Upper Semi-Continuity of Entropy in Non-Compact Settings

UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS

GODOFREDO IOMMI, MIKE TODD, AND AN´IBAL VELOZO

Abstract. We prove that the entropy map for countable Markov shifts of ﬁnite entropy is upper semi-continuous on the set of ergodic measures. Note that the phase space is non-compact. We also discuss the related problem of existence of measures of maximal entropy.

1. Introduction The entropy map of the continuous transformation T : X X defined on a metric space X, d is the map µ h T which is definedÑ on the space of p q ÞÑ µp q T invariant probability measures T , where hµ T is the entropy of µ (precise definitions´ can be found in SectionM2). The studyp ofq the continuity properties of the entropy map, with T endowed with the weak˚ topology, goes back at least to the work of BowenM [Bo1]. In general it is not a continuous map (see [Wa, p.184]). However, in certain relevant cases it can be shown that the map is upper semi-continuous, that is, if µ is a sequence in that converges to µ then p nqn MT lim supn h µn h µ . For example, if T is an expansive homemorphism of a compactÑ8 metricp q spaceď p thenq the entropy map is upper semi-continuous, see [Wa, Theorem 8.2] and [Bo1, De, Mi2]. It is also known that if T is a C8 map defined over a smooth compact manifold then, again, the entropy map is upper semi-continuous (see [N, Theorem 4.1] and [Y]). Lyubich [Ly, Corollary 1] proved that the entropy map is upper semi-continuous for rational maps of the Riemann sphere. In all the above examples the phase space is compact: in this article we will drop the compactness assumption on the underlying space. Markov shifts defined over finite alphabets have been used with remarkable suc- cess to study uniformly hyperbolic systems. Indeed, these systems possess finite Markov partitions and are, therefore, semi-conjugated to Markov shifts. See [Bo2] for an example of the wealth of results that can be obtained with this method. Following the 2013 work of Sarig [Sa2], countable Markov partitions have been constructed for a wide range of dynamical systems defined on compact spaces (see [Bu2, LM, LS, O]). The symbolic coding captures a relevant part, though not all, of the dynamics. We stress that this relevant part is often non-compact. For example, diffeomorphisms defined on compact manifolds have countable Markov partitions that capture all hyperbolic measures. These results have prompted a renewed interest in the ergodic properties of countable Markov shifts.

Date: March 1, 2019. 2010 Mathematics Subject Classiﬁcation. Primary 37A05; Secondary 37A35. We would like to thank Jerome Buzzi, Sylvain Crovisier, Omri Sarig and the referee for useful comments and remarks. G.I. was partially supported by CONICYT PIA ACT172001 and by Proyecto Fondecyt 1190194. 1 2 G. IOMMI, M. TODD, AND A. VELOZO

In this paper we study countable Markov shifts Σ,σ . More precisely, the shift on sequences of elements of a countable alphabet inp whichq the transitions, allowed and forbidden, are described by a directed graph (see Section 3 for definitions). The space Σis usually non-compact with respect to its natural topology. If the alphabet is finite, a so-called subshift of finite type, and so the space Σis compact, it is a classical result that the entropy map is upper semi-continuous [Wa, Theorem 8.2]. In this article we recover upper semi-continuity in the ergodic case for shifts of finite topological entropy defined on non-compact spaces. Our main result is: Theorem 1.1. Let Σ,σ be a two-sided countable Markov shift of finite topological p q entropy. If µn n,µ are ergodic measures in σ such that µn n converges in the weak* topologyp q to µ then M p q lim sup h µn h µ . n p qď p q Ñ8 That is, the entropy map is upper semi-continuous on the set of ergodic measures. This has the following corollary:

Corollary 1.2. Let Σ`,σ be a one-sided countable Markov shift of ﬁnite topo- p q logical entropy. If µn n,µ are ergodic measures in σ such that µn n converges in the weak* topologyp toq µ then M p q

lim sup h µn h µ . n p qď p q Ñ8 That is, the entropy map is upper semi-continuous on the set of ergodic measures. The strategy of the proof is the following. First note that if there exists a finite generating partition of the space such that the measure of its boundary is zero for every invariant measure, then the entropy map is upper semi-continuous. This is no longer true for countable generating partitions. Krieger [Kr] constructed a finite generating partition for each ergodic measure. More recently, Hochman [H1, H2] constructed finite partitions that are generating for every ergodic measure. We can not use the result by Krieger since we need the same partition for every measure and we can not use Hochman’s result since his partitions have a large boundary. We overcome this difficulty by constructing finite generating partitions with no boundary. If T : X X is a continuous transformation defined on a compact space for which the entropyÑ map is upper semi-continuous then there exists a measure of maximal entropy. Diffeomorphisms of class Cr, for any r 1, (in the compact setting), with no measure of maximal entropy have been constructedPr 8q by Misiurewicz [Mi1] and Buzzi [Bu1]. These are examples for which the entropy map is not upper semi-continuous. If the space X is non-compact, even if the entropy map is upper semi-continuous, measures of maximal entropy might not exist. Indeed, in the non-compact case the space of invariant probability measures might also be non-compact and therefore a sequence of measures with entropy converging to the topological entropy may not have a convergent subsequence. For instance, consider the geodesic flow on a non-compact pinched negatively curved manifold. Velozo [V] showed that in that setting the entropy map is always upper semi-continuous, but there are examples for which there is no measure of maximal entropy (see for example [DPPS]). In Section 4 we discuss conditions that guarantee the existence of measures of maximal entropy. We also address the relation between upper semi- continuity of the entropy map and thermodynamic formalism. More precisely, we UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 3 provide conditions for the existence of equilibrium measures. Finally, in Section 5 we apply our results to suspension flows. We would expect our results to pass to some systems which are coded by countable Markov shifts, but with the important caveat that it is not always straightfor- ward to compare topologies on the shift versus the coded space, so weak˚ convergence in one space may not imply weak˚ convergence in the other. 2. Entropy and generators This section is devoted to recall basic properties and definitions that will be used throughout the article. The reader is referred to [Do, P, Wa] for more details.

