Skew-products with concave fiber maps: entropy and weak ∗ approximation of measures and bifurcation phenomena
SSKEW-PRODUCTS WITH CONCAVE FIBER MAPS:ENTROPY AND WEAK ∗ APPROXIMATION OF MEASURESANDBIFURCATION PHENOMENA
L. J. DÍAZ, K. GELFERT, AND M. RAMSA
BSTRACT . We consider skew-products with concave interval fiber maps over a certainsubshift. This subshift naturally occurs as the projection of those orbits that stay in agiven neighborhood. It gives rise to a new type of symbolic space which is (essentially)coded. The fiber maps have expanding and contracting regions. As a consequence, theskew-product dynamics has pairs of horseshoes of different type of hyperbolicity. In somecases, they dynamically interact due to the superimposed effects of the (fiber) contractionand expansion, leading to nonhyperbolic dynamics that is reflected on the ergodic level(existence of nonhyperbolic ergodic measures).We provide a description of the space of ergodic measures on the base as an entropy-dense Poulsen simplex. Those measures lift canonically to ergodic measures for the skew-product. We explain when and how the spaces of (fiber) contracting and expanding ergodicmeasures glue along the nonhyperbolic ones. A key step is the approximation (in theweak ∗ topology and in entropy) of nonhyperbolic measures by ergodic ones, obtainedonly by means of concavity without involving the standard “blending-like" arguments.The description of homoclinic relations is also a key instrument.We also see that these skew-products can be embedded in increasing entropy one-parameter family of diffeomorphisms which stretch from a heterodimensional cycle to acollision of homoclinic classes. We study associated bifurcation phenomena that involve ajump of the space of ergodic measures and, in some cases, also of entropy. C ONTENTS
1. Introduction 21.1. Context and motivations 21.2. Summary of results 52. Statement of results 72.1. Topological (and hyperbolic) dynamics 72.2. Structure of the measure space and Lyapunov exponents 102.3. Weak ∗ and entropy approximation of ergodic measures 132.4. Bifurcation settings and homoclinic scenarios 152.5. Organization of the paper 16 Mathematics Subject Classification.
Key words and phrases. entropy, concave maps, coded systems, disintegration of measures, heterodimen-sional cycles, homoclinic classes, hyperbolic and nonhyperbolic ergodic measures, Lyapunov exponents, Poulsensimplex, skew-product, variational principle.This research has been supported [in part] by the Coordenação de Aperfeiçoamento de Pessoal de NívelSuperior - Brasil (CAPES) - Finance Code 001, by CNPq-grants and CNPq Projeto Universal (Brazil), by INCT-FAPERJ, and by National Science Centre grants 2014/13/B/ST1/01033 and 2019/33/B/ST1/00275 (Poland). Theauthors acknowledge the hospitality of IMPAN, IM-UFRJ, and PUC-Rio. The authors also thank D. Kwietniakfor helpful conversations. a r X i v : . [ m a t h . D S ] A p r L. J. DÍAZ, K. GELFERT, AND M. RAMS
3. Underlying structures: symbolic space and the IFS 163.1. Admissible compositions 163.2. Hyperbolic and parabolic periodic points 194. Coded systems 195. Underlying structures: Homoclinic classes 235.1. Homoclinic relations and classes 235.2. Proof of Proposition 5.1 245.3. Homoclinic relations for parabolic periodic points 286. Concave one-dimensional maps 296.1. Distortion control 296.2. Rescaling of moving intervals 306.3. Distance to fixed points 317. Density of periodic points – Proof of Theorem 2.6 327.1. Approximation by (hyperbolic) periodic points 327.2. Proof of Theorem 2.6 338. Partition of the spaces of ergodic measures – Proofs of Theorems 2.13 and2.15 358.1. Lifting measures 358.2. Projection, disintegration, and twin-measures 368.3. “Distance" between twin-measures 378.4. Hyperbolicity of measures – Proof of Theorem 2.13 378.5. Proof of Theorem 2.15 408.6. Frequencies 409. Accumulations of ergodic measures in M (Γ) : Proof of Theorem 2.19 419.1. Periodic approximation of ergodic measures 419.2. Skeletons 439.3. Weak ∗ and entropy approximation of ergodic measures: Proof of Theorem2.22 469.4. Proof of Theorem 2.19 519.5. Weak ∗ and entropy approximation of linear combinations of hyperbolicergodic measures with the same type of hyperbolicity 5110. Entropy-dense Poulsen structure of M (Σ) : Proof of Theorem 2.11 5211. Bifurcation exit scenarios 5311.1. Formal setting 5411.2. Bifurcation at t h : heterodimensional cycles 5611.3. Bifurcation at t c : collisions of sets 5612. Discussion: Homoclinic scenarios beyond concavity 59Appendix A. Wasserstein distance 61References 611. I NTRODUCTION
Context and motivations.
