LIMITING DISTRIBUTIONS FOR COUNTABLE STATE TOPOLOGICAL MARKOV CHAINS WITH HOLES

. We study the dynamics of countable state topological Markov chains with holes, where the hole is a countable union of 1-cylinders. For a large class of positive recurrent potentials and under natural assumptions on the surviving dynamics, we prove the existence of a limiting conditionally invariant distribution, which is the unique limit of regular densities under the renormalized dynamics conditioned on non-escape. We also prove the exis- tence of a Gibbs measure on the survivor set, the set of points that never enter the hole, which is an equilibrium measure for the punctured potential of the open system. We prove that the Gurevic pressure on the survivor set equals the exponential escape rate from the open system. These results extend to the non-compact setting results previously available for ﬁnite state topological Markov chains.

1. Introduction. The study of dynamical systems with holes is motivated by the study of systems out of equilibrium -systems in which mass or energy is allowed to escape. Since invariant measures cannot be supported on a bulk of the phase space in such systems, the emphasis becomes a search for physically relevant conditionally invariant measures, sometimes called quasi-stationary states, which are invariant under the dynamics conditioned on non-escape.
Given a measure space (X, B, m) and a nonsingular, measurable transformation T , one identifies a measurable subset H of the phase space X as the hole. Trajectories that are mapped into H disappear forever and one studies the dynamics conditioned on non-escape. DefiningX = X \ H to be the complement of the hole, we study the dynamics of the open systemT :X → X on the sequence of noninvariant domains,X n = n i=0 T −iX . A probability measure µ on X is called The scaling factor λ := µ(X 1 ) is sometimes referred to as the eigenvalue of the measure µ since the above relation can be iterated to yield µ(T −n (A) ∩X n ) = λ n µ(A), so that the conditionally invariant measure necessarily predicts an exponential rate of escape of mass from the open system. Unfortunately, the existence of conditionally invariant measures with any eigenvalue between 0 and 1 is ubiquitous [19] and so existence alone becomes meaningless.
Rather, one focuses on the limit points of the sequenceT n * m m(X n ) , where m is a reference measure of interest and the operatorT n * is defined byT n * m(A) = m(T −n (A) ∩X n ), ∀A ∈ B. Under certain assumptions, the limit of such a sequence is a conditionally invariant measure and is independent of the initial distribution drawn from a reasonable class of measures. It also predicts a unified exponential rate of escape for this same class of initial distributions. In this case, we call such a limiting distribution a physically relevant conditionally invariant measure.
In probabilistic Markov chains, this type of limiting distribution in the presence of holes (or absorbing states) is called the Yaglom limit and has been studied in [33] and more recently in [22]. The study of deterministic systems with holes was initiated by the work of Pianigiani and Yorke [30], and since extended to a number of hyperbolic systems, beginning with those that admit finite Markov partitions: expanding maps on R n [30,10]; Smale horseshoes [3]; finite state topological Markov chains [11]; Anosov diffeomorphisms [4,5,6,7]; and billiards with convex scatterers satisfying a non-eclipsing condition (which makes the open system an Axiom A diffeomorphism) [27]. These results were then extended to hyperbolic systems without Markov partitions, including piecewise expanding maps of the interval [24,8,26,12]; certain classes of unimodal maps [13,16]; and more general dispersing billiards [18,15], including those with corner points [14]. All the systems listed above admit physically relevant conditionally invariant limiting distributions and enjoy a unified exponential rate of escape for a large class of initial distributions.
Recently, there has been interest in open systems exhibiting polynomial rates of escape [20,2,21], and in particular their connection to slowly mixing systems from non-equilibrium statistical mechanics [34]. Such systems exhibit qualitatively different behavior from systems with exponential escape rates; for example, the limiting distributions obtained by pushing forward and renormalizing, i.e. the limit points ofT n * m m(X n ) , are not conditionally invariant measures, but rather singular invariant measures [17]. Indeed, no physically relevant conditionally invariant measures can exist for such systems due to the subexponental rate of escape.
Typically, such slowly mixing systems are studied via an induced map on the phase space: One chooses a subset Y ⊂ X and studies the return map T R : Y → Y , where R is the first return time to Y . The usual strategy is to prove results for the induced system T R , which has stronger hyperbolicity than the original map, and then pass those results back to the original system.
