The structure of the space of ergodic measures of transitive partially hyperbolic sets
Lorenzo J. Díaz, Katrin Gelfert, Tiane Marcarini, Michał Rams
TTHE STRUCTURE OF THE SPACE OF ERGODIC MEASURESOF TRANSITIVE PARTIALLY HYPERBOLIC SETS
L. J. D´IAZ, K. GELFERT, T. MARCARINI, AND M. RAMS
Abstract.
We provide examples of transitive partially hyperbolic dynamics(specific but paradigmatic examples of homoclinic classes) which blend dif-ferent types of hyperbolicity in the one-dimensional center direction. Thesehomoclinic classes have two disjoint parts: an “exposed” piece which is poorlyhomoclinically related with the rest and a “core” with rich homoclinic rela-tions. There is an associated natural division of the space of ergodic measureswhich are either supported on the exposed piece or on the core. We describethe topology of these two parts and show that they glue along nonhyperbolicmeasures.Measures of maximal entropy are discussed in more detail. We presentexamples where the measure of maximal entropy is nonhyperbolic. We alsopresent examples where the measure of maximal entropy is unique and non-hyperbolic, however in this case the dynamics is nontransitive. Introduction
An important task in ergodic theory is to describe the topology of the spaceof invariant and/or ergodic measures which are supported on a given invariantset. Here in many cases the weak ∗ topology is considered, though one also studiesconvergence in the weak ∗ topology and entropy. Recently there happened a certainrevival of this type of problems in the context of nonhyperbolic dynamical systems[15, 16, 11, 2], most of them revisiting the pioneering work of Sigmund on topologicaldynamical systems satisfying the specification property [26, 27].For a general continuous map F on a metric space Λ, consider the set of F -invariant Borel probability measures M (Λ) and denote by M erg (Λ) the subset ofergodic ones. If Λ is compact then M = M (Λ) is a Choquet simplex whose extremalelements are the ergodic measures. Density of ergodic measures in M implies thateither M is a singleton (when F is uniquely ergodic) or a nontrivial simplex whoseextreme points are dense. In the latter case, it is the so-called Poulsen simplex and by [22] has immediately a number of further strong properties such as arcwiseconnectedness. Sigmund [26, 27] addressed first the questions on the density ofergodic measures and also the properties of generic invariant measures. He showedthat for a map F satisfying the so-called periodic specification property the periodicmeasures (and thus the ergodic ones) are dense in M . Here a measure is periodic if it is the invariant probability measure supported on a periodic orbit. Moreover, Mathematics Subject Classification.
Key words and phrases. ergodic measure, heterodimensional cycle, homoclinic class and rela-tion, Lyapunov exponents, partially hyperbolic dynamics, skew product, transitive.This research has been supported [in part] by CNE-Faperj, CNPq-grants (Brazil), and NationalScience Centre grant 2014/13/B/ST1/01033 (Poland). The authors acknowledge the hospitalityof IMPAN, IM-UFRJ, and PUC-Rio. a r X i v : . [ m a t h . D S ] M a y L. J. D´IAZ, K. GELFERT, T. MARCARINI, AND M. RAMS the sets of ergodic measures and of measures with entropy zero are both residualin M . For an updated discussion and more references, see [15].Observe that Sigmund’s results [26, 27] immediately apply to any basic set ofa smooth Axiom A diffeomorphism. In a (more) general context, to address thegeneral question if the space M has dense extreme points or at least is connected,some natural requirements are to be satisfied. An important one is certainly topo-logical transitivity, which is however far from being sufficient as for example thereexist minimal systems with exactly two ergodic measures.Nowadays arguments which provide the connectedness of M are largely based onthe approximation of invariant measures by periodic measures or Markov ergodicmeasures supported on horseshoes (a specific type of basic set). This demandsthat the periodic orbits involved are hyperbolic and somehow dynamically relatedamong themselves. A natural relation introduced by Newhouse [23], and used inthis context, is the homoclinic relation , that is, the un-/stable invariant sets of theseorbits intersect cyclically and transversally.A natural strategy is to study the components of the space of measures whicheach are candidate to correspond to one of the “elementary” undecomposable piecesof the dynamics. One of the possibilities to define properly what is meant byelementary is the homoclinic class , that is, the closure of the hyperbolic periodicorbits which are homoclinically related to the orbit of a hyperbolic periodic point P and denoted by H ( P ). Note that one of the fundamental properties is thatthe dynamics on each class is topologically transitive. Basic sets of the hyperbolictheory mentioned above are the simplest examples of homoclinic classes.Notice that, when defining a homoclinic class, taking the closure can incorporateother orbits which are dynamically related but which are of different type of hyper-bolicity. In this way, homoclinic classes may fail to be hyperbolic, contain saddlesof different types of hyperbolicity (different u-index, that is, dimension of unstablemanifold), exhibit internal cycles, and support nonhyperbolic measures (also withpositive entropy). Homoclinic classes of periodic points of different indices mayeven coincide. Furthermore, there are examples where a homoclinic class H ( P ) ofa periodic point P properly contains another class H ( P (cid:48) ) of a periodic point P (cid:48) of the same index as P . Note that this precisely occurs if P (cid:48) ∈ H ( P ) was nothomoclinically related to P . One sometimes refers to H ( P (cid:48) ) as an exposed piece of H ( P ) [9]. This type of phenomenon is a key ingredient in this paper. This givesonly a rough idea what complicated structure these classes may have, see also [3,Chapter 10.4] for a more complete discussion.To be more precise for the following, we say that an ergodic measure µ is hyper-bolic if its Lyapunov exponents are nonzero. Moreover, almost all points have thesame number u = u ( µ ) of positive Lyapunov exponents and we call this number u the u -index of µ (analogously to hyperbolic periodic measures above). Given u , wedenote by denote by M erg ,u the set of ergodic measures of u-index u . Note that ingeneral one may have M erg ,u ( H ( P )) (cid:54) = ∅ for several values of u .For the following let us study the topological structure of M erg ,u ( H ( P )) for u be-ing the index of P . Assuming that H ( P ) is locally maximal and that all the saddlesof index u are homoclinically related, in [16] it is shown that M erg ,u ( H ( P )) is pathconnected with periodic measures being dense and that its closure is a Poulsen sim-plex. Note that M erg ,u ( H ( P )) may only capture some part of M erg ( H ( P )). Indeedthis occurs when H ( P ) contains saddles of different indices. Still in this context, TRUCTURE OF THE SPACE OF ERGODIC MEASURES 3 assume now that there coexists a saddle Q of index v (cid:54) = u and having the propertythat H ( Q ) ⊂ H ( P ) (in an extreme case, these classes can even coincide as sets) andassume that all the saddles of index v in H ( Q ) are homoclinically related with Q and consider M erg ,v ( H ( Q )). Though the interrelation between M erg ,u ( H ( P )) and M erg ,v ( H ( Q )) is not addressed in [16], note that, by the very definition, they aredisjoint. Nevertheless, their closures may intersect or may not. Indeed, the space M erg ( H ( P )) may be connected or may not. To address this point is precisely thegoal of this paper.We introduce a class of examples of saddles P and Q of different indices whosehomoclinic classes coincide H ( P ) = H ( Q ) = Λ such that Λ is the disjoint union oftwo invariant sets Λ ex (a compact set that is a topological horseshoe) and Λ core .Moreover, these sets satisfy the following properties: (i) P, Q ∈ Λ core and the closureof Λ core is the whole homoclinic class, (ii) every pair of saddles of the same indexin Λ core (respectively, Λ ex ) are homoclinically related, and (iii) no saddle in Λ core is homoclinically related to any one in Λ ex . We refer to Λ ex as the exposed piece ofΛ = H ( P ) = H ( Q ) and to Λ core as its core . We study the space M erg (Λ) and showthat it has an interesting topological structure: the set M erg (Λ) has three pairwisedisjoint parts M erg ,u (Λ), M erg ,v (Λ), v = u + 1 and u, v are the indices of P and Q ,and M erg (Λ ex ), such that M erg (Λ) = M erg ,u (Λ) ∪ M erg ,u (Λ) ∪ M erg (Λ ex ) ∪ M erg , nhyp (Λ)where M erg , nhyp (Λ) is the set of of nonhyperbolic ergodic measures of Λ. Note that M erg (Λ ex ) and M erg , nhyp (Λ) may intersect. Moreover, the sets closure( M erg ,u (Λ)),closure( M erg ,v (Λ)), and closure( M erg (Λ ex )), are Poulsen simplices whose intersec-tion is contained in M erg , nhyp (Λ), see Theorem 2.5. Figure 1 below illustrates theinterrelation between the measure space components. M erg ,u (Λ) M erg ,u +1 (Λ) M erg , nhyp (Λ) M erg (Λ ex ) δ P ex δ Q ex Figure 1.
The space M erg (Λ)Let us say a few additional words about the topological structure of the setΛ = H ( P ) = H ( Q ). There are two exposed saddles P ex , Q ex ∈ Λ ex of the sameindices such as P and Q , respectively, which are involved in a heterodimensionalcycle (i.e., the invariant sets of these saddles meet cyclically), Indeed, the intersec-tions of these invariant sets give rise to the exposed piece of dynamics that satisfyΛ ex = H ( P ex ) = H ( Q ex ) (cid:40) Λ. We are aware that on one hand this is a quite spe-cific dynamical configuration, on the other hand it provides paradigmatic examples.We also observe that this dynamical configuration resembles in some aspects theso-called Bowen eye (a two dimensional vector field having two saddle singularities
L. J. D´IAZ, K. GELFERT, T. MARCARINI, AND M. RAMS involved in a double saddle connection) in [14, 29] and the examples due to Kan ofintermingled basins of attractions (where an important property is that the bound-ary of an annulus is preserved) [18]. Finally, if we considered systems satisfyingsome boundary conditions or preserving a boundary, the conditions considered arequite general.A particular emphasize is given to the measures of maximal entropy. In somecases, In some cases, these measures can be nonhyperbolic. We give a (non-transitive) example where the unique measure of maximal entropy is nonhyperbolic.Finally, we state of results for step skew products (these examples have dif-ferentiable realizations as partially hyperbolic sets with one dimensional centraldirection) and throughout the paper we do not aim generality, on the contraryour goal is to make the construction in the simplest setting emphasizing the keyingredients behind the constructions.This paper is organized as follows. In Section 2 we state precisely our settingand our examples and state our main results. In Section 3 we study the “symme-tries” between certain measures and investigate entropy. In Section 4 we study theapproximation of “boundary measures”. In Section 5 we study the measures sup-ported in Λ core . In Appendix A we provide details on transitivity and homoclinicrelations in our examples and we analyze examples with nonhyperbolic measuresof maximal entropy.2.
Setting and statement of results
We now define precisely the dynamics that we will study. Consider C diffeo-morphisms f , f : [0 , → [0 ,
1] satisfying the following properties:(H1) The map f has (exactly) two fixed points f (0) = 0 and f (1) = 1, satisfies f (cid:48) (0) = β > f (cid:48) (1) = λ ∈ (0 , f has negative derivative and satisfies f (0) = 1 and f (1) = 0.The simplest (and also paradigmatic) example occurs when f ( x ) = 1 − x . f f Figure 2.
Fiber maps in (2.1)Let σ : Σ → Σ be the standard shift map on the shift space Σ = { , } Z oftwo-sided sequences, endowed with the usual metric. Consider the one step-skewproduct map F associated to σ and the maps f and f defined by(2.1) F : Σ × [0 , → Σ × [0 , , ( ξ, x ) (cid:55)→ (cid:0) σ ( ξ ) , f ξ ( x ) (cid:1) . TRUCTURE OF THE SPACE OF ERGODIC MEASURES 5
We consider the following F -invariant subsets of Λ def = Σ × [0 , ex def = Σ × { , } , Λ core def = (Σ × [0 , \ Λ exc = Σ × (0 , . We say that Λ ex is the exposed piece of Σ × [0 ,
1] and that Λ core is the core ofΣ × [0 ,
1] (these denominations are justified below). Note that Λ ex is a closedwhile Λ core is not. Moreover, F | Λ ex is topologically transitive. In fact, F | Λ ex isconjugate to a subshift of finite type, one may think this dynamical system asa horseshoe in a “plane”, in that plane any pair of saddles are “homoclinicallyrelated”. Remark 2.1 (Topological dynamics on Σ × [0 , . While the dynamics in Λ ex iscompletely characterized, in our quite general setting very few can be said about thedynamics of F in Λ core . The most interesting case certainly occurs when F | Λ core is topologically transitive. Below we will see more specific examples where thistransitivity indeed holds and, moreover, hyperbolic periodic orbits of positive andnegative Lyapunov exponent are both dense in Σ × [0 ,
1] and homoclinically related.We will see that nevertheless the measure space M (Λ ex ) is “semi-detached” from M (Λ core ).Consider now more specific hypotheses on the C diffeomorphisms interval maps f , f : [0 , → [0 , f ( x ) = 1 − x .(H3) The derivative f (cid:48) is decreasing. Considering the point c ∈ (0 ,
1) defined bythe condition f (cid:48) ( c ) = 1, it holds f ◦ f ( c ) > f ( c ) . (H4) The numbers λ and β given in (H1) satisfy κ def = λ (1 − λ ) β ( β − > . Observe that for fixed λ , the inequality in (H4) holds whenever β is close enoughto 1. Proposition 2.2.