This paper continues the project for understanding thestructure of the space of measures and entropies and for developing a thermodynamicformalism in systems with intermingled types of hyperbolicity, zero being in the inte-rior of the set of possible Lyapunov exponents. Our main focus is on partially hyperbolic
KEW-PRODUCTS WITH CONCAVE FIBER MAPS 3 systems, starting with a simplified but still very rich and inspiring setting of step skew-products with “hyperbolic base” and one-dimensional fibers. In particular, hyperbolicityis only determined by the hyperbolicity of the dynamics in the fibers. In previous works[21, 23], we consider those problems in the case when the fibers are circles and the fibermaps satisfy some natural topological assumptions (minimality, existence of a pair of con-tracting/expanding one-dimensional blenders, accessibility), moreover some arguments arebased upon synchronisation. Here we continue this study and propose a new approach us-ing essentially only concavity in a context where none of those ingredients is a priori available.Looking to the dynamics in the fibers, according to the sign of the fiber Lyapunov expo-nents, there is a natural partition of the space of ergodic measures into three parts: negative,zero, and positive exponent. We describe the topological structure of the measure spaceand investigate, in particular, how these “three pieces" are located. Here the existence, po-sition, and properties of measures with zero exponents, so-called nonhyperbolic measures,play a key role. This type of study was initiated with the pioneering work of Sigmund [48]assuming the specification property (a property essentially associated to hyperbolicity).We point out that our analysis does not invoke specification-like properties, which indeedare in no way available in our setting. The study of the space of measures in the spirit ofSigmund in nonhyperbolic settings has recently drawn attention, see for instance [11, 33]and the discussion in Section 2.3.Our starting point is a skew-product over a two-symbol shift space Σ = { , } Z withconcave interval fiber maps, a map f with two fixed points of different type of hyper-bolicity and a map f which forces some “interaction" between them. The two crucialhypotheses we always consider are the following:(H1) f : [0 , → [0 , and f : [ d, → [0 , , d ∈ (0 , , are differentiable increasingmaps such that f is onto, f ( d ) = 0 , f (cid:48) (0) > , f (cid:48) (1) ∈ (0 , , f ( x ) > x forevery x ∈ (0 , , f ( x ) < x for every x ∈ [ d, , and f (cid:48) (1) > .(H2) f (cid:48) is strictly decreasing and f (cid:48) is not increasing.(compare Figure 1). These hypotheses provide a setting which is on one hand very easyto describe and mixes contracting and expanding behaviors and on the other hand allowsto have a very rich dynamics. First note that f has two fixed points: being expandingand being contracting. The “interaction" above occurs, for instance, when the “forwardorbit" of the contracting point accumulates at the contracting point ; a very special casehappens when is contained in that orbit when a so-called heterodimensional cycle exists(see Remark 5.5 for further discussion).To define the actual skew-product we will study, consider any pair of strictly increasingdifferentiable extensions ˜ f , ˜ f : R → R of f , f to the real line and let(1.1) ˜ F : Σ × R → Σ × R , ( ξ, x ) (cid:55)→ ˜ F ( ξ, x ) def = ( σ ( ξ ) , ˜ f ξ ( x )) . We define the maximal invariant set Γ of ˜ F in Σ × [0 , ,(1.2) Γ def = (cid:92) n ∈ Z ˜ F n (Σ × [0 , . We study the topological dynamics and the ergodic properties of ˜ F on Γ . Observe that Γ is a locally maximal , compact, and ˜ F -invariant set whose dynamical properties do notdepend on the chosen extensions of f , f . We denote by F the restriction of ˜ F to Γ . Given a compact metric space X and a continuous map T : X → X , we say that a subset Y ⊂ X is locallymaximal if there exists an open neighborhood V ⊂ X of Y such that Y = (cid:84) n ∈ Z T n ( V ) . L. J. DÍAZ, K. GELFERT, AND M. RAMS
We will see that assumptions (H1) and (H2) imply the existence of pairs of horseshoesof fiber contracting and fiber expanding type, respectively. The dynamics of ˜ F on Γ can a priori exhibit two opposed behaviors: It may be hyperbolic or transitive (and there-fore not hyperbolic). In the latter case Γ contains pairs of horseshoes as above that areheteroclinically related in a cyclic way. f f d PQ F IGURE
1. Fiber maps f , f and the globally defined skew-product ˜ F To comment a little more our hypotheses, note that, after changing the reference inter-val accordingly, (H1) and the first hypothesis in (H2) persist under C perturbations (thesecond hypothesis in (H2) also persists if we would require that f (cid:48) is strictly decreasing).In some results we will also use the following slightly stronger version of (H2), assumingadditionally that f (cid:48) is strictly decreasing:(H2+) There exists M > such that for every x < y we have M − ( y − x ) ≤ log f (cid:48) i ( x ) − log f (cid:48) i ( y ) ≤ M ( y − x ) , i = 0 , Note that (H2+) also persists under C perturbations.We follow two a priori interdependent and complementary approaches. On the onehand, the system has associated a one-dimensional iterated function system (IFS). Thoughobserve that only certain concatenations are allowed (if x ∈ [0 , d ) then only f can beapplied) giving rise to a certain subshift (defined in (2.2)) which describes precisely thedynamics in between the saddles. This subshift, from the point of view of ergodic mea-sures, completely encodes ergodic and entropic properties of the skew-product. In general,this subshift is not of finite type and does not satisfy specification and there are no present-day tools available to study it. Although we are able to show that it is essentially coded andas such is, following the present day classification of shift spaces, just beyond the class oftransitive shifts with the specification property. As this coded shift appears naturally in a,to a certain extent, unusual context, it may serve as a good testing ground for the theory ofcoded systems. On the other hand, we can view Γ as a locally maximal invariant set of a“three-dimensional partially hyperbolic diffeomorphism with one-dimensional center”. Insuch a case, the study of the so-called homoclinic classes contained in Γ gives substantialdynamical information that we will explore.Let us discuss several motivation for studying the skew-products above. First, it canbe viewed as a plug in a semi-local analysis in a range of contexts, somewhat similar toso-called blenders (see, for instance, [10]). Here the expanding-and-covering property of The nonwandering set of ˜ F in Γ is the union of two hyperbolic sets. KEW-PRODUCTS WITH CONCAVE FIBER MAPS 5 blenders is replaced just by concavity. It provides a convenient model for higher dimen-sional dynamics where horseshoes of different type of hyperbolicity coexist or/and areintermingled, see for instance [17, 10, 26]. Also relevant is the fact that, for appropriatechoices (see, for instance, [25]) this plug models the bifurcation of the unfolding of het-erodimensional cycles (we will explore this in Section 11). In a way, the role played bythis pair is similar to the one of the quadratic family in the context of homoclinic bifurca-tions associated to tangencies, see [44, Chapter 6]. Such heterodimensional cycles appearin many contexts. We point out that an important context is, for instance, in the study ofLyapunov exponents of
SL(2 , R ) -matrix cocycles, following the approach in [23, Section11]. To see how such a plug appears, consider the projective action of two × real ma-trizes, one of them hyperbolic giving rise to a map similar to f and another one producing f . The case when there is some (admissible) concatenation sending to , when a het-erodimensional cycle occurs, is precisely the situation studied in the boundary case in [5,Theorem 4.1]. Our analysis includes such boundary situations, but also goes beyond.Let us also observe that the set Γ has a fractal nature that fits into the category of graph-and bony-like sets introduced in [39], that is, measurable (partially multi-valued) graphswhich are (with respect to certain measures) graphs from an ergodic point of view butcontain continua on a set of zero measure. This is intimately related with the (atomic)disintegration of ergodic measures that we also explore.Another related motivation is the point of view of IFSs. In general, the study of theirstatistical properties assumes some type of contraction or contraction-on-average. See forinstance [16] for an overview. For contracting-on-average IFS [7] establishes the unique-ness of the stationary measure. Examples which are beyond any contraction-like hypothe-ses are studied in [30] from the point of view of stationary measures (see also [1]). Notethat the IFS generated by { f , f } is genuinely non-contracting and [30, 1] can be seen asa boundary case of our setting, providing perhaps new perspectives.1.2. Summary of results.
