator deined on L 1 (, F, ν ) and moreover in the uniorm operator norm L n L L 1 (,F,ν ) 0, when n. Proo. The irst statement is a direct consequence o the Stone-Weierstrass Theorem. For any ϕ L 1 (, F, ν ) the extension o L n is given by L n (ϕ) L (exp( n )ϕ) which is well-deined due to Proposition 4. From this proposition we can also get the ollowing inequality L n (ϕ) dν = L (exp( n )ϕ) dν λ exp( n ) ϕ L 1 (,F,ν ) <. Since the distance in the uniorm operator norm between L n and L can be upper bounded by L n L L 1 (,F,ν ) = sup L n (ϕ) L (ϕ) dν 0< ϕ L 1 1 sup 0< ϕ L 1 1 λ sup 0< ϕ L 1 1 L (exp( n )ϕ) L (ϕ) dν ϕ (exp( n ) 1) dν 22

23 λ exp( n ) 1) sup 0< ϕ L 1 1 we can conclude that L n L L 1 (,F,ν ) 0, when n. ϕ dν, 10 Existence o the Eigenunctions We point out that given a continuous potential there exist always eigenprobabilities or L. However, does not always exist a positive continuous eigenunction h or L (see examples or instance in [9]). In this section we consider sequences o Borel probability measures (µ n ) n N deined by F E µ n (E) h n dν, (13) where n C γ () satisies n 0, and h n is the unique eigenunction o L n, which is assumed to have L 1 (, F, ν ) norm one. Since is compact we can assume up to subsequence that µ n µ P(, F ). We point out that or some parameters o the Hobauer model : {0, 1} R, there exists a positive measurable eigenunction h, but h dν is not inite (see [9]). From the deinition o µ n we immediately have that µ n ν. Notice that, in such generality, it is not possible to guarantee that µ ν. When this is true the Radon-Nikodym theorem ensures the existence o a non-negative unction dµ/dν L 1 (, F, ν ) such that or all E F we have dµ µ(e) = dν. (14) dν E In what ollows we give suicient conditions or this Radon-Nikodym derivative to be an eigenunction o L. Theorem 5. Let µ n as in (13), n Hölder approximating. I (h n ) n N is a relatively compact subset o L 1 (, F, ν ) then up to subsequence µ n µ, µ ν and L (dµ/dν ) = λ dµ/dν. Proo. Without loss o generality we can assume that h n converges to some non-negative unction h L 1 (, F, ν ). This convergence implies ϕh n dν ϕh dν 0, ϕ C(). Thereore µ n µ with µ ν and dµ/dν = h almost surely. E 23

24 Let us show that this Radon-Nikodym derivative is an non-negative eigenunction or the Ruelle operator L. From the triangular inequality ollows that L (h ) λ h L 1 (ν ) L (h ) L n (h ) L 1 (ν )+ L n (h ) λ h L 1 (ν ). The Proposition 6 implies that the irst term goes to zero when n goes to ininity. For the second term can estimate as ollows L n (h ) λ h L 1 (ν ) L n (h h n + h n ) λ h L 1 (ν ) L n (h h n ) + λ n h n λ h L 1 (ν ) L n L 1 (ν) h h n L 1 (ν) + λ n h n λ h L 1 (ν ). Since sup n N L n L 1 (ν) < + and h h n L 1 (,F,ν ) 0, when n, we have that the irst term in rhs also goes to zero when n goes to ininity. The second term in rhs above is bounded by λ n h n λ h L 1 (ν ) λ n h n λ h n L 1 (ν ) + λ h n λ h L 1 (ν ) = λ n λ + λ h n h L 1 (ν ). From Corollary 2 and our assumption ollows that the lhs above can be made small i n is big enough. Piecing together all these estimates we can conclude that L (h ) λ h L 1 (ν ) = 0 and thereore L (h ) = λ h, ν a.s.. Theorem 6. Let µ n as in (13) and suppose that µ n µ, n Holder approximating. I µ ν and h n (x) dµ/dν ν -a.s. then L (dµ/dν ) = λ dµ/dν. Proo. Notice that h n dν = 1 = dµ dν dν and h n (x) dµ/dν ν a.s.. The Schee s lemma implies that h n converges to dµ/dν in the L 1 (, F, ν ) norm. To inish the proo it is enough to apply the previous theorem. We now build an eigenunction or L without assuming converge o h n neither in L 1 (, F, ν ) or almost surely sense. We should remark that the next theorem applies even when no convergent subsequence o (h n ) n N do exists in both senses. Theorem 7. Let (h n ) n N be a sequence o eigenunctions on the unit sphere o L 1 (, F, ν ), n Holder approximating. I sup n N h n < +, then lim sup h n L 1 (, F, ν ) \ {0} and moreover L (lim sup h n ) = λ lim sup h n. Proo. Since we are assuming that sup n N h n < + then lim sup h n L 1 (, F, ν ). For any ixed x ollows rom this uniorm bound that the mapping M a lim sup h n (ax) 24