Assume that F defined in (2.1) satisfies the hypotheses (H1),(H2’), (H3), and (H4). Then F is topologically transitive. Moreover, every pair offiber expanding hyperbolic periodic orbits and every pair of fiber contracting hyper-bolic periodic orbits in Λ core are homoclinically related, respectively. Remark 2.3 (Discussion of hypotheses) . Homoclinic relations for skew productsare recalled in Appendix A, where also the above proposition is proved. Condition(H4) will provide so-called expanding itineraries which in turn imply the homoclinicrelations and their density for expanding points, while condition (H3) takes care ofso-called contracting itineraries and the corresponding homoclinic relations. Thus,we conclude transitivity. The proof follows largely blender-like standard argumentsused in [7]. Condition (H2’) is only used for simplicity and also to follow more closelythe model in [13]. The key facts remain true assuming only (H2), in particular wenever use the fact that for (H2’) the map f is an involution.We observe that (H3) and (H4) demand a certain “asymmetry” of the fiber map f . In Section A.4 we will provide a “symmetric” example which satisfies (H1) andfor which the associated skew product fails to be transitive and its only measure ofmaximal entropy is nonhyperbolic and supported on Λ ex . Remark 2.4 (Examples in Σ × S and Σ × S ) . We can produce a transitiveexample in Σ × S with properties analogous to the one in Proposition 2.2 as follows. L. J. D´IAZ, K. GELFERT, T. MARCARINI, AND M. RAMS
Obtain S by identifying the boundary points of [0 , g , g , g : S → S as follows • g ( x ) = f ( x ) if x ∈ [0 ,
1] and g ( x ) = f ( x −
1) if x ∈ [0 , • g ( x ) = f ( x ) = 1 − x if x ∈ [0 ,
1] and g ( x ) = 3 − x if x ∈ [0 , • g ( x ) = x + 1 mod 2 (or any appropriate map preserving { , } and inter-changing the interior of the intervals (0 ,
1) and (1 , ex = Σ × { , } and Λ core =Σ × ((0 , ∪ (1 , { g , g , g } does not satisfy the axiomsstated in [10] which would prevent the existence of exposed pieces of dynamics. Al-though the Axioms Transitivity and CEC (controlled expanding forward/backwardcovering) can be verified, the Axiom Accessibility is not satisfied (the points { , , } cannot “be reached from outside”).Note that the skew product on Σ × S generated by the fiber maps { g , g } as above is not transitive and has two open “transitive” components Λ − core andΛ +core contained in Σ × (0 ,
1) and Σ × (1 , × { , } . The additional map g in the previous example justmixes the two components Λ ± core while preserving the exposed piece. g g g g Figure 3.
Fiber maps of the example in Remark 2.4Let M be the space of all F -invariant measures and equip it with the weak ∗ topology. It is well known that it is a compact metrizable topological space [30,Chapter 6.1]. Denote by M erg = M erg (Σ × [0 , M erg (Λ ex ) the ergodic measures supported on Λ ex and by M erg (Λ core )the ergodic measures supported on Λ core . Observe that M erg = M erg (Λ core ) ∪ M erg (Λ ex ) . We will study this system by separately looking at measures supported on thesetwo sets. A crucial point for us is how these two components “glue”.Given X = ( ξ, x ) ∈ Σ k × [0 , (fiber) Lyapunov exponent of themap F at X which is defined by χ ( X ) def = lim n →±∞ n log | ( f nξ ) (cid:48) ( x ) | , where f nξ def = f ξ n − ◦ . . . ◦ f ξ , where we assume that both limits exist and are equal. Note that it is nothing butthe Birkhoff average of a continuous function. For every F -ergodic Borel probabilitymeasure µ the Lyapunov exponent is almost everywhere well defined and constant.This common value of exponents will be called the Lyapunov exponent of µ and TRUCTURE OF THE SPACE OF ERGODIC MEASURES 7 denoted by χ ( µ ). An ergodic measure µ is nonhyperbolic if χ ( µ ) = 0 and hyperbolic otherwise.Accordingly, we split the set of all ergodic measures in Λ core and consider thedecomposition M erg (Λ core ) = M erg ,< (Λ core ) ∪ M erg , (Λ core ) ∪ M erg ,> (Λ core )into measures with negative, zero, and positive fiber Lyapunov exponent, respec-tively. Analogously, we consider M erg (Λ ex ) = M erg ,< (Λ ex ) ∪ M erg , (Λ ex ) ∪ M erg ,> (Λ ex ) . Properties of the space of measures are summarized in the next theorem. Given N ⊂ M , its closed convex hull is the smallest closed convex set containing N . Theorem 2.5.
Assume that F defined in (2.1) satisfies the hypotheses (H1) and(H2). Then the space M (Σ × [0 , has the following properties: Periodic orbit measures are dense in the closed convex hull of M erg (Λ ex ) . Every hyperbolic measure M (Λ ex ) has positive weak ∗ distance from M (Λ core ) . Every nonhyperbolic measure M (Λ ex ) can be weak ∗ approximated by periodicmeasures in M erg (Λ core ) . Each of the components M erg ,(cid:63) (Λ ex ) , (cid:63) ∈ { < , , > } is nonempty.Moreover, if hypotheses (H2’), (H3), and (H4) additionally hold, then The set M erg ,(cid:63) (Λ core ) , (cid:63) ∈ { < , > } , is nonempty. The set M erg ,< (Λ core ) and the set M erg ,> (Λ core ) are arcwise connected,respectively. The fact that there are ergodic measures with zero Lyapunov exponent and pos-itive entropy in M erg (Λ core ) can be shown using methods in [1], we refrain from dis-cussing this here. We also refrain from studying how such measures are approachedby hyperbolic ergodic measures in M erg (Λ core ) as this is much more elaborate andwill be part of an ongoing project (see [10] for techniques in a slightly different buttechnically simpler context). f f ,t ( x ) = t (1 − x ) t Figure 4.
Porcupine-like horseshoes.
Remark 2.6 (Porcupine vs. totally spiny porcupine) . Let us compare the porcupine-like horseshoes corresponding to the interval maps in Figure 4 with the “totallyspiny porcupine” discussed here (corresponding to Figure 2). Porcupine-like horse-shoes were introduced in [12] as model for internal heterodimensional cycles in
L. J. D´IAZ, K. GELFERT, T. MARCARINI, AND M. RAMS horseshoes. Later these horseshoes were generalized and studied in a series of papersfrom various points of view: topological ([7, 8, 9], thermodynamical ([21, 9, 24, 25])and fractal ([13]) . This line of research is also closely related to the study of so-called bony attractors and sets (see [17] for a survey and references). One importantmotivation to study those models is that they serve as a prototype of partially hy-perbolic dynamics.Let us consider the map F t defined as in (2.1) but with the maps f , f ,t as inFigure 4 in the place of f , f in Figure 2. Let Γ t be the maximal invariant set of F t .In the above cited porcupine-like horseshoes, one also splits the maximal invariantset Γ t (which is nonhyperbolic and transitive) into two parts Γ t ex and Γ t core in thesame spirit as in (2.2) (and with analogous properties as in Proposition 2.2). Inthat case Γ t ex consists only of one fiber expanding point Q = (0 Z ,
0) and Γ t core isits complement that contains the fiber contracting point P = (0 Z , t splits into two components, each of them connectedbut at positive distance from each other, which are { δ Q } and M erg (Γ t core ) (see, inparticular, [21]). In the transition from a porcupine to a totally spiny porcupine(which occurs at t = 1), the space of ergodic measures becomes connected (statedin Theorem 2.5) and this happens as follows. The measures δ Q and δ P form partof the space of ergodic measures of an abstract horseshoe Λ ex . At the same time,the measure δ P detaches from M erg (Λ core ) which is a consequence of the fact thatthe saddle P is not homoclinically related to any saddle in Λ core , similarly for Q .The components M erg (Λ ex ) and M erg (Λ core ) become glued through nonhyperbolicmeasures. Theorem 2.7.
Assume that F defined in (2.1) satisfies the hypotheses (H1) and(H2). Then there is a unique measure µ exmax of maximal entropy log 2 in M erg (Λ ex ) and its Lyapunov exponent is given by χ ( µ exmax ) = 14 (log f (cid:48) (0) + log f (cid:48) (1) + log | f (cid:48) (0) | + log | f (cid:48) (1) | ) . Moreover, if the measure µ exmax is hyperbolic then there exists at least one measure ofmaximal entropy in M erg (Λ core ) . More precisely, if the measure µ exmax has positive(negative) Lyapunov exponent then there exists a measure of maximal entropy withnonpositive (nonnegative) exponent in M erg (Λ core ) . Note that the topological structure of M (Λ ex ) (items 1. and 4. in Theorem 2.7)are immediate consequences of the fact that the dynamics of F on Λ ex is conjugateto a subshift of finite type (see Section 3 for details).Note that the under the hypotheses of the above theorem, we do not know if themeasure of maximal entropy in Λ core is hyperbolic or not. Remark 2.8.
In view of Theorem 2.7, choosing the derivatives of the fiber mapsat 0 and 1 appropriately, one obtains one measure of maximal entropy µ exmax whichis nonhyperbolic. Note that condition (H4) is incompatible with such a choice, andhence it is unclear if the system is transitive (compare Proposition 2.2).Similar arguments apply to the examples discussed in Remark 2.4. It is inter-esting to compare to the results in [28] where maps with “sufficiently high entropy The term “porcupine” coined in [7] refers to the rich topological fiber structure of the homo-clinic class, which is simultaneously composed of uncountable many fibers which are continua anduncountable many ones which are just points. In this paper, all fibers are full intervals.
TRUCTURE OF THE SPACE OF ERGODIC MEASURES 9 measures” are always hyperbolic, though there a key ingredient is accessibility whichis missing here.In Appendix A.3, we provide examples where the system is transitive and exhibitsa nonhyperbolic measure of maximal entropy in M erg (Λ ex ), proving the followingtheorem. Theorem 2.9.
There are maps ˜ F defined as in (2.1) whose fiber maps ˜ f , ˜ f satisfy
1. ˜ f has (exactly) two fixed points ˜ f (0) = 0 and ˜ f (1) = 1 , with ˜ f (cid:48) (0) = 1 =˜ f (cid:48) (1) ,
2. ˜ f ( x ) = 1 − x ,such that ˜ F is topologically transitive and that every pair of fiber expanding hyper-bolic periodic orbits and every pair of fiber contracting hyperbolic periodic orbits in Λ core are homoclinically related, respectively. In particular, the unique measure ofmaximal entropy in M erg (Λ ex ) is nonhyperbolic. Note that in the above theorem this measure is also a measure of maximalentropy in M erg (Λ), however we do not know if there is some hyperbolic measureof maximal entropy in M erg (Λ). Mutatis mutandi , we can perform a version of the map ˜ F in Σ × S as inRemark 2.4.Finally, in Appendix A.4 we present an example with a unique measure of max-imal entropy which is nonhyperbolic and supported on Λ ex . However this examplefails to be transitive. Theorem 2.10.