In what follows, hyperbolicity refers only to expansion (resp.contraction) of the associated fiber maps and a closed F -invariant subset Λ ⊂ Γ is hy-perbolic of expanding type if there are constants C > and α > such that for every ( ξ, x ) ∈ Λ and for every n ≥ we have | ( f ξ n − ◦ . . . ◦ f ξ ) (cid:48) ( x ) | ≥ Ce nα . Hyperbolicity of contracting type is defined analogously considering backward iterates.A special case of a hyperbolic set is a hyperbolic (of either expanding or contractingtype) periodic orbit. In our setting, the orbit of a periodic point R = ( ξ, r ) = F n ( R ) iseither hyperbolic or parabolic , that is ( f ξ n − ◦ . . . ◦ f ξ ) (cid:48) ( r ) = 1 . There exist two designated fixed points for F ,(1.3) Q def = (0 Z , , P def = (0 Z , , which are hyperbolic of expanding and contracting type, respectively. The map f intro-duces an “interaction” between these two points and gives rise to rich topological dynamicsand ergodic properties.Given an ergodic probability measure µ (with respect to F ), its (fiber) Lyapunov expo-nent is χ ( µ ) def = (cid:90) log f (cid:48) ξ ( x ) dµ ( ξ, x ) . L. J. DÍAZ, K. GELFERT, AND M. RAMS
We say that µ is hyperbolic if χ ( µ ) (cid:54) = 0 and nonhyperbolic otherwise. The measure is hyperbolic of contracting type if χ ( µ ) < and hyperbolic of expanding type if χ ( µ ) > .In this way, the space M erg (Γ) of ergodic measures (with respect to F on Γ ) splits as(1.4) M erg (Γ) = M erg ,< (Γ) ∪ M erg , (Γ) ∪ M erg ,> (Γ) , into the sets of ergodic measures with negative, zero, and positive Lyapunov exponent, re-spectively. We explore the interplay between these three sets. Hypothesis (H1) implies that M erg ,< (Γ) and M erg ,> (Γ) both are nonempty and we show they are twin-like. Whetheror not M erg , (Γ) is empty depends on further analysis, and both possibilities can occur. Inthe case when the set M erg , (Γ) is nonempty, we prove that the closures of M erg ,< (Γ) and M erg ,> (Γ) nicely glue along it, see Theorem 2.19.We first analyze the projection Σ of the set Γ to the base Σ and the shift dynamicson it. We prove that Σ is coded (besides some dynamically irrelevant part which can beempty), see Theorem 2.1. We also prove that the set M (Σ) of invariant measures on Σ isan entropy-dense Poulsen simplex, see Theorem 2.11. We then turn to the skew-product F in order to study its topological properties and the sets supporting ergodic measures.Theorem 2.6 claims that the nonwandering set of F is the union of the homoclinic classesof P and Q and that their disjointness is equivalent to their hyperbolicity.The next step is to study the interplay between the set M erg (Σ) of ergodic measuresin Σ and M erg (Γ) . The set M erg (Γ) projects onto M erg (Σ) and any measure in M erg (Σ) can be lifted to M erg (Γ) . Theorem 2.13 characterizes hyperbolicity of the measures in M erg (Γ) in terms of these projections and lifts: either a measure M erg (Σ) lifts to a uniquemeasure in M erg (Γ) which turns out to be nonhyperbolic, or it lifts to exactly two measureswhich are hyperbolic of different type of hyperbolicity. Theorem 2.15 states that measuresin M erg (Γ) have atomic disintegration which is described in dynamical terms and providesthe graph-like structure of Γ in Corollary 2.16.Theorem 2.19 claims that every nonhyperbolic ergodic measure in M erg (Γ) is simul-taneously approached in the weak ∗ topology and in entropy by ergodic measures of con-tracting type and also by ergodic measures of expanding type. This result also extends toany nonergodic measure whose ergodic decomposition has measures of one type of hy-perbolicity, see Corollary 2.23. The gluing of M erg ,< (Γ) and M erg ,> (Γ) is described inCorollary 2.24 claiming that the set M erg (Γ) is arcwise connected if and only if M erg , (Γ) is nonempty. This provides meaningful information about the the gluing of M erg ,< (Γ) and M erg ,> (Γ) and it refines in a natural way the variational principle for the entropy,see Corollary 2.20. In the case when M erg , (Γ) is empty then the set of ergodic measuresconsists of two connected components M erg ,< (Γ) and M erg ,> (Γ) .Finally, we study the dynamics of globally defined maps ˜ F and discuss bifurcationscenarios. Let us give a rough idea of our constructions, the precise statements are given inSection 11. We present models of one-parameter families of maps ˜ F t , t ∈ [ t h , t c ] , inducedby families of interval maps { ˜ f , ˜ f ,t } . Here for the boundary parameters t h and t c the map ˜ f ,t is a “limit case” of our hypotheses (H1)–(H2). For each parameter t we consider themaximal invariant set Γ ( t ) associated to ˜ F t defined as in (1.2). These families completelyunfold heterodimensional cycles . The cycle occurs for t = t h . While usually in bifurcationtheory one considers parameters t close to the cycle parameter, here we study an entirerange of parameters going from essentially trivial dynamics at the cycle parameter t h (say,dynamics at the boundary of Morse-Smale systems) up to a