2.1. The weak˚ topology. Let X, d be a metric space, we denote by C X the p q bp q space bounded continuous functions ϕ : X R. Denote by X the set of Borel probability measures on the metric space ÑX, d . Mp q p q Deﬁnition 2.1. A sequence of probability measures µ deﬁned on a metric p nqn space X, d converges to a measure µ in the weak˚ topology if for every ϕ Cb X we havep q P p q

lim ϕdµn ϕdµ. n “ Ñ8 ż ż Remark 2.2. In this notion of convergence we can replace the set of test functions by the space of bounded Lipschitz functions (see [Kl, Theorem 13.16 (ii)]). That is, if for every bounded Lipschitz function ϕ : X R we have Ñ

lim ϕdµn ϕdµ. n “ Ñ8 ż ż then the sequence µ converges in the weak˚ topology to µ. p nqn If the space X, d is compact then so is X with respect to the weak˚ topol- p q Mp q ogy (see [Pa, Theorem 6.4]). For A X denote by int A, A and A the interior, the closure and the boundary of the setĂ A, respectively. The followingB result char- acterises the weak˚ convergence (see [Pa, Theorem 6.1]),

Theorem 2.3 (Portmanteau Theorem). Let X, d be a metric space and µn n,µ measures in X . The following statementsp areq equivalent: p q Mp q (a) The sequence µ converges in the weak˚ topology to the measure µ. p nqn (b) If C X is a closed set then lim supn µn C µ C . Ă Ñ8 p qď p q (c) If O X is an open set then lim infn µn O µ O . Ă Ñ8 p qě p q (d) If A X is a set such that µ A 0 then limn µn A µ A . Ă pB q“ Ñ8 p q“ p q Let T : X, d X, d be a continuous dynamical system, denote by T the space of T p invariantqÑp probabilityq measures. M ´ Proposition 2.4. Let T : X, d X, d be a continuous dynamical system de- ﬁned on a metric space, thenp qÑp q

(a) The space is closed in the weak˚ topology ([Wa, Theorem 6.10]). MT (b) If X is compact then so is T with respect to the weak˚ topology (see [Wa, Theorem 6.10]). M (c) The space T is a convex set for which its extreme points are the ergodic measures (seeM [Wa, Theorem 6.10]). It is actually a Choquet simplex (each measure is represented in a unique way as a generalized convex combination of the ergodic measures [Wa, p.153]). 4 G. IOMMI, M. TODD, AND A. VELOZO

2.2. Entropy of a dynamical system. Let T : X, d X, d be a continuous dynamical system. We recall the definition of entropyp qÑp of an invariantq measure µ (see [Wa, Chapter 4] for more details). P MT Definition 2.5. A partition of a probability space X, ,µ is a countable (finite or infinite) collection of pairwiseP disjoint subset of X whosep B unionq has full measure. Definition 2.6. The entropy of the partition is defined by P H : µ P log µ P , µ P p q “´P p q p q ÿPP where 0 log 0 : 0. “ It is possible that Hµ . Given two partitions and of X we define the new partition pPq“8 P Q : P Q : P ,Q P _ Q “t X P P P Qu 1 1 Let be a partition of X we define the partition T ´ : T ´ P : P and P n n 1 i P “ P P for n N we set : i ´0 T ´ . Recall that a sequence of real numbers P P “ “ P " ( an n is subadditive if for every n, m N we have an m an am. A classical resultp q by Fekete states thatŽ if a isP a subadditive sequence` ď of` non-negative real p nqn numbers then the sequence an n n converges to its infimum. Since the measure µ is T invariant, the sequencep H{ q n is subadditive. ´ µpP q Definition 2.7. The entropy of µ with respect to is defined by P 1 n hµ : lim Hµ n pPq “ Ñ8 n pP q Definition 2.8. The entropy of µ is defined by h T : sup h : a partition with H . µp q “ t µpPq P µpPqă8u If the underlying dynamical system considered is clear we write hµ instead of h T . µp q 2.3. Generators. We now recall the definition and properties of an important concept in ergodic theory, namely generators. Definition 2.9. Let T,X, ,µ be a dynamical system. A one-sided generating p B q n partition of T,X, ,µ is a partition such that n8 1 generates the sigma- algebra Pup top setsB of measureq zero. Analogously, if “T,X,P ,µ is an invertible dynamicalB system a two-sided generating partition Ť ofp T,X,B ,µq is a partition n P p B q such that n8 generates the sigma-algebra up to sets of measure zero. “´8 P B A classicalŤ result by Kolmogorov and Sinai [Si] states that entropy can be com- puted with either one- or two-sided generating partitions (see also [Wa, Theorem 4.17 and 4.18] or [Do, Theorem 4.2.2]). Theorem 2.10 (Kolmogorov-Sinai). Let T,X, ,µ be a dynamical system and a one-or a two-sided generating partition,p thenBh qT h µ, . P µp q“ p Pq The existence of generating partitions depends on the acting semi-group. Rohlin [Ro1, Ro2] proved that ergodic (actually aperiodic) invertible systems of finite entropy have countable two-sided generators. This was later improved by Krieger [Kr] (see also [Do, Theorem 4.2.3]). UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 5