There are maps F defined as in (2.1) whose fiber maps f , f satisfy f has (exactly) two fixed points f (0) = 0 and f (1) = 1 , with f (cid:48) (0) = 1 = f (cid:48) (1) , f ( x ) = 1 − x .such that F is not topologically transitive and has a unique measure of maximalentropy supported on Λ ex , which is nonhyperbolic. One of the key properties of the class of examples in the above theorem is that f is conjugate to its inverse f − by f . The proof of the result is based on ananalysis of random walks on R and of somewhat different flavor.3. Symmetric, mirror, and twin measures
Recalling well-known facts about shift spaces, we will see that there is a uniquemeasure of maximal entropy for F | Λ ex and we will deduce that, in the case thismeasure is hyperbolic, there is (at least) one “twin” measure in Λ core with thesame (maximal) entropy. The latter is either hyperbolic with opposite sign of itsexponent or nonhyperbolic.Recall that on the full shift σ : Σ → Σ there is a unique measure (cid:98) ν max ofmaximal entropy log 2 which is the ( , )-Bernoulli measure.To study the structure of the invariant set Λ ex , consider the “first level” rectan-gles C k def = { ξ ∈ Σ : ξ = k } and the subsets (cid:98) C L def = C × { } , (cid:98) C L def = C × { } , (cid:98) C R def = C × { } , (cid:98) C R def = C × { } , of Σ × [0 , A given by A def = . This matrix codes the transitions between the symbols { L , L , R , R } modellingthe transitions between the sets (cid:98) C L , (cid:98) C L , (cid:98) C R , (cid:98) C R by the map F . More precisely,note that the restriction of F | Λ ex is topologically conjugate to the subshift of finitetype σ A : Σ A → Σ A by means of a map (cid:36) : Λ ex → Σ A . Note that there is a uniquemeasure of maximal entropy ν exmax for σ A : Σ A → Σ A . Note that h ν exmax ( σ ) = log 2.Hence, by conjugation, the measure µ exmax = ( (cid:36) − ) ∗ ν exmax is the unique measure ofmaximal entropy log 2 for F : Λ ex → Λ ex .We define the following projectionΠ : Σ A → Σ , Π( . . . i − .i . . . ) def = ( . . . ξ − .ξ . . . ) ,ξ k def = (cid:40) i k ∈ { L , R } i k ∈ { L , R } . It is immediate to check that (cid:98) ν max = Π ∗ ν exmax . We say that the symbols i R , j L are the mirrors of i L and j R , respectively,for i and j in { , } and denote i R = ¯ i L and j L = ¯ j R . Given a sequence ξ =( . . . ξ − .ξ . . . ) ∈ Σ A , we define by ¯ ξ = ( . . . ¯ ξ − . ¯ ξ . . . ) the mirrored sequence of ξ .Note that ¯ ξ ∈ Σ A . Given a subset B ⊂ Σ A , we denote by ¯ B def = { ¯ ξ : ξ ∈ B } its mirrored set.Now we are ready to define symmetric sets and measures . Definition 3.1 (Symmetric sets and measures) . A measurable set B ⊂ Σ A is symmetric if B = ¯ B . We say that B is symmetric ν -almost surely if ν ( B ∆ ¯ B ) = 0.A measure ν ∈ M (Σ A ) is symmetric if ν ( ¯ B ) = ν ( B ) for every B ⊂ Σ A . If a measureis not symmetric then we call it asymmetric . A measure µ ∈ M (Λ ex ) is symmetric if (cid:36) ∗ µ is symmetric, otherwise we call it asymmetric .We denote by M symerg (Λ ex ) and M asymerg (Λ ex ) the sets of symmetric and asymmetricergodic measures in Λ ex , respectively.We will use the following lemma. Lemma 3.2.
Let ν ∈ M (Σ A ) be a symmetric measure. Then any set B ⊂ Σ A which is ν -almost symmetric satisfies ν ((Π − ◦ Π)( B )) = ν ( B ) . Proof.
Indeed, by ν -almost symmetry of B , setting C def = B ∩ ¯ B , D def = ¯ B \ C , and E def = B \ C , we have B = E ∪ C , ν ( C ) = ν ( ¯ C ) = ν ( B ) = ν ( ¯ B ), and ν ( D ) = ν ( E ) =0. Hence ν ( ¯ D ) = ν ( D ) = 0. Observing that(Π − ◦ Π)( B ) = B ∪ ¯ B = E ∪ C ∪ D the claim follows. (cid:3) Note that this measure is the Parry measure associated to the topological Markov chain σ A ,see [30, Theorem 8.10]. TRUCTURE OF THE SPACE OF ERGODIC MEASURES 11
Lemma 3.3.
For every ν ∈ M erg (Σ A ) , there exist at most one measure ¯ ν ∈ M erg (Σ A ) , ¯ ν (cid:54) = ν , such that Π ∗ ¯ ν = Π ∗ ν . There is no such measure if, and only if, ν is symmetric.Proof. It suffices to observe that the product σ -algebra of Borel measurable sets ofΣ A is generated by the semi-algebra generated by the family of all finite cylindersets { [ i k . . . i (cid:96) ] } . Note also that the mirror ¯ C of a cylinder C in Σ A is again a cylinderin Σ A . Now given ν ∈ M erg (Σ A ), define a measure ¯ ν by setting ¯ ν ( C ) def = ν ( ¯ C ) forevery cylinder C and extend it to the generated σ -algebra.By definition, we immediately obtain that Π ∗ ¯ ν = Π ∗ ν and that h ¯ ν ( σ A ) = h ν ( σ A ).To prove that ν and ¯ ν are the only ergodic measures satisfying Π ∗ ¯ ν = Π ∗ ν ,by contradiction assume that there exists (cid:98) ν ∈ M erg (Σ A ), ¯ ν (cid:54) = (cid:98) ν (cid:54) = ν satisfyingΠ ∗ (cid:98) ν = Π ∗ ν . Consider the measure (cid:101) ν def = ( ν + ¯ ν ). Note that (cid:101) ν is symmetric. Alsonote that Π ∗ (cid:101) ν = Π ∗ (cid:98) ν . Finally note that (cid:101) ν is singular with respect to (cid:98) ν and hencethere is a set B ⊂ Σ A satisfying (cid:101) ν ( B ) = 0 = (cid:98) ν ( B c ). Since (cid:101) ν is symmetric, we have (cid:101) ν ( ¯ B ) = (cid:101) ν ( B ). So we obtain (cid:101) ν ( ¯ B (cid:52) B ) = 0 and hence B is (cid:101) ν -almost symmetric.Hence, we have 0 = (cid:101) ν ( B )(by Lemma 3.2 ) = (cid:101) ν ((Π − ◦ Π)( B )) = Π ∗ (cid:101) ν (Π( B ))(since Π ∗ (cid:101) ν = Π ∗ (cid:98) ν ) = Π ∗ (cid:98) ν (Π( B )) = (cid:98) ν ((Π − ◦ Π)( B )) ≥ (cid:98) ν ( B ) = 1 , a contradiction. This proves that ¯ ν is uniquely defined.By definition, ν is symmetric if, and only if, ¯ ν = ν . (cid:3) Definition 3.4 (Mirror measure) . We call the measure ¯ ν provided by Lemma 3.3the mirror measure of ν . We call the measure (cid:36) − ∗ ¯ ν ∈ M (Λ ex ) the mirror measure of µ = (cid:36) − ∗ ν and denote it by ¯ µ .The following is an immediate consequence of Lemma 3.3 and the uniqueness ofthe measure of maximal entropy. Corollary 3.5.
The measure of maximal entropy µ exmax is symmetric. Lemma 3.6. If ¯ µ is a mirror measure of µ ∈ M erg (Λ ex ) then h ¯ µ ( F ) = h µ ( F ) .Moreover, we have χ (¯ µ ) + χ ( µ ) = N (0) log( f (cid:48) (0) · f (cid:48) (1)) + N (1) log( f (cid:48) (0) · f (cid:48) (1)) , where N (0) def = µ (Σ × { } ) and N (1) def = µ (Σ × { } ) .Proof. Let ν = (cid:36) ∗ µ . It suffices to observe that a sequence ξ is ν -generic if, andonly if, ¯ ξ is ¯ ν -generic and to do the straightforward calculation. (cid:3) Definition 3.7.
Given an ergodic measure µ ∈ M erg (Σ × [0 , (cid:101) µ ∈ M erg (Σ × [0 , (cid:101) µ (cid:54) = µ , is called a twin measure of µ if π ∗ (cid:101) µ = π ∗ µ .Note that the above immediately implies that if µ ∈ M erg (Λ ex ) is symmetricthen all its twin measures are in M erg (Λ core ). Lemma 3.8 (Existence of twin measures) . For every measure λ ∈ M (Σ ) thereexist a measure µ ∈ M (Σ × [0 , satisfying π ∗ µ = λ and χ ( µ ) ≥ and ameasure µ ∈ M (Σ × [0 , satisfying π ∗ µ = λ and χ ( µ ) ≤ .Moreover, if λ was ergodic then µ and µ can be chosen ergodic. Note that the measures µ and µ in the above lemma may coincide. Proof.
First observe that λ ∈ M (Σ ) is weak ∗ approximated by measures λ (cid:96) ∈ M (Σ ) supported on periodic sequences.For each such measure λ (cid:96) there exists a measure µ (cid:96) ∈ M (Σ × [0 , F -periodic orbit in Σ × [0 ,
1] and satisfies π ∗ µ (cid:96) = λ (cid:96) and χ ( µ (cid:96) ) ≥ λ (cid:96) is supported on the orbit of a periodic sequence ξ ∈ Σ ofperiod n . Recall that the fiber maps f and f and hence the map f nξ preservethe boundary { , } . Hence, this map f nξ has a fixed point x ∈ [0 ,
1] satisfying | ( f nξ ) (cid:48) ( x ) | ≥
1. Now observe that the orbit of ( ξ, x ) is F -periodic of period n andtaking the measure µ (cid:96) supported on it we have n log | ( f nξ ) (cid:48) ( x ) | = χ ( µ (cid:96) ).Now take any weak ∗ accumulation point µ of the sequence ( µ (cid:96) ) (cid:96) . Note that bycontinuity of π ∗ we have π ∗ µ = λ .If λ was ergodic, µ might not be ergodic. However, any ergodic measure inthe ergodic decomposition of µ also projects to λ and hence there must exist onemeasure µ (cid:48) in this decomposition satisfying χ ( µ (cid:48) ) ≥ χ ( · ) ≤ (cid:3) Corollary 3.9.
For every hyperbolic symmetric ergodic measure µ ∈ M erg (Λ ex ) there exists an ergodic twin measure (cid:101) µ ∈ M erg (Σ × (0 , , (cid:101) µ (cid:54) = µ , satisfying h (cid:101) µ ( F ) = h µ ( F ) .Proof. Assume that χ ( µ ) >
0, the other case χ ( µ ) < λ def = π ∗ µ .By Lemma 3.8, there exists a twin measure (cid:101) µ ∈ M erg (Σ × [0 , µ satisfying χ ( (cid:101) µ ) ≤
0. Note that h π ∗ (cid:101) µ ( σ ) ≤ h (cid:101) µ ( F ) and h π ∗ µ ( σ ) ≤ h µ ( F ). On the other hand,by [20]max { h µ ( F ) , h (cid:101) µ ( F ) } ≤ sup m : π ∗ m = π ∗ µ h m ( F ) = h π ∗ µ ( σ ) + (cid:90) h top ( F, π − ( ξ )) dπ ∗ µ ( ξ ) . Since π is 2-1, we have h top ( F, π − ( ξ )) = 0 for every ξ . Thus, we conclude h µ ( F ) = h (cid:101) µ ( F ) = h π ∗ µ ( σ ).By conjugation (cid:36) between F | Λ ex and σ A | Σ A , there can be at most one otherergodic measure in M erg (Λ ex ) which project to the same measure on Σ , namely (cid:36) − ∗ ¯ ν , where ν = (cid:36) − ∗ µ and ¯ ν is the mirror measure of ν . For symmetric µ , nosuch mirror exists. Hence, we must have (cid:101) µ ∈ M erg (Σ × (0 , (cid:3) By Corollary 3.5, the above applies in particular to µ exmax . Corollary 3.10.
If the measure of maximal entropy µ exmax ∈ M erg (Λ ex ) is hyperbolicthen there exists an ergodic twin measure of maximal entropy (cid:101) µ ∈ M erg (Λ core ) suchthat χ ( (cid:101) µ ) χ ( µ exmax ) ≤ .Proof of Theorem 2.7. As recalled already, there is a unique measure of maximalentropy for F | Λ ex and its Lyapunov exponents can be easily calculated. The factthat there may exist another measure of maximal entropy for F | Λ core follows imme-diately from Corollary 3.10. (cid:3) Approximations of boundary measures
This section discusses the approximation of measures in M (Λ ex ) by (ergodic)measures in M (Λ core ). In particular, we will complete the proof of Theorem 2.5.We will always work with the system satisfying hypotheses (H1) and (H2). TRUCTURE OF THE SPACE OF ERGODIC MEASURES 13
Recall again that M equipped with the weak ∗ topology it is a compact metrizabletopological space [30, Chapter 6.1]. Recall that X ∈ Σ × [0 ,
1] is a generic point of ameasure µ ∈ M (Σ × [0 , n ( δ X + δ F ( X ) + . . . + δ F n − ( X ) ) convergesto µ in the weak ∗ topology, where δ Y denotes the Dirac measure supported in Y .Recall that for every ergodic measure there exists a set of generic points with fullmeasure.Given δ ∈ (0 , / δ ) def = max i =0 , (cid:26) max z ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12)(cid:12) log | f (cid:48) i ( z ) || f (cid:48) i (0) | (cid:12)(cid:12)(cid:12)(cid:12) , max z ∈ [1 − δ, (cid:12)(cid:12)(cid:12)(cid:12) log | f (cid:48) i ( z ) || f (cid:48) i (1) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) . Note that ∆( δ ) → δ →
0. We state the following simple facts without proof.
Lemma 4.1.