completely chaotic dynamicsfor the parameter t c (with “full entropy” log 2 ). Figure 2 a) corresponds to the parameter t h where the invariant sets of the fixed points P and Q (of different type of hyperbolicity) KEW-PRODUCTS WITH CONCAVE FIBER MAPS 7 intersect cyclically giving rise to a heterodimensional cycle. Figures 2 b) and c) depict thetwo main possibilities for those complete unfoldings. Figure 2 b) depicts the case when theparameter t c corresponds to an intersection between the “strong unstable" and the stablesets of Q and Σ × { } ⊂ Γ ( t c ) . Figure 2 c) depicts the case when the parameter t c corresponds to an intersection between the unstable and the “strong stable” sets of P and Σ × { } ⊂ Γ ( t c ) . Hence in cases b) and c) the dynamics of Γ ( t c ) contains a horseshoe.See Remark 11.4 for a complete description of this figure. [1][0] a) A PQ fiber maps b) B C PQS fiber maps c) D ER PQ fiber maps F IGURE
2. Complete unfolding of a heterodimensional cycleFinally, the parameter t c can be seen as a parameter of collision between the set Γ ( t c ) and another ˜ F t c -maximal invariant set from outside Σ × [0 , , leading to a collision ofhomoclinic classes in the spirit of [29, 28]. We show how those collisions lead to ex-plosions of topological entropy and of the space of invariant measures of ˜ F t c | Γ ( t c) , seePropositions 11.6 and 11.7. 2. S TATEMENT OF RESULTS
Topological (and hyperbolic) dynamics.
Observe first that the restricted domain of f will require the consideration of admissible sequences ξ ∈ Σ . Denote by(2.1) π : Σ × [0 , → Σ L. J. DÍAZ, K. GELFERT, AND M. RAMS the natural projection defined by π ( ξ, x ) def = ξ and, recalling the definition of Γ in (1.2),define by(2.2) Σ def = π (Γ) the (compact and σ -invariant) set of admissible sequences. Our first result is the followingdescription of Σ . In symbolic dynamics a coded system is a transitive subshift which is theclosure of the union of an increasing family of transitive subshifts of finite type (SFT).Define(2.3) Σ het def = { ξ ∈ Σ : ξ = ( . . . τ . . . τ n . . . ) and ( f τ n ◦ . . . ◦ f τ )(1) = 0 } and(2.4) Σ cod def = Σ \ Σ het . Theorem 2.1.
Assume (H1). The subshift Σ is decomposed as Σ = Σ cod ∪ Σ het , where Σ cod is compact, σ -invariant, topologically mixing, and coded and Σ het is an atmost countable union of isolated points. Moreover, h top ( σ, Σ het ) = 0 and there is no σ -invariant probability measure supported on Σ het . For “generic choices" of maps f , f , the set Σ het is empty and hence Σ is coded, seeRemarks 2.2 and 2.3. Moreover, this set can be SFT or not, see Remark 2.7.Accordingly, the set Γ in general splits naturally into two subsets Γ cod def = π − (Σ cod ) and Γ het def = π − (Σ het ) . Recall that a point X ∈ Γ is non-wandering for (Γ , F ) if any neighbourhood of X (in Γ ) contains points whose forward orbit returns to it. The set of all non-wandering points,denoted by Ω(Γ , F ) , is F -invariant and closed. Note that Γ may contain points which failto be non-wandering points or recurrent points, see Remark 12.1. Indeed, if the set Σ het isnonempty then it consists of isolated points which are not recurrent, see Theorem 2.1.An important structure in differentiable dynamics is the one of homoclinic relationwhich we adapt to our setting. Following [45, Chapter 9.5], two hyperbolic periodic points A and B are heteroclinically related if the invariant sets of their orbits intersect cyclically.In our setting, there are two types of heteroclinic relations. If A and B have different typeof hyperbolicity we say that they form a heterodimensional cycle , otherwise we say thatthey are homoclinically related (in agreement with the terminology by Newhouse, thoughhere transversally is not needed). The latter defines an equivalence relation among hyper-bolic periodic points of the same type of hyperbolicity. Given a hyperbolic periodic point A , we call the closure of its equivalence class a homoclinic class and denote it by H ( A, F ) ,see Section 5. In some cases we will also consider the globally defined map ˜ F and use thecorresponding concepts. Remark 2.2 (The set Γ het and heterodimensional cycles) . The above defined set Γ het isindeed the set of points which are heteroclinic to the points P and Q defined in (1.3).In relation to what we defined above, note that any point (0 Z , x ) , x ∈ (0 , , is forwardasymptotic to P and backward asymptotic to Q . The nontrivial part to form a heterodimen-sional cycle associated to P and Q is precisely provided by points in Γ het (if nonempty).See Remark 5.5 for further details and Figure 3. Recall that any non-wandering point is recurrent and that the converse implication is in general false. Noticethe non-wandering set not only depends on the mapping but also on its domain and thus the non-wandering setsfor ˜ F and for F may differ. For instance, the set Γ het is wandering for F and is non-wandering for ˜ F . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 9 f f f (1)0 1 f (1) RPQ F IGURE