Theorem 2.11 (Krieger). Let T,X, ,µ be an ergodic invertible dynamical sys- p B q tem with hµ T then there exists a finite two-sided generator for µ. The p qă8 h T P cardinality of can be chosen to be any integer larger than e µp q. P Note that in Krieger’s result the generator depends upon the measure. There are some well known cases in which there exists a finite partition which is a generator for every measure. For example, if Σ,σ is a transitive two-sided subshift of finite type then the cylinders of length onep formq a two-sided generating partition for every invariant measure. Hochman proved that a uniform version of Krieger result can be obtained (see [H2, Corollary 1.2]). Theorem 2.12 (Hochman). Let T,X, be an invertible dynamical system with no periodic points and of finite entropy.p ThenBq there exists a finite two-sided partition that is a generator for every ergodic invariant measure. P 2.4. Upper semi-continuity of the entropy. The following well known result (see for example [Do, Lemma 6.6.7]) relates the existence of finite generating partitions with continuity properties of the entropy map. Proposition 2.13. Let T,X, be a dynamical system and be a finite two- p Bq P sided generator for every measure in T . If for every µ we have that µ 0 then the entropy map is upperC Ă semi-continuousM in . P C pBPq“ C Proof. Let µ and P . Note that since µ 0 the function ν ν P is continuous atPµC. Since theP P partition is finite,pB thePq“ function defined in Ñby p q P C ν H ν P log ν P , ν P Ñ p q “ ´ P p q p q ÿPP is continuous at µ. Note that µ 0 implies that µ n 0. Thus, the pBnPq“ pBP q“ function in defined by ν Hν is also continuous. Since the function defined in by C Ñ pP q C 1 n ν hν inf Hν ÞÑ pPq“ n n pP q is the infimum of continuous functions at µ we have that the map ν hν is upper semi-continuous at µ. Since is a uniform generating partitionÞÑ inpPqfor every ν we have h T h .P Thus, the map C P C ν p q“ ν pPq ν h T ÞÑ ν p q is upper semi-continuous at µ. Since µ was arbitrary the result follows. ! P C

Remark 2.14. Note that if in Proposition 2.13 we have T then the entropy map is upper semi-continuous in the space of invariant probabilityC “ M measures. Remark 2.15. The argument in Proposition 2.13 breaks down if we consider countable (infinite) generating partitions since, in that case, the map ν Hν need not to be continuous (see Remark 3.11 or [JMU, p.774]). Ñ pPq Remark 2.16. Expansive maps defined on compact metric spaces are examples of dynamical systems having finite generators as in Proposition 2.13 (see [Wa, Theorem 8.2]). We stress that while Theorem 2.12 provides a finite (uniform) generating partition, in general this does not satisfy the condition of zero measure boundary. 6 G. IOMMI, M. TODD, AND A. VELOZO

3. Countable Markov shifts In this section we deﬁne countable Markov shifts, both one-and two-sided and prove our main results.