For every δ ∈ (0 , / and every x ∈ [0 , δ ] we have e − ∆( δ ) ≤ f (cid:48) i ( x ) f (cid:48) i (0) , f (cid:48) i (1 − x ) f (cid:48) i (1) ≤ e ∆( δ ) and e − ∆( δ ) ≤ | f i ( x ) − f i (0) || x || f (cid:48) i (0) | , | f i (1 − x ) − f i (1) || − (1 − x ) || f (cid:48) i (1) | ≤ e ∆( δ ) . Proposition 4.2.
For every µ ∈ M (Λ ex ) satisfying χ ( µ ) = 0 there exists a sequence ( µ k ) k ⊂ M erg (Λ core ) of measures supported on periodic orbits which converge to µ in the weak ∗ topology.Proof. Let µ be an invariant measure supported in Λ ex and satisfying the hypothesis χ ( µ ) = 0 and let X = ( ξ, x ) ∈ Λ ex be a µ -generic point. Hence χ ( X ) = 0. Notethat ξ hence has infinitely many symbols 1 by our hypothesis f (cid:48) (0) (cid:54) = 1 (cid:54) = f (cid:48) (1).Hence, without loss of generality, we can assume that x = 1.Given the sequence ξ = ( . . . ξ − .ξ ξ . . . ), for n ≥ p n def = card (cid:8) i ∈ { , . . . , n − } : ξ i = 0 , card { j < i : ξ j = 1 } even (cid:9) ,q n def = card (cid:8) i ∈ { , . . . , n − } : ξ i = 0 , card { j < i : ξ j = 1 } odd (cid:9) ,r n def = card (cid:8) i ∈ { , . . . , n − } : ξ i = 1 , card { j < i : ξ j = 1 } even (cid:9) ,s n def = card (cid:8) i ∈ { , . . . , n − } : ξ i = 1 , card { j < i : ξ j = 1 } odd (cid:9) . Note that n = p n + q n + r n + s n . Let(4.2) φ ( n ) def = p n log | f (cid:48) (1) | + q n log | f (cid:48) (0) | + r n log | f (cid:48) (1) | + s n log | f (cid:48) (0) | . Observe that φ ( n ) = log | ( f nξ ) (cid:48) (1) | and hence χ ( X ) = 0 implieslim n →∞ φ ( n ) n = 0 . Let ψ ( n ) def = max i =1 ,...,n | φ ( i ) | and note that(4.3) lim n →∞ ψ ( n ) n = 0 . Let B = { , } . Given a fiber point x ∈ (0 ,
1) let us use the following notation ofits orbit under the fiber dynamics determined by the sequence ξ :(4.4) x i def = f iξ ( x ) . Partially affine case:
To sketch the idea of the proof, assume for a moment that f | I δ and f | I δ are affine, where I δ = [0 , δ ] ∪ [1 − δ,
1] for some small δ >
0. Notethat for given n and a point x ∈ [1 − δ,
1) satisfying(4.5) x i ∈ I δ for all i ∈ { , . . . , n − } we have(4.6) Dist( x i +1 , B )Dist( x i , B ) = e φ ( i +1) − φ ( i ) , where Dist( x, B ) denotes the distance of x from a set B . HenceDist( x n , B )Dist( x , B ) = e φ ( n ) . This implies(4.7) e − ψ ( n ) ≤ Dist( x i , B )Dist( x , B ) ≤ e ψ ( n ) for all i ∈ { , . . . , n } . Note that (4.5) is satisfied provided x was chosen to satisfy Dist( x , B ) < δe − ψ ( n ) .Note that e − ψ ( n ) may not converge to 0. For this reason, let us choose(4.8) δ ( n ) def = δe − { ψ ( n ) , √ n } . Note that(4.9) lim n →∞ δ ( n ) e ψ ( n ) = 0 . Let now n be a sufficiently large integer such that card { j ≤ n − ξ j = 1 } isodd. Note that this implies f nξ is orientation reversing. Let N ( n ) be the smallestpositive integer such that x def = f N ( n )0 (1 / ∈ [1 − δ ( n ) , N ( n ) ∼ | log δ ( n ) | , where the approximation is up to some universal multiplicative factor, independenton n . Note that f N ( n )0 is orientation preserving. We now apply the above argumentsto the chosen point x . We consider the sequence ( x i ) ni =0 as defined in (4.4). First,note that Dist( x , B ) ∼ δ ( n ) and with (4.7) we have δ ( n ) e − ψ ( n ) ≤ x i ≤ Dist( x , B ) e ψ ( n ) ≤ δ ( n ) e ψ ( n ) for all i ∈ { , . . . , n } . Further note that x n by our choice of n is close to 0. Then let M ( n ) be thesmallest positive integer such that f M ( n )0 ( x n ) ≥ /
2. Note that f M ( n )0 is orientationpreserving. Note that(4.11) M ( n ) ∼ | log δ ( n ) + ψ ( n ) | . Now consider the map g def = f M ( n )0 ◦ f nξ ◦ f N ( n )0 and note that it reverses orientation.Hence, there exists a point y in the fundamental domain [1 / , f (1 / g ( y ) = y . Note that by the estimates of N ( n ) and M ( n ) in (4.10) and (4.11) andour choice of δ ( n ) in (4.8) and by (4.3) we have(4.12) lim n →∞ N ( n ) + M ( n ) n = 0 . We now consider the (invariant) measure µ n,δ supported on the periodic orbit ofthe point Y = ( η, y ), where η = (0 N ( n ) ξ . . . ξ n − M ( n ) ) Z . It remains to show that TRUCTURE OF THE SPACE OF ERGODIC MEASURES 15 this measure is close to µ in the weak ∗ topology provided that n was big. Notethat we can write µ n,δ as µ n,δ = N ( n ) N ( n ) + n + M ( n ) µ ++ nN ( n ) + n + M ( n ) 1 n n − (cid:88) k =0 δ F N ( n )+ k ( Y ) + M ( n ) N ( n ) + n + M ( n ) µ , where µ and µ are some probability measures. Note that by (4.12) the first and thelast term converges to 0 as n tends to ∞ . The second term is close to µ because X was a µ -generic point and the orbit piece { F N ( n ) ( Y ) , F N ( n )+1 ( Y ) , . . . , F N ( n )+ n ( Y ) } is δ ( n ) e ψ ( n ) -close to the orbit piece { X, F ( X ) , . . . , F n ( X ) } . Recalling (4.9), thiscompletes the proof in the affine case. General case:
In the nonaffine case the proof goes similarly. Applying Lemma 4.1,we choose the number δ ( n ) in an appropriate way. First note that instead of (4.6)by this lemma we have Dist( x i +1 , B )Dist( x i , B ) ≤ e φ ( i +1) − φ ( i ) e ∆( δ ) . Arguing as above, let now δ ( n ) def = δe − { ψ ( n ) , √ n } e − n ∆( δe −√ n ) . Observe that with this choice, for every x ∈ [1 − δ ( n ) ,
1) for every i ∈ { , . . . , n } we have Dist( x i , B ) ≤ δ ( n ) e φ ( i ) e i ∆( δe −√ n ) ≤ δe −√ n provided that Dist( x j , B ) ≤ δe −√ n for all j ∈ { , . . . , i − } . By induction, we willget that for all i ∈ { , . . . , n } we haveDist( x i , B ) ≤ δe −√ n . Note that with the above definition of δ ( n ) the estimates of N ( n ) and M ( n ) in(4.10) and (4.11) remain without changes. And the rest of the proof is analogousto the partially affine case. (cid:3) We now prove the converse to Proposition 4.2.
Proposition 4.3.
For every µ ∈ M (Λ ex ) for which there exists a sequence ( ν k ) k ⊂ M (Λ core ) of measures which converge to µ in the weak ∗ topology we have χ ( µ ) = 0 . The proof of the above proposition will be an immediate consequence of thefollowing lemma. Recall the definition of ∆( · ) in (4.1). Lemma 4.4.
There exist constants K , K > such that for every δ ∈ (0 , / and every measure ν ∈ M (Λ core ) we have | χ ( ν ) | ≤ K ν (Σ × [ δ, − δ ]) + K ∆( δ ) . Proof.
Note that it is enough to prove the claim for ν ∈ M (Λ core ) being ergodic.Indeed, for a general invariant measure ν ∈ M (Λ core ) with ergodic decomposition ν = (cid:82) ν θ dλ ( ν θ ), applying the above claim to any (ergodic) ν θ in this decompositionwe have χ ( ν ) = (cid:90) χ ( ν ) dλ ( ν θ ) ≤ K ν (Σ × [ δ, − δ ]) + K ∆( δ ) with the analogous lower bound.Let us hence assume that ν ∈ M (Λ core ) is ergodic. Since ν is not supported onΛ ex , there exists δ (cid:48) ∈ (0 , δ ) such that ν (Σ × [ δ (cid:48) , − δ (cid:48) ]) >
0. Let X = ( ξ, x ) bea generic point for ν satisfying x ∈ [ δ (cid:48) , − δ (cid:48) ] and consider the sequence of points x i def = f iξ ( x ) for i ≥
0. Since ν is ergodic, there are infinitely many n ≥ x n ∈ [ δ (cid:48) , − δ (cid:48) ] and hence(4.13) 2 δ (cid:48) ≤ Dist( x n , B )Dist( x , B ) ≤ δ (cid:48) , where B = { , } . Because X is a generic point, given any ε , for n large enough wehave (cid:12)(cid:12)(cid:12) n log | ( f nξ ) (cid:48) ( x ) | − χ ( ν ) (cid:12)(cid:12)(cid:12) ≤ ε and also(4.14) (cid:12)(cid:12)(cid:12) n card { i ∈ { , . . . , n − } : x i ∈ [ δ, − δ ] } − ν (Σ × [ δ, − δ ]) (cid:12)(cid:12)(cid:12) ≤ ε. Applying Lemma 4.1, we have e ω ( i +1) e − ∆( δ ) ≤ Dist( x i +1 , B )Dist( x i , B ) ≤ e ω ( i +1) e ∆( δ ) if x i ∈ (0 , δ ] ∪ [1 − δ, ,e ω ( i +1) K − ≤ Dist( x i +1 , B )Dist( x i , B ) ≤ e ω ( i +1) K if x i (cid:54)∈ (0 , δ ] ∪ [1 − δ, , where K > ω ( i ) def = log | f (cid:48) ξ i (0) | if x i − ∈ (0 , δ ] , log | f (cid:48) ξ i (1) | if x i − ∈ [1 − δ, , . By a telescoping sum, we haveDist( x n , B )Dist( x , B ) = n − (cid:89) i =0 Dist( x i +1 , B )Dist( x i , B ) . We split the index set { , . . . , n − } = I ∪ I according to the rule that x i ∈ [ δ, − δ ]for all i ∈ I and x i (cid:54)∈ [ δ, − δ ] for all i ∈ I . Let φ ( n ) def = n (cid:88) i =1 ω ( i )and note that this function was also used in the previous proof, see (4.2). By theabove estimates, we hence have e φ ( n ) K − card I · e − ∆( δ ) card I ≤ Dist( x n , B )Dist( x , B ) ≤ e φ ( n ) K card I · e ∆( δ ) card I . This implies(4.15) e φ ( n ) ≤ Dist( x n , B )Dist( x , B ) K card I · e ∆( δ ) card I . Again applying Lemma 4.1, if Dist( x i , B ) < δ then we have | ( f ξ i +1 ) (cid:48) ( x i ) | ≤ e ω ( i +1) e ∆( δ )TRUCTURE OF THE SPACE OF ERGODIC MEASURES 17 and if Dist( x i , B ) ≥ δ then we have | ( f ξ i +1 ) (cid:48) ( x i ) | ≤ e ω ( i +1) L for some universal L >
1. Hence, decomposing the orbit piece ( x i ) n − i ≥ as aboveinto index sets I and I , we obtain | ( f nξ ) (cid:48) ( x ) | ≤ e φ ( n ) L card I · e ∆( δ ) card I with the analogous lower bound.By (4.14), we have card I ≤ n ( ν (Σ × [ δ, − δ ]) + ε ).Substituting the estimate for e φ ( n ) in (4.15) we obtain | ( f nξ ) (cid:48) ( x ) | ≤ Dist( x n , B )Dist( x , B ) ( KL ) card I · e δ ) card I ≤ δ (cid:48) ( KL ) card I · e δ ) card I , where we also used (4.13). Hence1 n log | ( f nξ ) (cid:48) ( x ) | ≤ n | log(2 δ (cid:48) ) | + log( KL ) (cid:0) ν (Σ × [ δ, − δ ]) + ε (cid:1) + 2∆( δ ) , with the analogous lower bound. Since | log | ( f nξ ) (cid:48) ( x ) | /n − χ ( ν ) | ≤ ε , passing n → ∞ and then ε → (cid:3) Proof of Theorem 2.5.