In many situations, the return maps can be constructed to admit countable (but not finite) Markov partitions and their dynamics can be studied via conjugacy to symbolic dynamics, i.e., topological Markov chains. Such techniques are very powerful and are by now classical in the study of dynamical systems. Unfortunately, while finite state topological Markov chains with holes have been well-studied ( [11], see also the recent book [9]), the corresponding results for countable state chains are so far unavailable. This motivates the present work: To study countable state topological Markov chains with holes and prove, under a natural set of assumptions and for a general class of potentials, results analogous to those proved for many of the hyperbolic systems listed above. Our hope is that this work will provide a standard reference to future studies of open systems, in particular to those seeking to expand the study of systems with subexponential rates of escape.
There are several complications in this setting. First, our space is not necessarily locally compact and so several of the function space arguments previously used in open systems have to be reformulated. Secondly, open systems are not topologically transitive or positive recurrent in the usual sense 1 of the literature (for example [31]), so one must formulate alternative conditions which generalize the notions of mixing and recurrence sufficiently to prove strong convergence results for the open system.
We introduce our topological Markov chain, formulate our assumptions and state our main theorems in Section 2. Section 3 contains the proofs of some preliminary facts about the spaces of functions we shall use while Section 4 contains the proof of Theorem 2.1 regarding convergence of a large class of initial densities to a conditionally invariant measure and a unified exponential rate of escape. In Section 5 we prove a variational principle, along with Theorem 2.2, linking the escape rate of the open system to the pressure of the closed system restricted to the survivor set, the singular set of points that never enters the hole.
2. Setting and main results. Let S denote the countable state space of the Markov chain, which we take to be a subset of N, and A the adjacency matrix, i.e. A i,j = 1 if the transition from state i to state j is permitted and 0 otherwise. The associated topological Markov chain is defined as the set of all admissible sequences, We denote by σ the one-sided shift on Σ, i.e. σ(x 0 , x 1 , x 2 , . . .) = (x 1 , x 2 , . . .).
We endow Σ with the usual separation time metric: Fix θ ∈ (0, 1) and for x, y ∈ Σ, define With this metric, the space (Σ, d θ ) is complete, but may not be compact when S is infinite. Indeed, it is not even locally compact unless #{k ∈ S | A s,k = 1} < ∞, for each s ∈ S. In what follows, we will not assume that Σ is locally compact.
We will denote cylinder sets in Σ in the usual way, A cylinder of length n is called an n-cylinder.
Cylinders are both open and closed as sets and form a basis for the metric topology on Σ. We will assume that our topological Markov chain is toplogically mixing: A topological Markov Chain satisfies the big images and preimages property (BIP) if there exists a finite set Λ ⊂ S such that ∀s ∈ S, ∃i, j ∈ Λ such that A s,i A j,s = 1.
Notice that we do not require any regularity between 1-cylinders so that ϕ may not be bounded on Σ. Using a locally Lipschitz ϕ, we define the associated transfer operator L ϕ acting on continuous functions by where S n ϕ = n−1 k=0 ϕ•σ k is the nth ergodic sum. We will assume that |L ϕ 1| ∞ < ∞. It follows from this plus BIP and mixing that: • the Gurevic pressure P G (ϕ) is finite [31, Theorem 1]; • ϕ is positive recurrent [32, Corollary 2] (see also [28]); • there exists a finite conformal Borel measure m, positive on cylinders, such that, dm dm•σ = e ϕ−P G (ϕ) [31, Theorem 4, Proposition 3] We will define Gurevic pressure in Section 5.3. Since m is finite, we normalize it to be a probability measure. We may also replace ϕ by ϕ − P G (ϕ) so without loss of generality, we may assume P G (ϕ) = 0.
When we introduce a hole H comprised of a countable union of 1-cylinders, we define the related 'punctured' potential ϕ H for the open system by ϕ H = ϕ onΣ and ϕ H = −∞ on H. Notice that ϕ H is still locally Lipschitz where defined. The associated transfer operator for the open system and its iterates are given bẙ where 1 A denotes the indicator function of a set A.
The importance of the operatorL ϕ from the point of view of the open system stems from the following relation. Recalling that we have normalized the pressure of ϕ to be 0, we observe, due to the conformality of m, so that the iterates ofL ϕ govern the rate of escape of mass with respect to m. We will use interchangeably the notation η(f ) = f dη for a given measure η on Σ.
2.3. Main results. We denote by L 1 (m) the set of (complex valued) integrable functions on Σ and by C 0 (Σ) the set of bounded continuous functions Σ → C. C 0 (Σ) is a Banach space equipped with the norm Similarly, define a space of bounded, locally Lipschitz functions on Σ by where f Lip := |f | ∞ + Lip(f ). Notice that with this definition, f · g Lip ≤ f Lip g Lip . Also, since m is a probability measure, C 0 (Σ) ⊂ L 1 (m). Since m is conformal with respect to ϕ, L ϕ andL ϕ are both bounded linear operators on L 1 (m). It follows from the fact that characteristic functions of cylinders are in Lip(Σ) as well as the assumption that |L ϕ 1| ∞ < ∞ thatL ϕ is a bounded linear operator on both C 0 (Σ) and Lip(Σ). Indeed, we will prove thatL ϕ has a spectral gap on Lip(Σ).