3.1. Two-sided countable Markov shifts. Let Σ,σ be a two-sided Markov shift defined over a countable alphabet . This meansp thatq there exists a matrix A S sij of zeros and ones such that “p qAÂ Z Σ x : sxixi 1 1 for every i Z . “ P A ` “ P Note that the matrix S"induces a directed graph on .( Let n N and r : n A P “ r1, . . . , rn , we say that r is an admissible word if sriri 1 1, for i p1, . . . , n qP1 .A We say that the system is transitive if, given x, y ` “, there existsP ant admissible´ u word starting at x and ending at y. Let r , . . . , r PbeA an admissible p 1 nq word and l Z, we define the corresponding cylinder set by P r1, ..., rn l : x Σ: xl r1,xl 1 r2, ..., xl n 1 rn . r s “t P “ ` “ ` ´ “ u If l 0 then we omit this in the notation, writing r1, ..., rn in place of r1, ..., rn 0. We“ endow Σwith the topology generated by ther cylinders sets. Noter that, withs respect to this topology, the space Σis non-compact. The shift map σ :Σ Σis Ñ defined by σ x i xi 1. Let σ be the set of σ invariant probability measures p p qq “ ` M ´ and σ σ the set of ergodic invariant probability measures. EachE Ă elementM a corresponds either to a transitive component of Σ, i.e., a P A maximal subset Σ1 of Σwhere there are x Σ1 with x a for some i, and which P i “ is transitive (note that Σ1,σ is also Markov); or, if not, to the wandering set. p q Since we will be dealing with ergodic measures µn converging to some µ, we will take a such that µ a 0, and we may assume that µn a 0 for all n. ErgodicityP A implies thatpr oursq measures ą are supported on the transitivepr sq ą component corresponding to a, and thus it is sufficient for our proofs to assume from here on that our system is transitive. The topological entropy of σ is defined by 1 htop σ : lim sup log χ a x r s p q “ n n σnx x p q Ñ8 ÿ“ sup h σ : µ sup h σ : µ , “ t µp q P Mσu“ t µp q P Eσu where a is an arbitrary symbol and χ a is the characteristic function of the A r s cylinder Pa . By transitivity this definition is independent of a . Gurevich formulatedr s this condition in [Gu1, Gu2] and proved that if Σ,σ PisA topologically mixing then in fact the limit exists (see also [DS, Remark 3.2]);p heq also proved the second and third equalities.1

3.2. Proof of Theorem 1.1. Given a we define P A k Σa : x Σ: σ x a for infinitely many positive and negative k Z . “t P Pr s P u Observe that Σa is a Borel σ-invariant subset of Σ. Therefore, the dynamical system σ :Σ Σ is well defined. Since Σ,σ is a finite entropy system, so is Σ ,σ . a Ñ a p q p a q

1If the system were not transitive, we could take the supremum of all these quantities over the transitive components. UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 7

Remark 3.1. Let µ be an ergodic σ-invariant probability measure such that P Eσ µ a 0, then by the Birkhoffergodic theorem we have µ Σa 1. This is thepr onlysq ą point in our proof where we use ergodicity of our measuresp q“ of interest. Moreover, since the system is transitive the set Σa is dense in Σ. The following class of countable Markov shifts that has been studied in [BBG, Ru1, Sa1] will be of importance in what follows. Definition 3.2. A loop graph is a graph made of simple loops which are based at a common vertex and otherwise do not intersect. A loop system is the two-sided countable Markov shift defined by a loop graph.

Lemma 3.3. The system Σa,σ is topologically conjugate to a loop system Σ,σ of ﬁnite entropy. p q p q

Proof. For every n N denote by Cn the set of non-empty cylinders in Σof the form ax ...x a , whereP x a for all i 1, . . . , n . Since the entropy of Σis finite, r 1 n s i ‰ Pt u the number of elements in Cn is finite. Let cn be the number of elements in Cn and write C A1 , ..., Acn . Construct a loop graph with exactly c loops of length n “t n n u n n 1, and denote by Σthe loop system associated to it. For convenience we denote the` vertex of the loop graph with the letter a. There is a one to one correspondence between Cn and the non-empty cylinders in Σof the form axa , where the word r s i x does not contain the letter a and it has length n. Denote by Bn to the cylinder i in Σassociated to An. Note that every x Σa is of the following form . . . ax 1ax0ax1a . . . , where P p ´ q xi are admissible words that do not contain the letter a. Observe that for each i i xi there is a unique corresponding cylinder Ani . Moreover, for each Ani there is i i a unique corresponding cylinder Bni in Σ. Finally, fo each Bni there is a unique b corresponding admissible word xi in Σ. Following this procedure we can define a b b b bijective map F :Σa Σby F . . . ax 1ax0ax1a ...... ax 1ax0 ax1 a . . . . Ñ p ´ q“p ´ q We will now prove that F is a homeomorphism. Let U be an open set in Σ. 1 Consider a point x U and define y F ´ x . To prove the continuity of F it P “ p q 1 b b b is enough to check that y is an interior point of F ´ U . Let ax ax a...ax a , p q r 1 2 m sh where h Z and no xi contains the letter a, be a cylinder contained in U such that x U. NoteP that P 1 b b b F ´ ax ax a...ax a ax ax a...ax a pr 1 2 m shq“r 1 2 m sh 1 Note that y ax ax a...ax a , ax ax a...ax a F ´ U and the cylinder set Pr 1 2 m sh r 1 2 m sh Ă ax1ax2a...axma h is open. Therefore, F is continuous. A similar argument gives r 1 s that F ´ is also continuous, therefore F is a homeomorphism. By construction we have that σ Σ F F σ Σa . Since Σa has finite entropy so does Σ. ! | ˝ “ ˝ | The following result was obtained by Boyle, Buzzi and Gómez[BBG, Lemma 3.7], they established the existence of a continuous embedding of a loop system into a subshift of finite type. Let us stress that the relevant part of the result is the continuity. Borel embeddings have been obtained in greater generality (see Hochman [H2, Corollary 1.2] or [H1, Theorem 1.5]). Theorem 3.4 (Boyle, Buzzi, Gómez). A loop system of finite topological entropy can be continuously embedded in an invertible subshift of finite type. 8 G. IOMMI, M. TODD, AND A. VELOZO