Item 1 is a well-known fact, see for example [26, Proposition2 item (a)]. This fact implies that M erg (Λ ex ) is a Poulsen simplex (see [22] or in theparticular case of the shift space [27]). Item 4 is then an immediate consequencefrom the facts that µ (cid:55)→ χ ( µ ) is continuous and that the Dirac measure on (0 Z ,
0) hasLyapunov exponent log f (cid:48) (0) > Z ,
1) has Lyapunovexponent log f (cid:48) (1) < M erg (Λ ex ) is path-connected.Item 2 follows from Proposition 4.3.Item 3 follows from Proposition 4.2.Let us now assume (H1), (H2’), (H3), and (H4). By Lemmas A.3 and A.5there exist hyperbolic periodic points in Λ core with positive and negative exponent,respectively. This proves item 5. By Proposition 2.2 we can apply 5.1. This impliesitem 6. (cid:3) The core measures
In this section we will investigate a bit further the topological structure of M erg (Λ core ). The overall hypotheses are again (H1) and (H2), and we will dis-cuss further additional conditions under which we are able to say more than in theprevious sections. Proposition 5.1.
Assume that every pair of fiber expanding hyperbolic periodicorbits in Λ core are homoclinically related. Then the set M erg ,> (Λ core ) is arcwise-connected. The analogous result holds true for fiber contracting hyperbolic periodicorbits in Λ core and the set M erg ,< (Λ core ) . A map F whose fiber maps f , f satisfy the hypotheses (H1), (H2’), (H3), and(H4) will satisfy the hypotheses of the above proposition.We will several times refer to a slightly strengthened version of [5, Proposition1.4] which, in fact, is contained in its proof in [5] and which can be seen as an ersatzof Katok’s horseshoe construction (see [19, Supplement S.5]) in the C dominatedsetting. We formulate it in our setting. Note that to guarantee that the approxi-mating periodic orbits are indeed contained in Σ × (0 ,
1) it suffices to observe that in the approximation arguments one can consider any sufficiently large (in measure µ ) set and hence restrict to points which are uniformly away from the “boundary”Σ × { , } . Indeed, the projection to [0 ,
1] of the support of µ can be the wholeinterval [0 ,
1] but it does not “concentrate” in { , } . Lemma 5.2.
Let (cid:63) ∈ { < , > } and µ ∈ M erg ,(cid:63) (Λ core ) .Then for every ρ ∈ (0 , there exist α > and a set Γ ρ ⊂ Σ × (2 α, − α ) anda number δ = δ ( ρ, µ ) > such that µ (Γ ρ ) > − ρ and for every point X ∈ Γ ρ thereis a sequence ( p n ) n ⊂ Σ × ( α, − α ) of hyperbolic periodic points such that: • p n converges to X as n → ∞ ; • the invariant measures µ n supported on the orbit of p n are contained in M erg ,(cid:63) (Λ core ) and converge to µ in the weak ∗ topology;Proof of Proposition 5.1. Similar results were shown before, though in slightly dif-ferent contexts (see [16] and [11, Theorem 3.2]). For completeness, we sketch theproof.Assume that µ , µ ∈ M erg ,> (Λ core ). By Lemma 5.2, µ i is accumulated by a se-quence of hyperbolic periodic measures ν in ∈ M erg ,> (Λ core ) supported on the orbitsof fiber expanding hyperbolic periodic points P in ∈ Λ core , i = 0 ,
1. Since, by hypoth-esis, P and P are homoclinically related, there exists a horseshoe Γ , ⊂ Λ core con-taining these two points. Hence, since M (Γ , ) is a Poulsen simplex [22, 27], there isa continuous arc µ : [1 / , / → M erg ,> (Γ , ) ⊂ M erg ,> (Λ core ) joining the mea-sures ν and ν . For any pair of measures ν n , ν n +1 , the same arguments apply and,in particular, there exists a continuous arc µ n : [1 / n +1 , / n ] → M erg ,> (Λ core )joining the measure ν n with ν n +1 . Using those arcs and concatenating their domains(or appropriate parts of), we can construct an arc ¯ µ n : [1 / n +1 , /
3] joining ν n +1 and ν . The same applies to the measures ν n , defining arcs ¯ µ n : [1 − / n , / → M erg ,> (Λ core ) joining ν n +1 and ν . Defining µ ∞ | (0 , : (0 , → M erg ,> (Λ core )by concatenating (appropriate parts of) the domains of those arcs, we completethe definition of the arc µ ∞ by letting µ ∞ (0) = lim n →∞ ¯ µ n (1 / n ) and µ ∞ (1) =lim n →∞ ¯ µ n (1 − / n ), joining µ and µ . Note that in the last step we assumethat µ , µ do not belong to the image of µ ∞ , if one of these measures belongs it isenough to cut the domain of definition of µ ∞ appropriately. (cid:3) Appendix A. Transitivity and homoclinic relations. Proof ofProposition 2.2
In this appendix we prove Proposition 2.2. Hence, we will always assume thathypotheses (H1), (H2’), (H3), (H4) are satisfied.A.1.
The underlying IFS.
Studying the iterated function system (IFS) associ-ated to the maps { f , f } , we use the following notations. Every sequence ξ =( . . . ξ − .ξ ξ . . . ) ∈ Σ is given by ξ = ξ − .ξ + , where ξ + ∈ Σ +2 def = { , } N and ξ − ∈ Σ − def = { , } − N . Given finite sequences ( ξ . . . ξ n ) and ( ξ − m . . . ξ − ), we let f [ ξ ... ξ n ] def = f ξ n ◦ · · · ◦ f ξ ◦ f ξ and f [ ξ − m ... ξ − . ] def = ( f ξ − ◦ . . . ◦ f ξ − m ) − = ( f [ ξ − m ... ξ − ] ) − . TRUCTURE OF THE SPACE OF ERGODIC MEASURES 19
A.1.1.
Expanding itineraries.
Under hypotheses (H1) and (H4) there are a positivenumber ε arbitrarily close to 0, a positive integer N ( ε ), and fundamental domains I ( ε ) = [ ε, f ( ε )] and I ( ε ) = [1 − ε, f (1 − ε )] of f having the following properties (A.1) f N ( ε )0 ( I ( ε )) = I ( ε ) and ( f N ( ε )0 ) (cid:48) ( x ) ≥ λ − κ > x ∈ I ( ε ) . In what follows we fix small ε > I , I , and N instead of I ( ε ), I ( ε ), and N ( ε ).Our construction now is analogous to the one in [8]. We sketch the main stepsfor completeness. Assuming additionally (H2’), given an interval H ⊂ f − ( I ) ∪ I we let N ( H ) = N if H ⊂ I and N ( H ) = N + 1 otherwise and consider theinterval f [0 N ( H ) ( H ). By construction, this interval is contained in [ δ ( ε ) , ε ], where δ ( ε ) = 1 − f (1 − ε ). Note that, by construction, δ ( ε ) < ε . Therefore there is a first M ( H ) such that f [0 N ( H ) M ( H ) ] ( H ) ∩ ( ε, f ( ε )] (cid:54) = ∅ . The expanded successor of H is the interval H (cid:48) def = f [0 N ( H ) M ( H ) ] ( H ). The expand-ing return sequence of H is the finite sequence 0 N ( H ) M ( H ) By construction theinterval H (cid:48) intersects the interior of I and is contained in [ δ ( ε ) , f ( ε )]. Also ob-serve that there is M such that M ( H ) ∈ { , . . . , M } for every subinterval H in f − ( I ) ∪ I . The following lemma justifies our terminology expanded successor. Lemma A.1 (Expanding itineraries [8, Lemma 2.3]) . For every closed subinterval H of f − ( I ) ∪ I and every x ∈ H it holds (cid:12)(cid:12)(cid:0) f [0 N ( H ) M ( H ) ] (cid:1) (cid:48) ( x ) (cid:12)(cid:12) ≥ κ > . Proof.
By (A.1) and the choice of H we have (cid:12)(cid:12) ( f [0 N ( H ) ] ) (cid:48) ( x ) (cid:12)(cid:12) ≥ κ for all x ∈ H .The assertion follows noting that f ( x ) = 1 − x , ( f [0 N ( H ) )( x ) ∈ [0 , ε ] if x ∈ H , and f (cid:48) ( y ) > y ∈ [0 , ε ]. (cid:3) Lemma A.1 and an inductive argument immediately implies the following:
Lemma A.2 ([8, Lemma 2.3]) . For every closed subinterval H of f − ( I ) ∪ I thereis a finite sequence ( ξ . . . ξ (cid:96) ( H ) ) such that (cid:12)(cid:12)(cid:0) f [ ξ ... ξ (cid:96) ( H ) ] (cid:1) (cid:48) ( x ) (cid:12)(cid:12) ≥ κ for every x ∈ H and f [ ξ ... ξ (cid:96) ( H )] ⊃ f − ( I ) .Proof. Write H and let H = H (cid:48) be its expanding successor. We argue recursively,if H contains f − ( I ) we stop the recursion, otherwise we observe that | H | ≥ κ | H | and consider the expanding successor H = H (cid:48) . Since H i ≥ κ i | H | there isa first i such that H i contains f − ( I ). We let ( ξ . . . ξ (cid:96) ( H ) ) be the concatenationof the successive expanding returns. (cid:3) Given a set H ⊂ [0 ,
1] denote its forward orbit by the IFS by O + ( H ) def = (cid:91) k ≥ (cid:91) ( ξ ... ξ k ) ∈{ , } k +1 f [ ξ ... ξ k ] ( H ) . A special case occurs when the set H is a point. Just note that, by the mean value theorem, there is z ∈ I ( ε ) with ( f N ( ε )0 ) (cid:48) ( z ) = | I ( ε ) | / | I ( ε ) | , that by monotonicity of the derivative of f (cid:48) we have ( f N ( ε )0 ) (cid:48) ( f ( z )) ≥ λ | I ( ε ) | / ( β | I ( ε ) | ) and that ( f N ( ε )0 ) (cid:48) ( x ) ≥ ( f N ( ε )0 ) (cid:48) ( f ( z )) for all x ∈ I ( ε ), and that for small ε we have | I ( ε ) | (cid:39) ( β − ε and | I ( ε ) | (cid:39) (1 − λ ) ε . Lemma A.3.
For every point p ∈ (0 , there are a small neighborhood I ( p ) of p and a finite sequence η . . . η r , r = r ( I ( p )) , such that (1) f [ η ...η r ] ( I ( p )) ⊃ I ( p ) , (2) (cid:12)(cid:12)(cid:0) f [ η ...η r ] ) (cid:48) ( x ) (cid:12)(cid:12) > for all x ∈ I ( p ) , and (3) O + ( I ( p )) = (0 , .Proof. Without loss of generality (considering some backward iterate of p and pos-sibly shrinking ε ) we can assume that p ∈ ( f − ( ε ) , ε ]. Let us suppose, for simplicitythat p (cid:54) = ε (the case p = ε would require a small additional step). In such a casewe can take I ( p ) ⊂ ( f − ( ε ) , ε ) and apply Lemma A.2 to H = I ( p ). This givesconditions (1) and (2) in the lemma. To get (3) note that we can assume that f [ η ...η r ] ( I ( p )) covers the fundamental domain f − ( I ). Therefore (cid:91) j ≥ f j ( f [ η ...η r ] ( I ( p ))) ⊃ ( ε, (cid:91) j ≥ f ◦ f j ( f [ η ...η r ] ( I ( p ))) ⊃ ( f (1) , f ( ε )) = (0 , f ( ε )) . Since f ( ε ) > ε the claim follows. (cid:3) A.1.2.