Then there exists a probability density g ∈ Lip(Σ), bounded away from 0 onΣ, such that: a)L ϕ g = λg and dµ := gdm defines a conditionally invariant probability measure forσ with eigenvalue λ; b) the rate of escape from the open system is exponential: log λ = lim n→∞ 1 n log m(Σ n ); c)L ϕ has a spectral gap on Lip(Σ): λ is a simple eigenvalue and the remainder of the spectrum ofL ϕ is contained in a disk of radius ρ < λ; d) If f ∈ Lip(Σ), then lim n→∞ λ −nL n ϕ f = c(f )g, for some constant c(f ), and convergence is in · Lip at an exponential rate.
We remark that for conditionally invariant measures to be physically relevant in the sense of Theorem 2.1(d), one must have convergence (and escape) occurring at an exponential rate. Thus the assumption of big images and preimages and the analogue for the open system, (H), that we have formulated are crucial from this point of view. If one weakens these assumptions to, for example, a positive recurrent potential with finite pressure, then the rate of convergence to equilibrium for the closed system may be subexponential (see [31]) and thus there will be no physically relevant conditionally invariant measures for the open system. For recent results regarding limiting distributions in open systems with subexponential rates of escape, see [17].
Next we turn to the survivor set,Σ ∞ = ∩ ∞ n=0 σ −n (Σ\H), the zero m-measure set of points that never enter H. While it may seem that λ is an artifact of the function space Lip(Σ) that we have chosen to work with, our next theorem demonstrates that this number (which depends on the potential ϕ) is intrinsic to the open system in that log λ equals the pressure on the survivor set.
is the measure-theoretic entropy of η. b) There exists ν ∈ M which is realized by the following limit, The measure ν is a Gibbs measure for the potential ϕ − log λ, enjoys exponential decay of correlations, and attains the supremum in the variational principle in (a), i.e. ν satisfies the escape rate formula, where m H is a positive Borel measure with support equal toΣ ∞ that is conformal for the potential d) The following criterion holds for convergence to g: if f ∈ Lip(Σ) and f ≥ 0, then where the covergence is in Lip(Σ) at an exponential rate.
The role played by the invariant Gibbs measure ν in linking the escape rate with the pressure on the survivor set is further justification for the assumption (H): Having big images is a necessary condition for the existence of a Gibbs measure [31,Theorem 8] (see also [32,Theorem 1]). Remark 2.3. Since λ n = µ(Σ n ), the characterization of ν in Theorem 2.2(b) can be restated in terms of the limit of conditional probabilities, 3. Function spaces and preliminary estimates. For the remainder of the paper, we fix a potential ϕ satisfying |L ϕ 1| ∞ < ∞ and assume (Σ, σ, H) satisfies the hypotheses of Theorem 2.1. Our main task will be to prove the existence of a spectral gap forL ϕ , from which most of our other results follow. We begin with a standard distortion estimate, whose proof we record for completeness.
Since σ n is injective on each n-cylinder, these estimates immediately imply that for any f ∈ C 0 (Σ), x ∈ Σ, where we have used the fact that m is a probability measure, so thatL ϕ is a bounded, linear operator on C 0 (Σ) with spectral radius at most 1. Unfortunately, we need greater regularity for our function spaces, so we will work in Lip(Σ). Since we are not assuming that Σ is compact or even locally compact, it may be that the unit ball of Lip(Σ) is not relatively compact in C 0 (Σ). Instead, we will use L 1 (m) as our weak norm. It is a standard fact that the closed unit ball of Lip(Σ) is relatively compact in L 1 (m). Before characterizing the spectrum ofL ϕ on Lip(Σ), we first prove some greater regularity properties of evolved densitiesL n ϕ f . 3.1. A fixed point for the normalized transfer operator. To obtain the spectral decomposition ofL ϕ , it will be convenient to first prove the existence of a positive eigenfunction g. To this end, we define a set of log-Lipschitz probability densities: Given C > 0, let , where | · | 1 denotes the L 1 (m) norm. Notice thatL 1 is a nonlinear operator. This section is devoted to the proof of the following proposition.