Lemma 3.5. Let A and B be disjoint open subsets of Σa. Suppose A1 and B1 are open subsets of Σ such that A1 Σa A and B1 Σa B. Then A1 and B1 are disjoint. X “ X “

Proof. Suppose that A1 and B1 are not disjoint. In this case we can ﬁnd a non- empty open set U A1 B1. Deﬁne V U Σa and observe that V A B. Since Σ is dense inĂ ΣweX have that V “U XΣ is non-empty, which contradictsĂ X a “ X a that A and B are disjoint. !

Given an open subset A of Σa we define A as the largest open subset in Σsuch that A Σ A. Similarly, for a closed subset B of Σ we define B as the smallest X a “ a closed subset of Σsuch that B Σa B. Thep existence of both A and B follows fromp Zorn’s Lemma. X “ q q p q Remark 3.6. If B Σ is closed set, then there exists a closed set B Σsuch Ă a 1 Ă that B1 Σa B. Since B B1, we conclude that B B1, where B is the closure ofXB in“ Σ. This impliesĂ that B B Σ B ΣĂ B, and therefore Ă X a Ă 1 X a “ B Σ B. Moreover B B. X a “ “ Lemma 3.7. Let A be an open and closed subset of Σ . Then A A, in particular q a Ă A A. “ p q Proof. Let B : Σa ! A. Observe that Σ ! A is open and that Σ ! A Σa B. p “ p qX “ By the definition of B it follows that Σ ! A B, or equivalently that Σ ! B A. p qĂ Ă Observe that A and B are disjoint open subsetsq of Σa, therefore we canq use Lemma 3.5 and obtain that Ap Σ ! B . All thisq togetherp implies that A A. p q! Ăp q Ă Lemma 3.8. Supposep that p R1, ..., RN is a partition of Σa psuchq that the sets R , with i 1,...,N , are openR “t and closed inu the topology of Σ . Then there exists i Pt u a a finite partition of Σ which induces the partition on Σa and µ 0 for every probability measureR on Σ such that µ Σ 1. R pBRq“ p aq“ p p We observe that there is an important step in the proof of this lemma which uses the density of Σa inΣ, which as in Remark 3.1 follows from topological transitivity.

N Proof. Let R1, ..., RN ,X , where X Σ ! i 1 Ri. The partition is well R “t u “ “ deﬁned since by Lemma 3.5 the sets R , ..., R are disjoint. Observe that the set 1 N Ť X is closedp and hasx emptyy interior (sincet Σ isu dense inx Σ). Therefore µ X a pB q“ µ X ! int X µ X µ Σ ! Σa x0. It followsy from Lemma 3.7 that µ Ri p q“ p qď p q“ pB q“ µ Ri ! Ri µ Ri ! Ri . As observed in Remark 3.6 we have that Ri Σa Ri. Thereforep q“ p q X x“ x x x µ Ri µ Ri ! Ri µ Ri Σa ! Ri µ Ri ! Ri 0. pB q“ p q“ pp X q q“ p q“ x x x x ! Proposition 3.9. Let µ be a sequence of invariant probability measures con- p nqn verging in the weak˚ topology to a measure µ. If µ Σ 1 and µ Σ 1, for p aq“ np aq“ every n N, then P lim sup hµn σ hµ σ . n p qď p q Ñ8 UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 9

Proof. It was shown in Lemma 3.3 that there exists a topological conjugacy F : Σ Σbetween Σ ,σ and Σ,σ . From Theorem 3.4 there exists a topologi- a Ñ p a q p q cal embedding G : Σ ã Σ0, where Σ0 is a subshift of ﬁnite type with alphabet Ñ 1, ..., N . The partition 1 , 2 , ..., N is a generating partition of Σ0,σ fort everyu invariant probabilityP “ tr measures r s (seer su [Wa, Theorem 8.2]). It followsp thatq 1 G´ is a generating partition for every σ-invariant measure in Σ. The Q “ P 1 continuity of G implies that G´ i is open and closed, in particular pr sqq 1 1 1 G´ i G´ i ! int G´ i . B pr sq “ pr sq pr sq “ H 1 Therefore . Let F ´ . Then is a generating partition for Σa,σ , andBQ moreover“H theR “ elementsQ in areR open and closed. From Lemmap 3.8q BR “H R we construct a partition of Σthat induces when restricted to Σa. Since by assumption the measuresRµ and µ give fullR measure to Σ , Lemma 3.8 implies p nqn a that µ 0. Moreoverp h σ h σ, and h σ h σ, for every pBRq“ µp q“ µp Rq µn p q“ µn p Rq n N. Indeed, for every σ-invariant probability measure ν on Σsuch that ν Σa 1 P p q“ we have thatp the systems Σ,σ,ν and Σ ,σp ,ν are isomorphic.p Therefore p q p a |Σa |Σa q 1 n hν Σ,σ hν Σa,σ lim Hν , n p q“ p q“ Ñ8 n pR q since is a generating partition for Σa,σ . Note that for every n N and R p q P n we have Ri1,...,ın P R ν ν Σ ν . p p pRi1,...,ın q“ pRi1,...,ın X aq“ |Σa pRi1,...,ın q Thus h σ h σ, . Therefore, Proposition 2.13 implies that ν p q“ ν p pRq p