3. Example when Γ het is nonempty ( Γ het contains the point R = ((0 − N . N ) , and its F -orbit). Remark 2.3.
By a Kupka-Smale genericity-like result, generically there are no heterodi-mensional cycles associated to P and Q and hence Γ het is empty. Remark 2.4 (Basic sets and horseshoes) . In what follows, we call basic any set which islocally maximal, compact, F -invariant, topologically transitive, and has uniform contrac-tion (or uniform expansion) in the fiber direction. When the basic set is uncountable (not aperiodic orbit) and topologically mixing, we call it a horseshoe . Remark 2.5 (Dynamics of Γ cod ) . In general, assuming only (H1), the topological dynam-ics of the map F on Γ cod may exhibit a huge variety (also reflected on the ergodic level).A more detailed discussion is done in Section 12. Assuming additionally (H2), then hy-perbolic periodic points of the same type of hyperbolicity are homoclinically related (seeProposition 5.1) and hence we have only two homoclinic classes. Still, under hypothesis(H2), we observe two main “opposed” types of dynamical behaviors that can occur:1) The map F | Γ cod is transitive and hence not hyperbolic (in this case, we may haveeither Γ het = ∅ or Γ het (cid:54) = ∅ ).2) The map F | Γ cod is not transitive and the set of nonwandering points of F | Γ cod ishyperbolic and consists of two disjoint horseshoes, one expanding and the otherone contracting in the fiber direction. In this case, Γ cod also contains wanderingpoints, for instance the subset { Z } × (0 , . Moreover, there are also a somewhat intermediate scenario:3) The map F | Γ cod is not transitive and Γ cod contains “touching" or “overlapping"components of different type of hyperbolicity.To be able to describe any further structure, in what follows, besides (H1) and (H2) wewill invoke the (uniform concavity) hypothesis (H2+) strengthening (H2). Under condi-tions (H1)–(H2+), we can prove that every non-wandering point of F in Γ cod (or, in viewof Theorem 2.1, any non-isolated non-wandering point in Γ ) is approximated by hyperbolicperiodic points. Indeed, we have a more accurate statement: Note that this situation is compatible with Γ het (cid:54) = ∅ , in which case we have that Ω( F ) is hyperbolic but Γ het ⊂ Ω( ˜ F ) and hence Ω( ˜ F ) is not hyperbolic. Theorem 2.6.
Assume (H1)–(H2+). The non-wandering set
Ω(Γ , F ) is contained in Γ cod and satisfies Ω(Γ , F ) = H ( P, F ) ∪ H ( Q, F ) = closure { A ∈ Γ : A hyperbolic and periodic } . Moreover, the sets H ( P, F ) and H ( Q, F ) both are hyperbolic if and only if they are dis-joint. Remark 2.7. If H ( P, F ) is hyperbolic, then it is locally maximal. Thus, by the spectraldecomposition theorem, if H ( P, F ) is hyperbolic, then H ( P, F ) is a basic set and itsdynamics is conjugate to a SFT. Hence, in this case, Σ itself is a SFT.If H ( P, F ) is not hyperbolic, then Σ may be a SFT or not. For instance, one may havethat H ( P, F ) ∩ H ( Q, F ) is just a parabolic periodic orbit and Σ is SFT. On the other hand,if Q ∈ H ( P, F ) (and hence the latter set is not hyperbolic) then Σ is not SFT. See alsothe discussion in Section 12.We also have the following twinning of basic sets, which we state without proof. Proposition 2.8 (Twin-basic sets) . Assume (H1)–(H2). For every basic set Γ + ⊂ Γ withuniform fiber expansion there exists a basic set Γ − ⊂ Γ with uniform fiber contractionsuch that F | Γ − and F | Γ + are topologically conjugate and that π (Γ − ) = π (Γ + ) . Remark 2.9.
Observe that the IFS of the inverse fiber maps { f − , f − } , after a changeof coordinates x (cid:55)→ − x , again satisfies (H1)–(H2). Moreover, if the original systemsatisfied (H2+) then the inverse system will also do so. Hence, for many statements, thereis a certain symmetry with respect to time reversal.The ergodic counterpart of the results above is done in the following section, see inparticular Corollary 2.16.2.2. Structure of the measure space and Lyapunov exponents.