Note that when #(S) = ∞, L K is not compact in L 1 (m) since the log-Lipschitz constant does not control the L ∞ norm of a function. Thus there are unbounded functions in L K . As the next several lemmas will show, however,L 1 (L K ) ⊂ L K is bounded in C 0 (Σ) and so it is possible to define an invariant subset of L K which is compactly embedded in L 1 (m).
. For each n ∈ N, note thatσ −n (x) and σ −n (y) are in 1-1 correspondence sinceσ n is a bijection fromΣ n ∩ [j 0 , j 1 , . . . j n−1 , i] to its image for each (n + 1)-cylinder [j 0 , . . . , j n−1 , i]. So we may enumerate the elements ofσ −n (x) = {u j } j∈J andσ −n (y) = {v j } j∈J so that u j and v j lie in the same (n + 1)-cylinder for each j, and J is the relevant index set. Now we estimate, where we have used the fact that d θ (u j , v j ) = θ n d θ (x, y). The lemma follows by noting that Thus we may apply (3.4) toL ϕ f . Next we turn our attention to proving an upper bound onL ϕ f (x) for x ∈ Σ\G Λ H . For each y ∈σ −1 (x), let [i y ] denote the 1-cylinder containing y. Due to (H)(a), σ([i y ]) ⊃ [j] for some j ∈ Λ H . Thus we may organizeσ −1 (x) according to these images. Define for each j ∈ Λ H , For each j ∈ Λ H , choose a point z j ∈ [j]. Note that each y ∈σ −1 (x) may be contained in more than one set P j (x). But for each y ∈ P j (x), there exists w y ∈ [i y ] such thatσ(w y ) = z j . Now using the regularity of f and bounded distortion, we estimateL where for the last inequality, we have applied (3.4) Proof. First notice that for f ∈ L K , , and we have used the fact that for two sequences of positive terms, Thus using Lemma 3.1, On the other hand, using this estimate together with (3.5) yields, which implies that L ϕ f dm ≥ 1/(B + 1).
As mentioned earlier, L K is not necessarily compact in L 1 (m) when S is infinite. However, motivated by Lemmas 3.4 and 3.5, we set C = C 1 /C 2 and define a subset of L K as follows.
Proof. As already noted in the proof of Lemma 3.4,L ϕ (M K ) ⊂ M K . Indeed, L 1 is continuous on L K sinceL ϕ is continuous on Lip(Σ) and the normalization factor, |L ϕ f | 1 , is uniformly bounded away from 0 for f ∈ L K by Lemma 3.5. Thus Proposition 3.7. L ∞ K is a convex, compact subset of L 1 (m). Proof. Recall that the closed balls B R = {f ∈ Lip(Σ) | f Lip ≤ R} are compact in L 1 (m). Thus, to prove that L ∞ K is compact, it suffices to show that L ∞ K is bounded in Lip(Σ) and closed in L 1 (m).
where in the last inequality, we have used the estimate |e z − 1| ≤ |z|e |z| as in the proof of (3.2) in Lemma 3.1. Combining this with the bound on |f | ∞ , we conclude , it follows that f ∈ Lip(Σ) and thus f is continuous. Fix (f n ) n∈N ⊆ L ∞ K such that f n → f pointwise a.e. and in L 1 (m). Then Σ f dm = 1.
Since m(Σ \ G) = 0 and m is positive on cylinders, G is dense in Σ. Since 0 ≤ f n (x) ≤ C , it follows from the continuity of f and the density of G that Since f is continuous, we may take the limit as k → ∞ to conclude that f (z) ≤ f (x)e Kd θ (x,z) .
We have shown that for x, z ∈ [i], then Taking the appropriate suprema proves Lip(log(tf + (1 − t)h)) ≤ K.
Collecting these results, we are ready to prove Proposition 3.2.
Proof of Proposition 3.2. By Lemma 3.6, the restrictionL 1 : K is a convex, compact subset of L 1 (m), it follows from the Schauder-Tychonoff theorem thatL 1 has a fixed point in L ∞ K . Let g ∈ L ∞ K be such thatL 1 g = g. It follows thatL ϕ g = λg, where and λ ≥ C 2 by Lemma 3.5. Since g ∈ L ∞ K , we have |g| ∞ ≤ C and by Claim 1 in the proof of Proposition 3.7, we have Lip(g) ≤ Ke K C . Thus g ∈ Lip(Σ). We begin by proving a version of the standard dynamical Lasota-Yorke or Döblin-Fortet inequality, which provides a bound on the essential spectral radius ofL ϕ .