lim sup hµn σ lim sup hµn σ, hµ σ, hµ σ . n p p q“ n p Rqď p Rq“ p q Ñ8 Ñ8 p p ! Proof of Theorem 1.1. The above argument proves that if µ is a sequence of p nqn ergodic measures that converges in the weak˚ topology to the ergodic measure µ then limn hµ σ hµ σ . Indeed, there exists a such that µ a 0. Ñ8 n p qď p q P A pr sq ą Since limn µn a µ a 0, there exists N N such that for every n N Ñ8 pr sq “ pr sq ą P ą we have that µn a 0. We then use Proposition 3.9 to get the result. ! pr sq ą 3.3. One-sided countable Markov shifts. Let Σ`,σ be a one-sided Markov shift defined over a countable alphabet . This meansp thatq there exists a matrix A S sij of zeros and ones such that “p qAÂ N Σ` x : sxixi 1 1 for every i N . “ P A ` “ P The shift map σ :Σ` " Σ` is defined by σ x i xi 1.( The entropy of σ is Ñ p p qq “ ` defined by htop htop σ : sup hµ σ : µ σ . Note that, as in the two-sided setting, it is possible“ top q give“ a definitiont p q ofP E entropyu computing the exponential growth of periodic orbits, but for the purposes of this article the above definition suffices.

Proof of Corollary 1.2. Denote by Σ,σ the natural extension of Σ`,σ and by p q p q and ` the corresponding sets of ergodic measures. There exists a bijection E E π : ` such that for every µ ` we have that hµ hπ µ (see [Do, Fact E E E p q 4.3.2] andÑ [Sa3, Section 2.3]). P “ 10 G. IOMMI, M. TODD, AND A. VELOZO

Lemma 3.10. Let µ ,µ ` such that µ converges weak˚ to µ. Then p nqn P E p nqn π µ converges weak˚ to π µ . p p nqqn p q Proof. Note that in the notion of weak˚ convergence we can replace the set of test functions by the space of bounded Lipschitz functions (see Remark 2.2). That is, if for every bounded Lipschitz function ϕ :Σ R we have Ñ

lim ϕdµn ϕdµ, n “ Ñ8 ż ż then the sequence µn n converges in the weak˚ topology to µ. A result by Daon [Da, Theorem 3.1] impliesp q that for every Lipschitz (the result also holds for weakly H¨older and summable variations) function ϕ :Σ R there exists a cohomologous Ñ Lipschitz function ψ :Σ R that depends only on future coordinates. The transfer function can be chosen boundedÑ and uniformly continuous. The function ψ can be canonically identiﬁed with a Lipschitz function ρ :Σ` R. Thus, Ñ ϕdπ µ ψdπ µ ρdµ . p nq“ p nq“ n żΣ żΣ żΣ` Therefore,

lim ϕdπ µn lim ρdµn ρdµ ϕdπ µn . n p q“n “ “ p q Ñ8 żΣ Ñ8 żΣ` żΣ` żΣ The result now follows. !

Let µ ,µ be such that µ converges weak˚ to µ. Lemma 3.10 implies n P Eσ p nqn that π µn n converges weak˚ to π µ . Moreover, for every n N we have that p p qq p q P hπ µ hµn and hπ µ hµ. Since, by Theorem 1.1, the entropy map is upper p nq “ p q “ semi-continuous in Σ,σ we obtain the result. ! p q Remark 3.11. The finite entropy assumption in Theorem 1.1 and in Corollary 1.2 is essential as the following example shows. Let Σ`,σ be the full shift on a countable alphabet, note that h σ . Denote by p theq partition formed by the length topp q “ 8 P one cylinders. This is a generating partition. Let h R` be a positive real number and a be the sequence defined by a h forP every n 1. Consider the p nqn n “ log n ą following stochastic vector a a a p( : 1 a , n , n ,..., n , 0, 0,... , n “ ´ n n n n where the term a n appears´ n times. Let µ be the Bernoulli¯ measure defined n{ n by p( . Note that the sequence µ converges in the weak˚ topology to a Dirac n p nqn measure δ1 supported on the fixed point at the cylinder C1. Note that 1 H 1 a log 1 a a log a a log . µn pPq “ ´p ´ nq p ´ nq´ n n ´ n n Therefore,

lim Hµ h 0 Hδ . n n 1 Ñ8 pPq“ ą “ pPq The above example shows that the map ν H need not to be continuous Ñ ν pPq for countable generating partitions. Moreover, since the measures µn are Bernoulli we have that hµn σ Hµn , and therefore the above argument shows that the entropy map is notp q“ upper semi-continuous:pPq

lim hµ σ h 0 hδ σ . n n 1 Ñ8 p q“ ą “ p q UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 11