Contracting itineraries.
For the contracting itineraries we will now in partic-ular focus on (H3), which plays the role of (H4) in the previous subsection. Recallthat c ∈ (0 ,
1) is given by the condition f (cid:48) ( c ) = 1. Note that, since f (cid:48) is decreasing,we have f (cid:48) ( f ( c )) < υ def = 1 f (cid:48) ( f ( c )) > . In what follows, for notational simplicity let g def = f − and g def = f − (= f ) andbelow consider the IFS generated by { g , g } .Next lemma is a variation of [8, Lemma 2.6], where an important difference isthat in our case g is not expanding. Lemma A.4 (Contracting itineraries) . Let H be a closed subinterval of [ c, f ( c )] .Then there are a subinterval H of H and a sequence ξ . . . ξ k such that g [ ξ ...ξ k ] ( H ) ⊃ [ f ( c ) , f ( c )] and | g (cid:48) [ ξ ...ξ k ] ( x ) | ≥ υ for every x ∈ H . Proof.
Note that for every x ∈ [ f ( c ) , f ( c )] it holds | g (cid:48) [01] ( x ) | ≥ υ . Condition f ◦ f ( c ) > f ( c ) implies that g [01] ( x ) > f ( c ). Thus, there is a first i ≥ g [010 i ] ( x ) ∈ [ f ( c ) , f ( c )]. Note that | g (cid:48) [010 i ] ( x ) | ≥ υ . Now the result followsarguing as in Lemma A.2. (cid:3) Define the backward orbit O − ( · ) by the IFS of a set in the natural way. Arguingas in the expanding case, we have the following version of Lemma A.3. Lemma A.5.
For every point p ∈ (0 , there are a small neighborhood J ( p ) of p and a finite sequence ( ν . . . ν r ) , r = r ( J ( p )) , such that (1) f [ .ν ...ν r ] ( J ( p )) ⊃ J ( p ) , (2) (cid:12)(cid:12)(cid:0) f [ .ν ...ν r ] ) (cid:48) ( x ) (cid:12)(cid:12) > for all x ∈ J ( p ) , and (3) O − ( J ( p )) = (0 , . TRUCTURE OF THE SPACE OF ERGODIC MEASURES 21
A.1.3.
Almost forward and backward minimality.
Corollary A.6 (Almost minimality) . For every x ∈ (0 , the sets O + ( x ) and O − ( x ) are both dense in [0 , .Proof. Fix any x ∈ (0 , p ∈ (0 ,
1) and anarbitrarily small neighborhood J ( p ) of it. By Lemma A.5 item (3) we have that x ∈ O − ( J ( p )) and hence J ( p ) ∩ O + ( x ) (cid:54) = ∅ . The proof of the forward minimalityis analogous using Lemma A.3 item (3). (cid:3) A.2.
Transitive dynamics. Homoclinic relations.
To prove that F is topo-logically transitive, we use the notion of a homoclinic class adapted to the skewproduct setting. For that we need some definitions. Observe that if P = ( ξ, p ) is aperiodic point of F of period k + 1, then ξ = ( ξ . . . ξ k ) Z and f [ ξ ... ξ k ] ( p ) = p . Notethat 1 k + 1 log | f (cid:48) [ ξ ... ξ k ] ( p ) | = χ ( P ) . If χ ( P ) (cid:54) = 0 then we call P (fiber) hyperbolic . There are two types of such points: if χ ( P ) > P (fiber) expanding , otherwise χ ( P ) < P (fiber)contracting . We denote by Per hyp ( F ) the set of all fiber hyperbolic periodic pointsof F and by Per > ( F ) and Per < ( F ) the (fiber) expanding and (fiber) contractingperiodic points, respectively. Clearly, Per hyp ( F ) = Per > ( F ) ∪ Per < ( F ). Givena fiber hyperbolic periodic point P we consider the stable and unstable sets of itsorbit O ( P ) denoted by W s (cid:0) O ( P ) , F (cid:1) and W u (cid:0) O ( P ) , F (cid:1) .Two periodic points P , P ∈ Per hyp ( F ) of the same type of hyperbolicity (thatis, either both points are fiber expanding or both are fiber contracting) with differentorbits O ( P ) and O ( P ) are homoclinically related if the stable and unstable sets oftheir orbits intersect cyclically: W s (cid:0) O ( P ) , F (cid:1) ∩ W u (cid:0) O ( P ) , F (cid:1) (cid:54) = ∅ and W u (cid:0) O ( P ) , F (cid:1) ∩ W s (cid:0) O ( P ) , F (cid:1) (cid:54) = ∅ . A point X (cid:54)∈ O ( P ) is a homoclinic point of P if X ∈ W s (cid:0) O ( P ) , F (cid:1) ∩ W u (cid:0) O ( P ) , F (cid:1) . Observe that our definitions do not involved any transversality assumption (indeedin our context of a skew product such a transversality does not make sense, see also[6, Section 3] for more details on homoclinic relations for skew products). However,due to the fact that the maps f and f have no critical points, the homoclinicpoints behave as the transverse ones in the differentiable setting.The homoclinic class H ( P ) of a fiber hyperbolic periodic point P is the closureof the orbits of the periodic points of the same type as P which are homoclinicallyrelated to P . As in the differentiable setting, the set H ( P ) coincides with theclosure of the homoclinic points of P . This set is transitive.Let us introduce some notation. For (cid:63) ∈ { < , > } , definePer core ,(cid:63) ( F ) def = Per (cid:63) ( F ) ∩ Λ core and Per ex ,(cid:63) ( F ) def = Per (cid:63) ( F ) ∩ Λ ex . Proposition A.7 (Homoclinic relations) . Let (cid:63) ∈ { < , > } . (1) Every pair of points R , R ∈ Per core ,(cid:63) ( F ) are homoclinically related. (2) Every pair of points R , R ∈ Per ex ,(cid:63) ( F ) are homoclinically related and theirhomoclinic classes coincide with Λ ex . (3) No point in
Per core ,(cid:63) ( F ) is homoclinically related to any point of Per ex ,(cid:63) ( F ) . (4) The set Σ × [0 , is the homoclinic class of any R ∈ Per core ,(cid:63) ( F ) . As aconsequence, the set Per core ,(cid:63) ( F ) is dense in Σ × [0 , .Proof. As the arguments in this proof are similar to the ones in [7, Section 2] we willljust sketch them. We prove (1) for fiber contracting periodic points only. Fix P =(( ξ . . . ξ k ) Z , p ) and R = (( η . . . η (cid:96) ) Z , r ), r, p ∈ (0 , I ( r )containing r such that I ( r ) ⊂ W s loc ( r, f [ η ...η (cid:96) ] ). By Corollary A.6, there is ρ . . . ρ m such that f [ ρ ...ρ m ] ( p ) ∈ I ( r ). Take X = (( ξ . . . ξ k ) − N .ρ . . . ρ m ( η . . . η (cid:96) ) N , p ). Byconstruction, X ∈ W u ( O ( P ) , F ) ∩ W u ( O ( R ) , F ). Reversing the roles of P and R we obtain a point in W u ( O ( R ) , F ) ∩ W s ( O ( P ) , F ), proving that P and R arehomoclinically related.The proof of (2) is an immediate consequence of the fact that F Λ ex can be seenas an “abstract horseshoe”.To prove item (3) note that O ± (0) , O ± (1) ⊂ { , } . This prevents any periodicpoint with fiber coordinate 0 or 1 to be homoclinically related to points in Λ core .We prove item (4) for expanding points only. Fix an expanding periodic point R ∈ Λ core . Consider any point X = ( ξ, x ), x ∈ (0 , m ≥ and δ > ξ − m . . . ξ m ) and I ( δ ) = ( x (cid:48) − δ, x (cid:48) + δ ), where x (cid:48) = f [ ξ − m ...ξ − . ] ( x ).Consider now I (cid:48) ( δ ) = f [ ξ − m ...ξ ...ξ m ] ( I ( δ )). Applying Lemma A.3 to I ( δ ) (cid:48) , we get η . . . η r such that f [ η ...η r ] ( I ( δ ) (cid:48) ) covers I ( δ ) and f [ ξ − m ...ξ ...ξ m η ...η r ] is expandingon I ( δ ) (cid:48) . This provides an expanding periodic point P δ,m close to X . Note that P δ,m → X as δ → m → ∞ . By item (1) this point is homoclinically relatedto R . As a consequence, we have X ∈ H ( R, F ). (cid:3) A.3.
The parabolic case.
In this section we will prove Theorem 2.9. For that wesee how the constructions above can be modified to construct examples where theset Λ has an ergodic measure of maximal entropy which is nonhyperbolic. For thiswe modify the map f satisfying conditions (H1), (H3), and (H4) to get a new map˜ f such that the points 0 and 1 are parabolic (0 is repelling and 1 is attracting)and consider the skew product (cid:101) F associated to ˜ f and f ( x ) = x − Proof of Theorem 2.9.
We start with a map f satisfying hypotheses (H1), (H3),and (H4) and consider exactly as in Appendix A.1 the fundamental domains I ( (cid:15) ) =[ ε, f ( ε )] and I ( (cid:15) ) = [1 − ε, f (1 − ε )) (for small ε >
0) and the natural number N ( ε ) with f N ( ε )0 ( I ( ε )) = I ( ε ). Note that the estimate in (A.1) holds. We definefor a subinterval H of f − ( I ( ε )) ∪ I ( ε ) the number N ( H ) ∈ { N ( ε ) , N ( ε ) + 1 } .Similarly we define M ( H ) ∈ { , . . . , M } ( M independent of H ).Assume not that f satisfies (H2’). Let a def = min { f − ( ε ) , − f N ( ε )+10 ( ε ) } , b def = f N ( ε )+10 ( ε ) . Note also that the definition of the expanding successors only involves iterates inthe set [ f − ( ε ) , f N ( ε )+10 ( ε )] ∪ f (cid:0) [ f − ( ε ) , f N ( ε )+1 ( ε )] (cid:1) = [ a , b ] ⊂ [ δ, − δ ] , for some small δ >
0. We now fix very small τ (cid:28) δ and consider a new map ˜ f such that(i) ˜ f = f in [ δ, − δ ],(ii) ( ˜ f ) (cid:48) (0) = 1 and 0 is repelling,(iii) ( ˜ f ) (cid:48) (1) = 1 and 1 is attracting, ˜ f has no fixed points in (0 , TRUCTURE OF THE SPACE OF ERGODIC MEASURES 23 see Figure 5. Note that for this new map ˜ f we can define expanding returns in I ( ε ) as before. Note also that every point x ∈ (0 ,
1) has some forward and somebackward iterate in I ( ε ) by the IFS associated to { ˜ f , f } (here we use that 0 isrepelling, 1 is attracting and ˜ f has no fixed points in (0 , { ˜ f , f } . This concludes thepart corresponding to the expanding itineraries. f ˜ f f δ − δ Figure 5.
Fiber maps: The parabolic caseIt remains to check that the arguments corresponding to the contracting itinerariesin Appendix A.1.2 also hold. Recall the definition of the point c in hypothesis(H3), see also (A.2). Let g = f − and g = f − . Note that if δ is small wecan assume that δ < g ( c ) < − g ( c )] < − δ . Note that for closed intervals H ⊂ [ c, f ( c )] the definition of their expanding successor only involves iterations inthe set [ g ( c ) , − g ( c )]. Since in this interval ˜ g = g we obtain versions of Lem-mas A.4 and A.5 for the IFS associated to { ˜ g , g } . In the same way we recoverCorollary A.6 for the IFS associated to { ˜ f , f } .We can now consider the skew product (cid:101) F associated to ˜ f , f and prove Propo-sition A.7 for ˜ F , obtaining, in particular, that the set Σ × [0 ,
1] is a homoclinicclass of ˜ F . By Theorem 2.7 the unique measure µ exmax of maximal entropy log 2 in M erg (Λ ex ) is nonhyperbolic. (cid:3) A.4.
Nontransitive case with a unique measure of maximal entropy.
Inthis section we prove Theorem 2.10 by presenting an example which is not transitiveand for which there exists just one measure of maximal entropy, which is nonhy-perbolic. This measure is supported on Λ ex and there is no measure of maximalentropy in Λ core . Proof of Theorem 2.10.