The novelty of the inequality in this setting is the presence of the integral factors appearing in both terms of the inequality. This will enable us to link the essential spectral radius (not just the spectral radius) to the escape rate of mass from the open system. Note that the presence of these L 1 terms is distinct from the analogous inequalities for topological Markov chains derived from finite state spaces; such inequalities exploit a C 0 bound due to the compactness of Σ that is not available in the present setting. |f (x)| dm(x). (4.1) We first estimate the C 0 -norm ofL ϕ f .
Using  Here we made use of large images and the fact that if y ∈Σ n , then [y 0 , . . . , y n−1 ] ⊆ Σ n−1 . We conclude that Next, suppose x, v ∈ Σ with x 0 = v 0 . For each y ∈σ −n (x), let z y,n be as in (4.1). Since x 0 = v 0 , there is a one-to-one correspondence between y ∈σ −n (x) and w ∈σ −n (v) that preserves n-cylinders. Now we estimate, Summing the series as in (4.2), dividing by d θ (x, y) and taking the appropriate suprema, we obtain Adding this to (4.3) concludes the proof of the lemma with C 0 = 3e 2C d κ −1 .
The previous proposition and the relative compactness of the closed unit ball of Lip(Σ) in L 1 (m) are nearly enough to prove thatL ϕ is quasi-compact. Due to the loss of mass caused by the hole, it is still necessary to show that the spectral radius ofL ϕ is strictly larger than the contraction provided by Proposition 4.1. This will be done in Section 4.2. In the next section, we prepare some groundwork by investigating further properties of L ∞ K .

4.1.
Regularity of log-Lipschitz functions. In order to make a Perron-Frobenius argument and show thatL ϕ has a spectral gap, we first show that functions in L ∞ K become positive under the action ofL n ϕ (Proposition 4.3). This will imply that eigenfunctions ofL ϕ in L ∞ K are bounded away from 0 (Corollary 4.4). We begin by proving a combinatorial result about admissible sequences in the open system. Then there exists a finite set Λ H ⊆ S and N * ∈ N such that Λ H ⊆ Λ H and ∀j, k ∈ Λ H , ∀n ≥ N * ∃a 1 , . . . , a n ∈ Λ H such that A j,a1 A a1,a2 · · · A an−1,an A an,k = 1.
To complete the proof of the proposition, we show that the subshift Σ Λ H = {x ∈ Σ | x i ∈ Λ H } is topologically mixing. Recall that a topologically transitive TMC is topologically mixing if and only if there exist states j, k and relatively prime integers p, q such that σ −p ([j]) ∩ [j] = ∅ and σ −q ([k]) ∩ [k] = ∅ (see [1,Section 4.2]).
For the case j ∈ Λ H and k ∈ Λ H \Λ H , we argue differently due to the asymmetry of (H)(a). Since k ∈ Λ H \ Λ H , k must belong to Λ i, for some i, ∈ Λ H . Thus there exists a 1 , . . . a n ∈ Λ i, such that A i,a1 A a1,a2 · · · A an,k = 1. Then appending this sequence to Λ j,i , since i, j ∈ Λ H , yields the required sequence in Λ H connecting j to k. The case when j, k ∈ Λ H \ Λ H follows by combining the other two cases and using again (H)(a). Therefore, Σ Λ H is topologically transitive.
Note that the set Λ H given by Lemma 4.2 is not unique: we simply choose one finite set Λ j,k for each pair j, k ∈ Λ H . We consider the set Λ H so constructed as fixed for the remainder of the paper. Proof Due to the log-regularity of f , it follows that f (y) ≥ 1 2 e −K for all y ∈ [i 0 ]. By (H), there exists N i0 such thatσ n ([i 0 ]) =Σ for all n ≥ N i0 . Thus for each n ≥ N i0 and x ∈Σ, there exists w x,n ∈ [i 0 ] such that σ n (w x,n ) = x. We may increase N i0 to ensure that N i0 ≥ N * , where N * is from Lemma 4.2. Thus using (H)(a) and Lemma 4.2, we may choose w x,n so that Corollary 4.5. There exists C 3 > 0 such that for all n ∈ N, where λ ∈ (0, 1) is the eigenvalue corresponding to the fixed point g chosen in Proposition 3.2.
Proof. Fix n ∈ N. By the previous corollary, Since λ −nL n ϕ g(x) = g(x), we obtain the two inequalities: Notice that Corollary 4.4 together with the proof of Corollary 4.5 imply that if g, h ∈ L ∞ K are two fixed points forL 1 such thatL ϕ g = λg andL ϕ h = αh, then α = λ. The next corollary is used in the proof thatL ϕ has a spectral gap.

4.2.