In other words, the entropy map could fail to be upper semi-continuous for general dynamical systems deﬁned on non-compact spaces, even if there exists a uniform (countable) generating partition with no boundary. The construction of the above example is based on examples constructed by Walters ([Wa, p.184]) and by Jenkin- son, Mauldin and Urba´nski [JMU, p.774]. This example also shows that the entropy map is not upper semi-continuous even if we consider a sequence of measures for which their entropy is uniformly bounded.

4. Measures of maximal entropy A continuous map T : X, d X, d deﬁned on a compact metric space for which the entropy map is upperp q semi-continuous Ñ p q has a measure of maximal entropy. Indeed, from the variational principle there exists a sequence of ergodic invariant probability measures µn n such that limn hµn T htop T . Since the space of invariant measures p q is compact, thereÑ8 existsp anq“ invariantp measureq µ which is MT an accumulation point for µn n. It follows from the fact that the entropy map is upper semi-continuous thatp q

htop T lim hµ T hµ T . n n p q“ Ñ8 p qď p q Therefore, µ is a measure of maximal entropy. For countable Markov shifts the variational principle holds (see [Gu1, Gu2]) and the entropy map is upper semi- continuous (see Theorem 1.1 and Corollary 1.2), however the space σ is no longer compact. Despite this, under a convergence assumption we can proveM the existence of measures of maximal entropy. Indeed, Corollary 1.2 provides a new proof of the following result by Gurevich and Savchenko [GS, Theorem 6.3]. Proposition 4.1. Let Σ,σ be a finite entropy countable Markov shift. Let µ p q p nqn be a sequence of ergodic measures such that limn hµ σ htop σ . If µn n Ñ8 n p q“ p q p q converges in the weak˚ topology to an ergodic measure µ then h σ h σ . µp q“ topp q It might happen that there is a sequence µn n with limn hµ σ htop σ , p q Ñ8 n p q“ p q but µn n does not converge in the weak˚ topology. Examples of countable Markov shiftsp withq this property have been known for a long time. In [Ru2] a simple construction of a Markov shift of any given entropy with no measure of maximal entropy is provided. In [Gu3, GN] examples are constructed of finite entropy countable Markov shifts having a measure of maximal entropy µmax and sequences of ergodic measures µn n, νn n with limn hµ σ limn hν σ htop σ p q p q Ñ8 n p q“ Ñ8 n p q“ p q such that µn n converges in the weak˚ topology to µmax and νn n does not have any accumulationp q point. p q The continuity properties of the entropy map also have consequences in the study of thermodynamic formalism. Let Σ`,σ be a transitive one-sided countable p q Markov shift of finite entropy and ϕ :Σ` R a continuous bounded function of Ñ summable variations. That is n8 1 varn ϕ , where “ p qă8 var ϕ : sup ϕ x ϕ y : x y ,i 1, . . . , n . np q “ třp q´ p q i “ i Pt uu Sarig (see [Sa3] for a survey on the topic) defined a notion of pressure in this context, the so called Gurevich pressure, that we denote by P ϕ . He proved the following variational principle p q

P ϕ sup h σ ϕdµ : µ sup h σ ϕdµ : µ p q“ µp q` P Mσ “ µp q` P Eσ " ż * " ż * 12 G. IOMMI, M. TODD, AND A. VELOZO

A measure µ σ such that P ϕ hµ σ ϕdµ is called equilibrium measure. It directly followsP M from Corollaryp 1.2q“thatp q` ş Proposition 4.2. Let Σ`,σ be a ﬁnite entropy countable Markov shift and ϕ : p q Σ R a bounded function of summable variations. Let µn n be a sequence of Ñ p q ergodic measures such that limn hµ σ ϕdµn P ϕ . If µn n converges Ñ8 n p q` “ p q p q in the weak˚ topology to an ergodic measure µ then P ϕ h σ ϕdµ. ` ş ˘p q“ µp q` Note that the thermodynamic formalism of a two-sided countableş Markov shift can be reduced to the one-sided case (see [Sa3, Section 2.3]).