Given a compact metricspace X and a continuous map T on X , we denote by M ( X ) the space of T -invariant Borelprobability measures on X and by M erg ( X ) the subspace of ergodic measures. We equip M ( X ) with the Wasserstein distance which we denote by W (we recall its definition andsome of its properties in Appendix A). We denote by h ( ν ) the metric entropy of a measure ν ∈ M ( X ) .The space M ( X ) equipped with the weak ∗ topology is a Choquet simplex whose ex-treme points are precisely the ergodic measures (see [51, Chapter 6.2]). If M ( X ) is not asingleton and if the set of ergodic measures M erg ( X ) is dense in its closed convex hull M ( X ) , one refers to it as a Poulsen simplex (see also [42]). Moreover, one says that M ( X ) is an entropy-dense Poulsen simplex if for every µ ∈ M ( X ) , every neighbourhood U of µ in M ( X ) , and every ε > there exists ν ∈ M erg ( X ) ∩ U such that h ( ν ) > h ( µ ) − ε . To see this, take any small neighborhood to which hyperbolicity extends. Any point X whose orbit is in thisneighborhood will be accumulated by periodic points which by Proposition 5.1 belong to H ( P, F ) . Hyperbolicitythen implies that X ∈ H ( P, F ) . Indeed, in this case there are “admissible words" ( ξ . . . ξ k ) such that f [ ξ ... ξ k ] (1) = ε k → + as k → ∞ . This then implies that ( ξ . . . ξ k m is admissible if and only if m ≥ m ( ε k ) , where m ( ε k ) → ∞ as k → ∞ , the latter preventing Σ to be a SFT. To prove it, use the key property that hyperbolic periodic points appear in contracting–expanding pairs andthat periodic hyperbolic points of the same type are homoclinically related, these facts being a consequence ofthe concavity assumption (see Proposition 5.1). Recall that the convex hull of a set N ⊂ M ( X ) is the smallest convex set containing N , denoted by conv( N ) ,and that the closed convex hull of N is the smallest closed convex set containing N , denoted by conv( N ) . By[49, Theorem 5.2 (i)–(ii)], we have conv( N ) = conv( N ) , where N denotes the weak ∗ closure of N . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 11
Remark 2.10 (Entropy map) . For the step skew-product map F , the entropy map µ (cid:55)→ h ( µ ) is upper semi-continuous on M (Γ) . Indeed, the map ˜ F when seen as a partiallyhyperbolic diffeomorphism with one-dimensional central bundle is h -expansive (see [19]).Hence [12] implies upper semi-continuity. Thus, in our setting, being an entropy-densePoulsen simplex means that every µ ∈ M (Σ) can be approximated weak ∗ and in entropyby ergodic measures. Theorem 2.11.
Assume (H1)–(H2). The space M (Σ) is an entropy-dense Poulsen simplex. Remark 2.12.
Denote by π ∗ µ def = µ ◦ π − the pushforward of a measure µ by the projection π defined in (2.1). Note that for every µ ∈ M erg (Γ) , the measure π ∗ µ is ergodic. On theother hand, given ν ∈ M erg (Σ) , for every µ ∈ M (Γ) satisfying π ∗ µ = ν , every ergodiccomponent µ (cid:48) of µ satisfies π ∗ µ (cid:48) = ν .We observe the following well-known fact (2.5) h ( π ∗ µ ) = h ( µ ) for every µ ∈ M (Γ) . Theorem 2.13.
Assume (H1)–(H2+). There are continuous functions κ , κ : (0 , ∞ ) → (0 , ∞ ) which are increasing and satisfy lim D → κ ( D ) = 0 = lim D → κ ( D ) such that, given any ν ∈ M erg (Σ) , one of the following two cases occurs: a) There exist exactly two measures µ , µ ∈ M erg (Γ) such that π ∗ µ = ν = π ∗ µ .In this case, both measures are hyperbolic and have fiber Lyapunov exponents withdifferent signs. More precisely, the Wasserstein distance D def = W ( µ , µ ) > between µ and µ satisfies D = (cid:90) x dµ ( ξ, x ) − (cid:90) x dµ ( ξ, x ) and (2.6) − κ ( D ) ≤ χ ( µ ) ≤ − κ ( D ) < < κ ( D ) ≤ χ ( µ ) ≤ κ ( D ) . b) There exists only one measure µ ∈ M (Γ) such that π ∗ µ = ν . In this case, µ isergodic and satisfies χ ( µ ) = 0 . Corollary 2.16 below will complement Theorem 2.13 stating a complete relation be-tween the spaces of ergodic measures in Γ and Σ . For this we need some further defini-tions.Notice that the fixed points Q and P in (1.3) are contained in the common fiber { Z } × [0 , . It turns out that in our step skew-product setting this picture repeats on other periodicfibers. Given ξ ∈ Σ we write ξ = ξ − .ξ + to denote its forward and backward one-sidedsequences. For ξ ∈ Σ def = π (Γ) , define its spine by J ξ def = (cid:0) { ξ } × [0 , (cid:1) ∩ Γ = π − ( ξ ) ∩ Γ . Spines are intimately related to the homoclinic structure and there are two possibilities:either J ξ is a continuum of the form J ξ = { ξ } × I ξ , where I ξ = [ x ξ + , x ξ − ] or J ξ = Just observe that h top ( F, π − ( ξ )) = 0 for every ξ ∈ Σ . Hence, by the Ledrappier-Walters formula [40], h ( ν ) = sup µ : π ∗ µ = ν h ( µ ) . { ξ } × { x ξ } is a singleton. In the latter case, we define x ξ + = x ξ − def = x ξ . In particular, thefollowing two sets are σ -invariant Σ spine def = { ξ ∈ π (Γ) : J ξ is a continuum } , Σ sing def = { ξ ∈ π (Γ) : J ξ is a singleton } (2.7)When ξ ∈ Σ spine is periodic then J ξ is bounded by two hyperbolic points ( ξ, x ξ ± ) . When ξ ∈ Σ sing is periodic then J ξ = ( ξ, x ξ ) is a point on a parabolic periodic orbit. See alsoSection 3.2. Remark 