Quasi-compactness ofL ϕ . We can use Corollary 4.4 to compute both the spectral radius and the essential spectral radius ofL ϕ : Lip(Σ) → Lip(Σ). This proves once and for all that the essential spectral radius, ρ ess (L ϕ ), is strictly less than ρ(L ϕ ). It turns out this computation also gives us the escape rate. Proof. First, we prove that lim sup n→∞ m(Σ n ) 1/n ≤ ρ(L ϕ ). To see this, it is enough to integrateL n ϕ 1: Taking n-th roots and letting n → ∞ shows that lim sup m(Σ n ) 1/n ≤ ρ(L ϕ ). Next, from Proposition 4.1, we have the following bound for f ∈ Lip(Σ), from which it follows that L n ϕ Lip ≤ 2C 0 m(Σ n−1 ) and thus ρ(L ϕ ) ≤ lim inf n→∞ m (Σ n−1 ) 1/n . We conclude that lim n→∞ m(Σ n ) 1/n exists and equals ρ(L ϕ ).

DEMERS, IANZANO, MAYER, MORFE AND YOO
By Corollary 4.4, the left hand side is not zero. Taking n-th roots and letting n → ∞, we conclude that Since λ is an eigenvalue, it is bounded above by ρ(L ϕ ). Thus, λ = ρ(L ϕ ) = lim n→∞ m(Σ n ) 1/n . In addition, we see that if n is sufficiently large. It follows from Hennion's Theorem [23], Proposition 4.1, and the relative compactness of the unit ball of Lip(Σ) in L 1 (m) that ThusL ϕ is quasi-compact.
The previous theorem implies that mass escapes at an exponential rate. In particular, which is item (b) of Theorem 2.1.

4.3.
A spectral gap forL ϕ . Before proving thatL ϕ has a spectral gap, we prove the following useful result. If there exists a sequence (n k ) ⊆ N and α > 0 such that α −n kL n k ϕ f → f uniformly, then α = λ and f ∈ L ∞ K . Proof. Since f ∈ Lip(Σ) and inf{f (x) | x ∈Σ} > 0, it follows that Lip(log f ) < ∞ (notice also that f H ≡ 0 since (L n ϕ f ) H ≡ 0 and f is the uniform limit of such functions). By Lemma 3.3, Lip(log α −n kL n k ϕ f ) ≤ θ n k Lip(log f ) + C d , and thus ∃M ∈ N such that Next, note that f ∈ M K , where M K is defined before Lemma 3.4. Thus by Lemmas 3.4 and 3.5, we have |L 1 f | ∞ ≤ C , and thusL 1 f ∈ L ∞ K . By Lemma 3.6, L n 1 f ∈ L ∞ K for all n ∈ N. Notice that by uniform convergence, Thus f ∈ L ∞ K . Finally, let s = inf{f (x) | x ∈Σ} > 0 and S = sup{f (x) | x ∈Σ} ≤ C . For each k ∈ N and x ∈Σ, Using Corollary 4.5, if α < λ, then the first inequality in (4.5) implies the infimum of α −n kL n k ϕ f onΣ becomes arbitrarily large, contradicting the uniform convergence to f , which has integral 1. On the other hand, if α > λ, then the second inequality in (4.5) implies that the limit of α −n kL n k ϕ f is 0, again contradicting the fact that f dm = 1. Thus α = λ.
The following proposition proves λ is the only eigenvalue on the circle {z ∈ C | |z| = ρ(L ϕ )}. Moreover, it follows from this argument that the eigenspace corresponding to λ is one-dimensional. A key strategy in the proof is adapted from [30]. Proposition 4.9. Suppose h ∈ Lip(Σ, C) is an eigenfunction ofL ϕ with corresponding eigenvalue λe i2πφ with λ = ρ(L ϕ ). Then h = zg for some z ∈ C and φ = 0.
Fix α 1 > 0 such that To see that J is open, let t ∈ J and set δ = inf{f t (x) | x ∈Σ} > 0. If , and we conclude that f s ∈ J.
We claim that J is also closed. Suppose t ∈ R is a limit point of J.
Thus, since L ∞ K is closed in L 1 (m), we conclude that f t ∈ L ∞ K . Since λ −n kL n k ϕ f t → f t uniformly, it follows by Corollary 4.6 that inf{f t (x) | x ∈Σ} > 0. Therefore t ∈ J and we conclude that J is closed. It follows that J = R. Suppose x ∈Σ. We have shown that which implies that f 0 (x) = g(x). On the other hand, if x ∈ H, then h(x) = 0 and thus f 0 (x) = 0. We conclude that f 0 = g.