Let us consider a C orientation preserving homeomor-phism φ : R → (0 ,
1) satisfying(A.3) φ ( y ) = 1 − φ ( − y )and lim y →∞ φ (cid:48) ( y + 1) φ (cid:48) ( y ) def = 1 ∈ (0 , ∞ ) . (For example, φ ( y ) = π arctan y + satisfies the conditions above.) Now define f : [0 , → [0 , f ( x ) def = φ ( φ − ( x ) + 1) if x ∈ (0 , , x = 0 , x = 1 , and f : [0 , → [0 ,
1] by f ( x ) = 1 − x . Note that f is a C map which satisfies(A.4) f (cid:48) (0) = 1 = f (cid:48) (1) . Moreover, note that f ◦ φ = φ ◦ θ , where θ : R → R denotes the unit translationon the real line defined by θ ( y ) def = y + 1. The symmetry assumption (A.3) meansthat f ◦ φ = φ ◦ γ where γ : R → R is defined by γ ( y ) def = − y . Observe that φ ( − φ − ( x )) = 1 − x implies(A.5) f f = f f − , that is, f is conjugate to its inverse by f . This provides us fiber maps f , f satisfying item 1. and 2. in the theorem. Proposition A.8. F is not topologically transitive.Proof. It suffices to prove that for any x ∈ [0 ,
1] the set O + ( x ) def = { f [ ω ... ω n ] ( x ) : ω i ∈{ , } , i ∈ { , . . . , n }} is not dense in [0 , x ∈ { , } . Thus, inwhat follows, we let x ∈ (0 , n ∈ N , consider some finite sequence ( ω . . . ω n ) ∈ { , } n . First, recallthat f ( x ) = x we can replace this sequence by one in which we eliminated allblocks 11. Hence, without loss of generality, we can assume that the sequence( ω . . . ω n ) does not contain two consecutive 1s. Assume first that this sequencecontains an even number of symbols 1, that is, we can divide it into a finite numberof pieces of the form 0 k (cid:96) m . By (A.5), f [0 k (cid:96) m ] ( x ) = f k + m − (cid:96) ( x ). Hence, wehave f [ ω ...ω n ] ( x ) = f j ( x ) for some integer j . Similarly, if this sequence containsan odd number of symbols 1, we can write ω . . . ω n = 0 k ω (cid:48) with ω (cid:48) containingan even number of symbols 1. As f [0 k ( x ) = f − k (1 − x ), applying the previousargument, we have that f [ ω ...ω n ] ( x ) = f j (1 − x ) for some integer j .This proves that the full forward orbit of x by the IFS, O + ( x ), is containedin two sets { f j ( x ) : j ∈ Z } and { f j (1 − x ) : j ∈ Z } , each of which has just twoaccumulation points: 0 and 1. This proves the proposition. (cid:3) Proposition A.9. F has a unique measure of maximal entropy, which is nonhy-perbolic.Proof. By Theorem 2.7, the measure µ exmax is unique and nonhyperbolic by ourchoice (A.4). Hence, it is enough to prove that there cannot exist a measure ofmaximal entropy supported on Σ × (0 , µ . Itsprojection to Σ must be the measure (cid:98) ν max of maximal entropy for σ : Σ → Σ , thatis, the (1 / , / ν = (cid:98) ν max .The measure µ admits a disintegration, that is, there exists a family { µ ξ : ξ ∈ Σ } of probabilities such that ξ (cid:55)→ µ ξ is measurable and every µ ξ is supported on { ξ } × (0 ,
1) and satisfies µ ( E ) = (cid:90) µ ξ ( E ) dν ( ξ ) TRUCTURE OF THE SPACE OF ERGODIC MEASURES 25 for any measurable set E . With a slight lack of precision we will consider each µ ξ as a measure on (0 , µ ξ under our dynamics, we will use the followingresult whose proof we postpone. Recall our notation f nω def = f ω n − ◦ . . . ◦ f ω . Lemma A.10.
For ν -almost every ω ∈ Σ , for every ε > and for every measure µ supported on (0 , we have lim n →∞ n n (cid:88) i =1 ( f iω ) ∗ µ (( ε, − ε )) = 0 . We postpone the proof of the above lemma to the following subsection. Assumingthat the above lemma was proven, we can now complete the proof of the theorem.In particular, we can, for a ν -generic ω , apply Lemma A.10 to the measure µ = µ ω .Thus, recalling that µ is F -invariant, for every n ≥ µ (Σ × ( ε, − ε )) = F ∗ µ (Σ × ( ε, − ε )) = 1 n n (cid:88) i =1 ( F i ) ∗ µ (Σ × ( ε, − ε ))= 1 n n (cid:88) i =1 (cid:90) ( f iω ) ∗ ( µ ω )(( ε, − ε )) dν ( ω )= (cid:90) n n (cid:88) i =1 ( f iω ) ∗ ( µ ω )(( ε, − ε )) dν ( ω ) . Now, by Lemma A.10, taking the limit n → ∞ and apply the dominated conver-gence theorem, we obtain µ (Σ × ( ε, − ε )) = 0. As ε is arbitrary, this implies µ (Σ × (0 , µ was supportedon Σ × (0 , (cid:3) This proves the theorem. (cid:3)
A.4.1.
Random walks – Proof of Lemma A.10.
To proof Lemma A.10 we needto introduce several auxiliary objects in order to reduce it to well-known results.Heuristically, a ν -typical ω ∈ Σ can be treated as a random process with nomemory, and then the dynamics generated by f iω is given by a certain randomwalk. The result we will prove below is a version of a well-known statement that arandom walk does not stay in any bounded region.For what we study below, we will consider the one-sided shift space Σ +2 only andby a slight abuse of notation continue to denote the (1 / , / ν . We consider a ν -typical ω = ( ω ω . . . ) ∈ Σ +2 and interpret the values ω i asrandom variables, with ν giving their joint distribution. That is, each ω i takes val-ues 0 and 1 with probabilities 1/2 each, independently of any other ω j ’s. Denoting ω n = ω . . . ω n , let Ω n be the σ -algebra generated by the cylinders [ ω ] , . . . , [ ω n ].We first introduce the following auxiliary IFS of maps g , g . Let Ω = (0 , ×{ +1 , − } and define g , g : Ω → Ω by g ( x, +1) def = ( f ( x ) , +1) , g ( x, − def = ( f − ( x ) , − , and g ( x, +1) def = ( x, − , g ( x, − def = ( x, +1) . Consider the projections π , π : Ω → (0 ,
1) defined as follows π ( x, k ) def = (cid:40) x if k = +1 , − x if k = − , π ( x, k ) def = x. One immediately checks that π ◦ g i = f i ◦ π , i = 0 ,
1, that is, the original IFS { f , f } on (0 ,
1) is a factor of the IFS { g , g } on Ω under π . Note that, given x ∈ (0 , n ≥ π ◦ g [ ω n ] )( x, +1) = f [ ω n ] ( x ) , that is, we can consider (0 ,
1) as (0 , × { +1 } , apply the maps g i instead of f i andthen project back the results by π and get the same result as if we applied maps f i and never left (0 , f and (A.5).To model the claimed random walk, we consider now (0 , × { +1 } instead of(0 ,
1) and apply the maps g i instead of f i and then project the results by φ − ◦ π .That is, let R i ( x ) = R i ( x, ω ) def = ( φ − ◦ π ◦ g [ ω i ] )( x, +1) . This defines a random walk on R . The main aim of this section is to proof thefollowing result. Lemma A.11.
For every probability measure µ on R and for any bounded A ⊂ R , ν -almost surely we have lim n →∞ n n (cid:88) i =1 ( R i ) ∗ µ ( A ) = 0 . The above result now will provide the
Proof of Lemma A.10.
Note that, for any ε > π − (( ε, − ε )) = π − (( ε, − ε )) = ( ε, − ε ) × { , } . Hence, by (A.6) for A = φ − (( ε, − ε )) we have( f [ ω i ] ) ∗ µ ( ε, − ε ) = ( π ◦ g [ ω i ] ) ∗ µ ( ε, − ε ) = ( R i ) ∗ µ ( A ) . Applying now Lemma A.11 implies the lemma, proving Lemma A.10. (cid:3)
A.4.2.
Random walks – Analysis of the random walk R i . This random process hasa complicated behavior. We will introduce a sequence of simpler auxiliary randomprocesses which will help to prove Lemma A.11.Without loss of generality, we assume ω = 0. Given ω , let n i , i ≥
0, enumeratethe positions at which in the sequence ω there appears the symbol 0. With thisnotation, we have the following relation R n i ( x, ω ) = φ − ◦ π ◦ g [ ω ni ] )( x, +1)= (cid:40) ( φ − ◦ π ◦ g [ ω ni − ] )( x, +1) + 1 if { k ∈ { n i − , . . . , n i } : ω k = 1 } is even , ( φ − ◦ π ◦ g [ ω ni − ] )( x, +1) − { k ∈ { n i − , . . . , n i } : ω k = 1 } is odd . An elementary calculation shows that the number of 1’s between any two con-secutive 0’s is even with probability 2 / /
3. That is,the random variable n i − n i − takes an even value with probability 1/3 and anodd value with probability 2/3, moreover this random variable is independent from TRUCTURE OF THE SPACE OF ERGODIC MEASURES 27 Ω n i − . Considering then the subsequence ( n i ) i , we will now pass from the randomwalk R i to the following “induced” walk S i , which is defined by S i ( x ) def = ( φ − ◦ π ◦ g [ ω ni ] )( x, +1) . The latter is a random walk on the real line composed by translations S i ( x ) = (cid:40) S i − ( x ) + 1 if { k ∈ { n i − , . . . , n i } : ω k = 1 } is even ,S i − ( x ) − { k ∈ { n i − , . . . , n i } : ω k = 1 } is odd , where each step being independently and identically distributed: in the same di-rection as the previous one with probability 2/3 and in the opposite direction withprobability 1/3 (with the convention that the ‘zeroth step’ was in the positive di-rection).Since S i does not encode explicitly the information in which direction the walkis moving (it does not carry the second coordinate), we will instead consider thefollowing auxiliary walk. Let U i be a random walk on R × {− , +1 } given by U i ( x, j ) def = (cid:40) ( x + j, j ) with probability 2 / , ( x − j, − j ) with probability 1 / . which is just S i adding the information about the direction of the last step: there ex-ists a measure preserving isomorphism under which the first coordinate of U i ( x, +1)is equal to S i ( x ).Recall that we want to show that the evolution of a measure under the applicationof the fiber maps of the IFS is eventually moving to the boundary of (0 , ±∞ for the walk lifted by φ − to R . For that reason, let us now consideran “induced” walk that only looks at times immediately after we moved in positivedirection. Let V + i denote the random walk which is the first return of U i to R ×{ +1 } .That is, V + i ( x ) def = (cid:40) x + 1 with probability 2 / ,x − k, k = 0 , , . . . with probability 2 k / k +2 . At last we got an usual random walk. Note that the walk V + i is recurrent. In-deed, an elementary calculation gives that its expected displacement is zero, thatis E ( V + i ( x ) − x ) = 0. Given A ⊂ R , let us define A ( y ) = 1 if y ∈ A and A ( y ) = 0otherwise.The Chung and Erd¨os Theorem [4] immediately implies Lemma A.12.
For any bounded A ⊂ R , ν -almost surely lim n →∞ n n (cid:88) i =1 A ( V + i (0)) = 0 . Proof.
By [4, Theorem 3.1], for any a, b ∈ Z we havelim i →∞ P ( V + i (0) = a ) P ( V + i (0) = b ) = 1 . Hence, for any bounded set A ⊂ Z for every a ∈ A we havelim i →∞ P ( V + i (0) = a ) ≤ | A | , where | A | denotes the cardinality of A and hencelim i →∞ P ( V + i (0) ∈ A ) = 0 . In particular, for every a ∈ Z we have(A.7) lim i →∞ P ( V + i (0) = a ) = 0 . Let T k denote the k th return time of V + i (0) to 0. Note that T k equals the sumof k independent copies of T . Claim 1. E ( T ) = ∞ .Proof. By contradiction, assume that E ( T ) would be finite. Hence, by the stronglaw of large numbers, almost surely we would have lim n →∞ n T n = E ( T ). Hence,by Egorov’s theorem, for any ε ∈ (0 ,
1) there would exist N ≥ k ≥ N with probability at least 1 − ε we would have k T k ≤ E ( T ) + ε . This wouldimply k ( E ( T )+ ε ) (cid:88) i =1 P ( V + i (0) = 0) = (cid:90) dω k ( E ( T )+ ε ) (cid:88) i =1 { } ( V + i (0)) ≥ (1 − ε ) k ( E ( T ) + ε ) , which would imply lim sup i →∞ P ( V + i (0) = 0) >
0. Contradiction with (A.7). (cid:3)
Analogously, given a ∈ Z , let T a denote the first hitting time of V + i (0) at a .Observe that the return times of V + i + T a (0) to a have the same distribution as thereturn times of V + i (0) to 0. Hence, if T ak denotes the k th return time of V + i (0) to a ,then analogously to the above claim we conclude E ( T a ) = ∞ . Hence, by the stronglaw of large numbers almost surely we have k T ak → ∞ , which in turn implies0 = lim sup k →∞ kT ak = lim sup n →∞ n n (cid:88) i =1 { a } ( V + i (0)) . Writing now A = (cid:80) a ∈ A a , we obtainlim n →∞ n n (cid:88) i =1 A ( V + i (0)) = 0almost surely. This proves the lemma. (cid:3) Looking at the random walk U i , we get the corresponding statement for the set A × { +1 } , that is lim n →∞ n n (cid:88) i =1 A ×{ +1 } ( U i (0 , +1)) = 0 . The proof for A × {− } is similar, only instead of V + i we need to take the firstreturn to R × {− } . Recalling that the projection of U i to the first coordinate isjust S i , we obtain the following corollary. Corollary A.13.