2.14 (Boundary graphs) . Observe that the two naturally associated functions ξ (cid:55)→ x ξ + and ξ (cid:55)→ x ξ − are measurable (see [50, Proposition 3.1.21 and Theorem 5.3.1]). Wealso observe that our construction also provides that ( ξ, x ξ + ) ∈ H ( Q, F ) and ( ξ, x ξ − ) ∈ H ( P, F ) for every ξ ∈ Σ cod , see Proposition 5.7. The graphs of the functions ξ (cid:55)→ x ξ + and ξ (cid:55)→ x ξ − bound precisely Γ and the region in between plays a role somewhat similar to a Conley pairin the study of recurrent sets (see [45, Chapter IX]). This structure also resembles the one ofa trapping region in skew-products studied, for instance, in [39, 37, 32], and the structureof the two graphs provides a spiny (bony in the terminology of [39, 37]) structure. Insome (necessarily nonhyperbolic) cases, the spines are contained in the nonwandering set Ω(Γ , F ) .In our setting, concavity forces that ergodic measures are only supported on the “bound-ary of that region" in the following sense. The direct product structure provides a disinte-gration for every measure in M (Γ) : if µ ∈ M erg (Γ) then ν def = π ∗ µ ∈ M erg (Σ) and thereis a family of measures { µ ξ } ξ ∈ Σ supported on [0 , such that for every measurable set B ⊂ Γ we have µ ( B ) = (cid:90) Σ µ ξ ( B ∩ J ξ ) dν ( ξ ) . The family { µ ξ } ξ ∈ Σ is called the disintegration of µ . Observe that we have µ ξ ( I ξ ) = 1 for ν -almost every ξ . This disintegration is atomic if µ ξ are atomic ν -almost everywhere. Theorem 2.15 (Atomic disintegration) . Assume (H1)–(H2+). The disintegration of any µ ∈ M erg (Γ) is atomic. Moreover, • µ is hyperbolic if and only if π ∗ µ (Σ spine ) = 1 , • µ is nonhyperbolic if and only if π ∗ µ (Σ sing ) = 1 .More precisely, letting I ξ = [ x ξ + , x ξ − ] and denoting by δ x the Dirac measure at x , • if µ ∈ M erg ,> (Γ) , then µ = (cid:82) Σ δ x ξ + dπ ∗ µ ( ξ ) , • if µ ∈ M erg ,< (Γ) , then µ = (cid:82) Σ δ x ξ − dπ ∗ µ ( ξ ) , • if µ ∈ M erg , (Γ) , then x ξ ± = x ξ for π ∗ µ -almost every ξ and µ = (cid:82) Σ δ x ξ dπ ∗ µ ( ξ ) . The above implies that, besides the natural partition in (1.4), we have a partition of M erg (Σ) as follows. As the sets Σ spine and Σ sing both are σ -invariant, if ν ∈ M erg (Σ) isergodic then only one of them has full measure. In this way, the space of ergodic measuressplits accordingly into the two subsets M (cid:63) erg (Σ) def = { ν ∈ M erg (Σ) : ν (Σ (cid:63) ) = 1 } , (cid:63) ∈ { spine , sing } . Hence the projection π ∗ provides bijections between the spaces M erg ,> (Γ) , M erg ,< (Γ) ,and M spineerg (Σ) and between M erg , (Γ) and M singerg (Σ) . Moreover, as “there is no entropy KEW-PRODUCTS WITH CONCAVE FIBER MAPS 13 in the fibers" (see (2.5)), this also extends to the entropy of the measures. We collect thesefacts in the following corollary, without further explicit proof.
Corollary 2.16 (Twin-measures and symmetry of measure spaces) . Assume (H1)–(H2+).Then for every µ ∈ M erg (Γ) and ν ∈ M erg (Σ) satisfying ν = π ∗ µ we have h ( ν ) = h ( µ ) . Moreover, exactly one of the following two possibilities holds: a) We have µ ∈ M erg ,< (Γ) ∪ M erg ,> (Γ) and ν ∈ M spineerg (Σ) . In this case, thereexists exactly one other ergodic measure µ (cid:48) also satisfying π ∗ µ (cid:48) = ν . Moreover,the Lyapunov exponents of µ and µ (cid:48) satisfy (2.6) and, in particular, µ and µ (cid:48) haveopposite type of hyperbolicity. b) We have µ ∈ M erg , (Γ) and ν ∈ M singerg (Σ) , and π − ∗ ν = { µ } . In item a) in the above corollary, we call the measures µ and µ (cid:48) twin-measures . Similarphenomena are observed in [20, 46]. Remark 2.17.
As indicated above, there is a close relation of our results with the ex-istence of bony attractors in, for example, [39, 37], though here we face two essentiallydifferent properties: On one hand, we do not have “trapping regions" (key ingredient in[37]) and our system has rather a saddle-type nature. On the other hand, we have to dealwith admissible sequences and restricted domains of the fiber maps. By Corollary 2.16,given µ ∈ M erg , (Γ) , with respect to ν = π ∗ µ the set Γ is a bony graph in the sense thatits intersection with ν -almost every fiber is a single point (that is ν (Σ sing ) = 1 ) but is acontinuum “otherwise" (that is, Σ spine is nonempty but has zero measure). It is possiblethat ν has full support Σ , justifying the comparison with bony graphs.2.3. Weak ∗ and entropy approximation of ergodic measures. The following theoremabout approximation of an ergodic measure in entropy and in the weak ∗ topology is well-known for hyperbolic measures of diffeomorphisms (see Remark 2.21 below). Thus, westate the result below only for nonhyperbolic ergodic measures.To fix some terminology, given a compact F -invariant set Ξ , denote by h top ( F, Ξ) the topological entropy of F on a set Ξ (see [51] for its definition). Remark 2.18.
As there is no entropy in the fibers (see (2.5)), the variational principle fortopological entropy of F on a compact F -invariant set Ξ immediately implies that h top ( σ, π (Ξ)) = h top ( F, Ξ) . We will call an invariant probability measure periodic if it is supported on a periodicorbit.