It follows that h 1 = (α 2 − α 1 )g. Moreover, , which implies that h 2 = βg for some β ∈ R. It follows that h = (α 2 − α 1 + iβ)g and Finally, we prove that λ is an eigenvalue ofL ϕ of algebraic multiplicity 1. Proof. It follows directly from Proposition 4.9 that the dimension of the eigenspace E λ corresponding to λ is one. Thus if λ has a non-trivial Jordan block, there must exist h ∈ Lip(Σ) such that (L ϕ − λ)h = g. Iterating this equation, we obtain Then g(x) > 0 and combining the two previous inequalities, we must have Therefore, (L ϕ − λ1)h = 0 from which we conclude that h ∈ E λ , i.e. h is a multiple of g. Thus λ has no Jordan block.
Since ρ ess (L ϕ ) < ρ(L ϕ ) = λ, Propositions 4.9 and 4.10 together imply thatL ϕ has a spectral gap. We are now ready to complete the proof of Theorem 2.1.
Proof of Theorem 2.1. Let g be the eigenfunction ofL ϕ with eigenvalue λ given by Proposition 3.2. The fact that g ∈ L ∞ K together with Corollary 4.4 prove the initial statement of Theorem 2.1.
The fact thatL ϕ g = λg together with the conformality of the measure m as expressed in (2.1) proves item (a) of the theorem. Item (b) follows from (4.4) as already noted in Section 4.2. Item (c) is proved in Propositions 4.9 and 4.10.
It remains to prove item (d) of the theorem. SinceL ϕ has a spectral gap, we let Π λ : Lip(Σ) → Lip(Σ) be the projection onto E λ , which is simply the span of g, and writeL where R is a bounded linear operator on Lip(Σ) with spectral radius strictly less than λ and R • Π λ = Π λ • R = 0.
Define W = Π −1 λ ({0}). Since Π λ is a projection, we have the following decomposition Lip(Σ) = E λ ⊕ W. It follows that every Lipschitz function f has a unique decomposition where c(f ) ∈ C and w ∈ W. We shall see that the linear functional c has special properties. Note that continuity of Π λ implies W is a closed subspace of Lip(Σ). Also note that both E λ and W are invariant underL ϕ . Proof. Fix > 0 such that ρ(R) + < λ. Choose M ∈ N such that R n 1/n Lip < ρ(R) + if n ≥ M . If f ∈ Lip(Σ) and n ≥ M , then Lemma 4.12. c : Lip(Σ) → C is a bounded linear functional.
Proof. For f ∈ Lip(Σ), write f = c(f )g + w. For each n ∈ N, This together with the fact that Π λ is a bounded linear operator proves the lemma.
it follows thatL n ϕ f |L n ϕ f |1 ∈ W for all n ∈ N. Since W is closed, any limit point of is in W. We conclude that g is not a limit point of the sequence.

5.
Gibbs measure and variational principle. We next turn our attention to the survivor set,Σ ∞ = ∞ n=0 σ −n (Σ), and the proof of Theorem 2.2. We will begin by showing that the functional c(·) induces a measure m H , supported onΣ ∞ , for which Σ f dm H = c(f ) whenever f ∈ Lip(Σ). We will then show that m H is conformal with respect to the potential ϕ H − log λ. Recall that 1 A denotes the indicator function of the set A. Proof. Let C denote the collection of cylinders in Σ. Since S is infinite, C need not be a semi-algebra in general. We define a semi-algebra by first introducing the notion of a generalized n-cylinder. Given i 0 , i 1 , . . . , i n−2 ∈ S and k ∈ N ∪ {0}, we define a generalized n-cylinder by, Note that if k = 0, then [i 0 , i 1 , . . . , i n−2 ; k] = [i 0 , i 1 , . . . , i n−2 ] so that every (n−1)cylinder is a generalized n-cylinder. Let C ⊃ C denote the set of all generalized n-cylinders in Σ. Now define a set function τ : C → R ≥0 by τ (E) = c(1 E ) for all E ∈ C . Note that τ is well-defined since 1 E ∈ Lip(Σ) for each E ∈ C . Also, τ is non-negative since c(f ) is real and non-negative whenever f is.
Suppose {E n } n∈N is a disjoint collection of sets in C with ∪ n∈N E n ∈ C . We claim that c(1 ∪En ) = n∈N c(1 En ). To see this, note that where the interchange of the sum and integral is justified by the monotone convergence theorem. Note that the sequence (a k,n ) k∈N , where a k,n = λ −k ΣL k ϕ 1 En dm, is uniformly (in both n and k) bounded above by C 3 , where C 3 is from Corollary 4. It follows that τ (∪E n ) = n∈N τ (E n ) and so τ is countably additive. It follows as a consequence of the Caratheodory extension theorem that τ extends uniquely to a positive Borel measure m H on Σ (see for example [29,Chapter 2]).