5. Suspension flows

Let Σ,σ be a finite entropy countable Markov shift and let τ :Σ R` be a locallyp q Hölder potential bounded away from zero. Consider the spaceÑY “ x, t Σ R:0 t τ x , with the points x,τ x and σ x , 0 identified for eachtp qPx Σˆ. Theďsuspensionď p qu flow over Σwithp roofp qq functionp p τq isq the semi-flow P Φ ϕt t R on Y defined by ϕt x, s x, s t whenever s t 0,τ x . Denote by“p qtheP space of flow invariantp probabilityq “ p ` measures.q Let` Pr p qs MΦ τ : µ : τdµ . (5.1) Mσp q “ P Mσ ă8 " ż * It follows directly from results by Ambrose and Kakutani [AK] that the map R: , defined by Mσ Ñ MΦ µ Leb R µ p ˆ q|Y , p q“ µ Leb Y p ˆ qp q where Leb is the one-dimensional Lebesgue measure, is a bijection. Denote by EΦ the set of ergodic flow invariant measures. Let F : Y R be a continuous function. Ñ Define ∆F :Σ R by Ñ τ x p q ∆ x : F x, t dt. F p q “ p q ż0 Kac’s Lemma states that if ν is an invariant measure that can be written as P MΦ µ Leb ν ˆ , “ µ Leb Y p ˆ qp q where µ , then P Mσ ∆F dµ F dν Σ . “ τdµ żY ş Σ The following Lemma describes the relationş between weak˚ convergence in Φ with that in . M Mσ Lemma 5.1. Let ν ,ν be flow invariant probability measures such that p nq P MΦ µn Leb µn Leb νn ˆ and ν ˆ “ τdµn “ τdµn where µn n,µ σ are shiftş invariant probabilityş measures. If sequence νn n p q P M p q converges in the weak˚ topology to ν then

µn n converges in the weak˚ topology to µ and lim τdµn τdµ. p q n “ Ñ8 ż ż UPPER SEMI-CONTINUITY OF ENTROPY IN NON-COMPACT SETTINGS 13

Proof. Assume ﬁrst that νn n converges in the weak˚ topology to ν. Let f :Σ R be a bounded continuousp function.q Following Barreira, Radu and Wolf [BRW] thereÑ exists a continuous function F : Y R such that Ñ τ x p q f x ∆ x : F x, t dt. (5.2) p q“ F p q “ p q ż0 Indeed, deﬁne F x, t : Y R by p q Ñ f x t F x, t : p qψ1 , p q “ τ x τ x p q ˆ p q˙ 1 where ψ : 0, 1 0, 1 is a C function such that ψ 0 0, ψ 1 1 and ψ1 0 r s Ñ r s p q“ p q“ p q“ ψ1 1 0. Note that since τ is bounded away from zero, F x, t is continuous and bounded.p q“ Therefore p q

lim F dνn F dν. n “ Ñ8 ż ż By Kac’s Lemma we have that ∆ dµ ∆ dµ lim F n F . (5.3) n τdµ “ τdµ Ñ8 ş n ş In particular if f 1 is the constant function equal to one, we obtain “ ş ş τdµ lim n 1. (5.4) n τdµ “ Ñ8 ş Let f :Σ R be a bounded continuousş function, then it follows from equations (5.2), (5.3)Ñ and (5.4) that

lim f dµn f dµ. n “ Ñ8 ż ż Therefore µn converges weak˚ to µ. ! p q Proposition 5.2. If νn n is a sequence of ergodic measures in Φ converging to an ergodic measure ν,p thenq E

lim sup hνn Φ hν Φ . n p qď p q Ñ8 Proof. Let ν be a sequence in that converges in the weak˚ topology to p nqn EΦ ν Φ. Note that there exists a sequence of ergodic measures µn n σ and µ P E ergodic such that p q P M P Mσ µ Leb µ Leb ν n ˆ and ν ˆ . n “ µ Leb Y “ µ Leb Y p n ˆ qp q p n ˆ qp q µ1 Leb By Abramov’s formula [Ab], for any ν1 Φ, with ν1 µ Lebˆ Y we have that P M “ p 1ˆ qp q

hµ1 σ hν1 Φ p q . p q“ τdµ1

Recall that by Lemma 5.1 we have that νnş ν in the weak˚ topology implies that Ñ µn µ in the weak˚ topology and limn τdµn τdµ. Since htop σ by TheoremÑ 1.1 or Corollary 1.2 we have thatÑ8 “ p qă8 ş ş

lim sup hµn σ hµ σ . n p qď p q Ñ8 14 G. IOMMI, M. TODD, AND A. VELOZO

Therefore,

hµn σ lim supn hµn σ hµ σ lim sup hνn Φ lim sup p q Ñ8 p q p q hν Φ . n p q“ n τdµn “ τdµ ď τdµ “ p q Ñ8 Ñ8 ş ş ş !

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Facultad de Matematicas,´ Pontificia Universidad Catolica´ de Chile (PUC), Avenida Vicuna˜ Mackenna 4860, Santiago, Chile E-mail address: [email protected] URL: http://www.mat.uc.cl/~giommi/

Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland E-mail address: [email protected] URL: http://www.mcs.st-and.ac.uk/~miket/

Department of Mathematics, Yale University, New Haven, CT 06511, USA. E-mail address: [email protected] URL: https://gauss.math.yale.edu/~av578/