For any bounded A ⊂ R , ν -almost surely lim n →∞ n n (cid:88) i =1 A ( S i (0)) = 0 . TRUCTURE OF THE SPACE OF ERGODIC MEASURES 29
Recall that S i takes into account only the steps of the initial walk when thesymbol 0 appeared. Vaguely speaking, S i takes only into account whether n i − n i − is even or odd. Let Ω (cid:48) be the σ -algebra generated by S i . We will consider thefollowing auxiliary random variable d i defined by d i def = (cid:40) n i − n i − if n i − n i − is odd, n i − n i − − n i − n i − is even.Below we will argue that d i is independent of Ω (cid:48) . The remaining information(needed to recover Ω = (cid:84) n Ω n ) is in exact values of n i − n i − . Knowing ( S i ) ni =1 and ( d i ) ni =1 , we can recover ( R i ) ni =1 . Note that d i being independent from Ω (cid:48) meansthat we can decompose the measure ν (the (1 / , / , Ω))as µ s × µ d , where µ s is the distribution of ( S i ) and µ d is the joint distribution ofthe i.i.d. random variables d i . Hence then we can conclude that if for µ s -almostevery realization ( S i ) for µ d -almost every realization ( d i ) an event holds, then itholds for ν -almost every ω . Lemma A.14.
The random variable ( d i ) is independent of Ω (cid:48) and has finite ex-pectation E ( d i ) .Proof. Whether n i − n i − is even or odd, d i always has the same distribution P ( d i = 2 k + 1 | n i − n i − even) = P ( n i − n i − = 2 k + 1 | n i − n i − odd) = 34 k +1 , which follows from an elementary calculation. Hence, in particular, the expectedvalue of d i is finite. This proves the lemma. (cid:3) With the above, we now return to the random walk R i . Corollary A.15.
For any bounded A ⊂ R , ν -almost surely, lim n →∞ n n (cid:88) i =1 A ( R i (0)) = 0 . Proof.
Note that R n (0) = R n i − (0) = S i − (0) for every n ∈ { n i − , . . . , n i − } .Hence, we have 1 n i n i − (cid:88) k =0 A ( R k (0)) = 1 n i i − (cid:88) (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0)) . In particular, to show the claim, it is enough to consider the specific subsequencefrom above proving that0 = lim i →∞ n i n i − (cid:88) k =0 A ( R k (0)) = lim i →∞ n i i − (cid:88) (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0)) . Thus, we havelim sup n →∞ n n − (cid:88) k =0 A ( R k (0)) = lim sup i →∞ n i n i − (cid:88) k =0 A ( R k (0))= lim sup i →∞ n i i − (cid:88) (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0))= lim sup i →∞ in i · (cid:80) i − (cid:96) =0 A ( S (cid:96) (0)) i · (cid:80) i − (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0)) (cid:80) i − (cid:96) =0 A ( S (cid:96) (0)) ≤ lim sup i →∞ in i · lim sup i →∞ (cid:80) i − (cid:96) =0 A ( S (cid:96) (0)) i · lim sup i →∞ (cid:80) i − (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0)) (cid:80) i − (cid:96) =0 A ( S (cid:96) (0))=: L · L · L . Claim.
Almost surely, we have that L and L are finite and L = 0 . With this claim and also using Lemma A.14, obtain that ν -almost surely we have L · L · L = 0 and we conclude that ν -almost surelylim sup n →∞ n n − (cid:88) k =0 A ( R k (0)) = 0 , proving the corollary. What remains is to prove the claim.To estimate these latter terms, first observe that almost surely n i i = ( n i − n i − ) + . . . + ( n − i → E ( n i − n i − ) . Hence E ( n i − n i − ) = (cid:88) i ≥ (cid:16) d i P ( n i − n i − even) + ( d i + 1) P ( n i − n i − odd) (cid:17) ≥ E ( d i )and therefore L ≤ ( E ( d i )) − < ∞ . Moreover, this calculation also gives(A.8) E ( n i − n i − ) ≤ E ( d i ) + 1 . By Corollary A.13, we have L = 0.To estimate L , first observe that, as the expected value of d i is finite, if we fix( S i ) then, by the law of large numbers, almost surely the average of { d i : S i (0) ∈ A } and the average of { d i : S i (0) / ∈ A } converge to the same limit E ( d i ). Thus, almostsurely lim i →∞ (cid:80) i − (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) A ( S (cid:96) (0)) (cid:80) i − (cid:96) =0 A ( S (cid:96) (0)) = lim i →∞ (cid:80) i − (cid:96) =0 ( n (cid:96) +1 − n (cid:96) ) i ≤ E ( d i ) + 1 , where the latter follows from (A.8). Thus, L is finite. This proves the claim. (cid:3) The statement for random walk starting from 0 can be generalized to any startingdistribution which allows us to finally prove Lemma A.11.
Proof of Lemma A.11.
Fix µ . For any ε > N such that µ ([ − N, N ]) > − ε . As φ − ◦ π ◦ g i is a translation, if R n (0) / ∈ B N ( A ) then( R n ) ∗ µ ( A ) < ε . Hence,lim sup n →∞ n ( R n ) ∗ µ ( A ) ≤ ε + lim n →∞ n B N ( A ) ( R n (0)) = ε. TRUCTURE OF THE SPACE OF ERGODIC MEASURES 31
Passing with ε to 0 ends the proof. (cid:3) References [1] Jairo Bochi, Christian Bonatti, and Lorenzo J. D´ıaz. Robust criterion for the existence ofnonhyperbolic ergodic measures.
Comm. Math. Phys. , 344(3):751–795, 2016.[2] Jairo Bochi, Christian Bonatti, and Katrin Gelfert. Dominated Pesin theory: Convex sum ofhyperbolic measures.
To appear in: Israel J. Math. [3] Christian Bonatti, Lorenzo J. D´ıaz, and Marcelo Viana.
Dynamics beyond uniform hyperbol-icity , volume 102 of
Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2005.A global geometric and probabilistic perspective, Mathematical Physics, III.[4] Kai Lai Chung and Paul Erd¨os. Probability limit theorems assuming only the first moment.I.
Mem. Amer. Math. Soc., , No. 6:19, 1951.[5] Sylvain Crovisier. Partial hyperbolicity far from homoclinic bifurcations.
Adv. Math. ,226(1):673–726, 2011.[6] Lorenzo J. D´ıaz, Salete Esteves, and Jorge Rocha. Skew product cycles with rich dynamics:from totally non-hyperbolic dynamics to fully prevalent hyperbolicity.
Dyn. Syst. , 31(1):1–40,2016.[7] Lorenzo J. D´ıaz and Katrin Gelfert. Porcupine-like horseshoes: transitivity, Lyapunov spec-trum, and phase transitions.
Fund. Math. , 216(1):55–100, 2012.[8] Lorenzo J. D´ıaz, Katrin Gelfert, and Micha(cid:32)l Rams. Almost complete Lyapunov spectrum instep skew-products.
Dyn. Syst. , 28(1):76–110, 2013.[9] Lorenzo J. D´ıaz, Katrin Gelfert, and Micha(cid:32)l Rams. Abundant rich phase transitions in step-skew products.
Nonlinearity , 27(9):2255–2280, 2014.[10] Lorenzo J. D´ıaz, Katrin Gelfert, and Micha(cid:32)l Rams. Nonhyperbolic step skew-products: er-godic approximation.
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire , 34(6):1561–1598, 2017.[11] Lorenzo J. D´ıaz, Katrin Gelfert, and Micha(cid:32)l Rams. Topological and ergodic aspects of par-tially hyperbolic diffeomorphisms and nonhyperbolic step skew products.
Tr. Mat. Inst.Steklova , 297(Poryadok i Khaos v Dinamicheskikh Sistemakh):113–132, 2017.[12] Lorenzo J. D´ıaz, Vanderlei Horita, Isabel Rios, and Martin Sambarino. Destroying horseshoesvia heterodimensional cycles: generating bifurcations inside homoclinic classes.
Ergodic The-ory Dynam. Systems , 29(2):433–474, 2009.[13] Lorenzo J. D´ıaz and Tiane Marcarini. Generation of spines in porcupine-like horseshoes.
Nonlinearity , 28(11):4249–4279, 2015.[14] Andrea Gaunersdorfer. Time averages for heteroclinic attractors.
SIAM J. Appl. Math. ,52(5):1476–1489, 1992.[15] Katrin Gelfert and Dominik Kwietniak. On density of ergodic measures and generic points.
To appear in: Ergodic Theory Dynam. Systems .[16] Anton Gorodetski and Yakov Pesin. Path connectedness and entropy density of the space ofhyperbolic ergodic measures. In
Modern theory of dynamical systems , volume 692 of
Contemp.Math. , pages 111–121. Amer. Math. Soc., Providence, RI, 2017.[17] Yulij S. Ilyashenko and Ivan Shilin. Attractors and skew products. In
Modern theory ofdynamical systems , volume 692 of
Contemp. Math. , pages 155–175. Amer. Math. Soc., Prov-idence, RI, 2017.[18] Ittai Kan. Open sets of diffeomorphisms having two attractors, each with an everywheredense basin.
Bull. Amer. Math. Soc. (N.S.) , 31(1):68–74, 1994.[19] Anatole Katok and Boris Hasselblatt.
Introduction to the modern theory of dynamical sys-tems , volume 54 of
Encyclopedia of Mathematics and its Applications . Cambridge UniversityPress, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza.[20] Franccois Ledrappier and Peter Walters. A relativised variational principle for continuoustransformations.
J. London Math. Soc. (2) , 16(3):568–576, 1977.[21] Renaud Leplaideur, Krerley Oliveira, and Isabel Rios. Equilibrium states for partially hyper-bolic horseshoes.
Ergodic Theory Dynam. Systems , 31(1):179–195, 2011.[22] Joram Lindenstrauss, Gunnar H. Olsen, and Yaki Sternfeld. The Poulsen simplex.
Ann. Inst.Fourier (Grenoble) , 28(1):vi, 91–114, 1978.[23] Sheldon E. Newhouse. Lectures on dynamical systems. In
Dynamical systems (Bressanone,1978) , pages 209–312. Liguori, Naples, 1980. [24] Vanessa Ramos and Jaqueline Siqueira. On equilibrium states for partially hyperbolic horse-shoes: uniqueness and statistical properties.
Bull. Braz. Math. Soc. (N.S.) , 48(3):347–375,2017.[25] Isabel Rios and Jaqueline Siqueira. On equilibrium states for partially hyperbolic horseshoes.
Ergodic Theory Dynam. Systems , 38(1):301–335, 2018.[26] Karl Sigmund. On dynamical systems with the specification property.
Trans. Amer. Math.Soc. , 190:285–299, 1974.[27] Karl Sigmund. On the connectedness of ergodic systems.
Manuscripta Math. , 22(1):27–32,1977.[28] Ali Tahzibi and Jiagang Yang. Strong hyperbolicity of ergodic measures with large entropy.
Preprint arXiv:1606.09429 , To appear in: Trans. Amer. Math. Soc. [29] Floris Takens. Heteroclinic attractors: time averages and moduli of topological conjugacy.
Bol. Soc. Brasil. Mat. (N.S.) , 25(1):107–120, 1994.[30] Peter Walters.
An introduction to ergodic theory , volume 79 of
Graduate Texts in Mathemat-ics . Springer-Verlag, New York-Berlin, 1982.
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