Proof. We know this formula holds when f is the characteristic function of a generalized cylinder. As a first step, suppose that f = n∈N a n 1 En , where {a n } n∈N ⊆ C is bounded and {E n } n∈N is a disjoint collection of sets in C . We claim that the proposition holds in this case. Indeed, since n∈N a n 1 En is bounded by sup n {|a n |}, it follows from the dominated convergence theorem that Since λ −kL k ϕ 1 En ≤ C 3 , another application of the dominated convergence theorem gives: Thus, Σ f dm H = lim k→∞ λ −k Σk f dm. The proof of the proposition will be complete once we prove the following claim: Any Lipschitz function f on Σ is the uniform limit of a sequence (g n ) n∈N satisfying g n = k∈N a k 1 E n,k , where {E n,k } k∈N is the collection of all n-cylinders and {a k } k∈N ⊆ C lies in a disk of radius |f | ∞ .
Fix f ∈ Lip(Σ). For each n, k ∈ N, let y E n,k be an arbitrary point in E n,k . Define g n : Σ → R by It follows that g n → f uniformly in Σ.
We can now compute Σ f dm H as where Σ g n dm H = lim k→∞ λ −k Σk g n dm by the first step of the proof. We claim that we can interchange the limits.
Fix n ∈ N and observe that |g n | ∞ ≤ |f | ∞ . Then Thus, λ −k Σk g n dm → lim k→∞ λ −k Σk g n dm uniformly (with respect to n). It follows that we can interchange the limits in (5.1) to obtain Next, we show that m H is conformal for the renormalized punctured potential ϕ H − log(λ). Since the measures agree on n-cylinders, they agree as Borel measures on Σ. If f : Σ → R is Borel measurable and A is a Borel set, then Proof. ν is a probability measure since ν(Σ) = c(g) = 1. Moreover, since the support of m H isΣ ∞ and g is bounded away from 0, the support of ν is alsoΣ ∞ . Now fix a nonempty cylinder E n = [i 0 , . . . , i n−1 ]. Then by Lemma 5.4, ν(σ −1 (E n )) = σ −1 (En) g dm H = λ −1 EnL ϕ g dm H = En g dm H = ν(E n ).
The following proposition implies that in fact ν is a Gibbs measure with respect to the potential ϕ.
Proposition 5.6. There exists a constant C G > 0 such that for any n-cylinder E n ⊂Σ n and any y ∈ E n , C −1 G exp(S n ϕ(y))λ −n ≤ ν(E n ) ≤ C G exp(S n ϕ(y))λ −n .
Lower bound. For any x ∈Σ and k > n, L k ϕ (1 En g)(x) = y∈σ −k (x)∩En g(y)e S k ϕ(y) ≥ e −C d e Snϕ(w) y∈σ −k (x)∩En g(y) g(σ n (y)) · g(σ n (y))e S k−n ϕ•σ n (y) where C g = sup{g(x) | x∈Σ} inf{g(x) | x∈Σ} . The second-to-last equality follows from the fact that the restriction Thus, it follows as in the proof of Proposition 4.3 that we can find N ∈ N and B N > 0 such thatL N ϕ (1 σ([in−1]) g) ≥ B N . Then using Corollary 4.5, we have for k > N , . It follows that ν(σ([i n−1 ])) > 0. Indeed, due to (H), we can remove the dependence of the lower bound on i n−1 : Set where the last expression is positive since Λ H is finite and the support of ν isΣ ∞ . It follows that ν(E n ) ≥ C −1 g e −C d e Snϕ(w) λ −n κ, which is the required lower bound.
By our observation above of the injectivity of σ n on n-cylinders, we can make the substitution z = σ n (y) and obtain Therefore, we conclude that ν(E n ) ≤ C g e Snϕ(w)+C d λ −n .

Proof of Theorem 2.2.
We begin by defining the Gurevich pressure of a locally Lipschitz continuous potential ϕ. Define the partition function Z n (ϕ, a) on states a ∈ S by Z n (ϕ, a) = σ n (x)=x 1 [a] (x)e Snϕ(x) .
The Gurevic pressure always exists (though it may equal ∞) and is independent of the choice of state a. When |L ϕ 1| ∞ < ∞, the Gurevic pressure is finite [31,Theorem 1].
We now verify the items of Theorem 2.2. Item (c) follows from Lemmas 5.3 and 5.4 together with Proposition 5.5. Proposition 5.5 also proves the first half of (b). It follows from Proposition 5.6 together with [31,Theorem 8], that ν is an equi-