Theorem 2.19 (Hyperbolic approximation of nonhyperbolic measures) . Assume (H1)–(H2+). Then for every measure µ ∈ M erg , (Γ) , every ε E > , and every ε H ∈ (0 , h ( µ )) there exist a basic set Γ + ⊂ Γ with uniform fiber expansion and a basic set Γ − ⊂ Γ withuniform fiber contraction such that their topological entropies satisfy h top ( F, Γ + ) , h top ( F, Γ − ) ∈ [ h ( µ ) − ε H , h ( µ ) + ε H ] . Moreover, every measure µ ± ∈ M (Γ ± ) is ε E -close to µ in the Wasserstein metric. Inparticular, there are hyperbolic measures µ + , µ − ∈ M erg (Γ) with χ ( µ + ) ∈ (0 , ε E ) and χ ( µ − ) ∈ ( − ε E , and h ( µ ± ) ∈ [ h ( µ ) − ε H , h ( µ ) + ε H ] . If h ( µ ) > then Γ ± are horseshoes, otherwise they are hyperbolic periodic orbits.In particular, every measure in M erg (Γ) is weak ∗ accumulated by hyperbolic periodicmeasures. The following is an immediate consequence of Theorem 2.19, stated without proof.
Corollary 2.20 (Variational principle for entropy) . Assume (H1)–(H2+). Then h top ( F, Γ) = sup µ ∈ M erg ,< (Γ) h ( µ ) = sup µ (cid:48) ∈ M erg ,> (Γ) h ( µ (cid:48) ) . Observe that in our setting, one can apply the methods in [8] to verify that indeednonhyperbolic measures with positive entropy may exist.
Remark 2.21 (Katok’s horseshoes) . For every hyperbolic measure µ ∈ M erg (Γ) , it iswell-known that in our “partially hyperbolic setting" there exists a sequence of horse-shoes (with uniform hyperbolicity) (Γ n ) n such that M (Γ n ) converges weak ∗ to µ andalso h top ( F, Γ n ) converges to h ( µ ) . Here one can use Katok’s horseshoe constructionseither assuming C [36, Chapter S.4] or assuming C regularity plus domination [15, 31].In particular, every periodic measure in Γ n is weak ∗ -close to µ .Comparing Theorem 2.19 with previous results, the analogous result was obtained in[21] for transitive step skew-products with fiber maps being circle diffeomorphisms assum-ing the existence of so-called expanding/contracting blending intervals and forward/back-ward minimality of the underlying IFS. Those are quite strong (though natural) propertiesand they describe the somewhat intermingled structure of two types of hyperbolicity. Theconstructions in [21] was extended in [24, 52] to some partially hyperbolic diffeomor-phisms with minimal strong foliations, following the strategy outlined in [21, Section 8.3].Here a priori we do not have these hypotheses and in many cases they indeed fail. Here, theabove approaches are somewhat replaced by our concavity hypothesis having a differentflavour.The main tool to prove Theorem 2.19 are the so-called skeletons that rely only on er-godic properties, see Definition 9.9. Their existence in fact only requires (H1), see Propo-sition 9.11. Assuming additionally (H2), the main step towards the proof of Theorem 2.19is the following result. Theorem 2.22 (Shadowplay) . Assume (H1)–(H2). For every µ ∈ M erg (Γ) there existsa sequence of basic sets (Υ n ) n ⊂ Γ such that for every sequence ( µ n ) n , µ n ∈ M (Υ n ) ,the corresponding sequence ( ν n ) n , ν n = π ∗ µ n , converges weak ∗ to π ∗ µ and h top ( F, Υ n ) converges to h ( π ∗ µ ) = h ( µ ) . Theorem 2.19 will be an almost immediate consequence of Theorem 2.22. Indeed,it only remains to show the convergence of the measures µ n to µ , which follows fromTheorem 2.13 (invoking additionally (H2+)).Finally, returning to the structure of the space of ergodic measures, in correspondenceto Theorem 2.11, we state how the entropy-dense Poulsen structure of M (Σ) lifts to M (Γ) . For that we consider the notation M erg , ≤ (Γ) = M erg ,< (Γ) ∪ M erg , (Γ) and M erg , ≥ (Γ) = M erg , (Γ) ∪ M erg ,> (Γ) . Corollary 2.23.
Assume (H1)–(H2+). Any µ ∈ M (Γ) having an ergodic decomposition µ = (cid:82) µ (cid:48) d λ ( µ (cid:48) ) with ergodic measures µ (cid:48) ∈ M erg , ≥ (Γ) is weak ∗ accumulated by pe-riodic measures in M erg ,> (Γ) . Moreover, there is a sequence of ergodic measures in M erg ,> (Γ) which converges weak ∗ and in entropy to µ .The analogous result holds for measures in M erg , ≤ (Γ) and accumulation and conver-gence in M erg ,< (Γ) . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 15
Corollary 2.23 can be seen as an extended version of [9, Theorem 2] where it is assumedthat in the ergodic decomposition µ = (cid:82) µ (cid:48) d λ ( µ (cid:48) ) almost every measure is in M erg ,> (Γ) .A key point in [9] is the study of local unstable manifolds using Pesin theory, which herewe replace by concavity. This allows us to incorporate nonhyperbolic ergodic measuresinto the decomposition.In general, in comparable settings, convex combinations of ergodic measure of differenttype of hyperbolicity may not be approached by ergodic ones (weak ∗ and in entropy).For example, this applies to the pair of the two measures of maximal entropy of differenttype of hyperbolicity in the case of proximality in [23, Corollary 3]. Even though, [23]presents many similarities with this paper, it relies on some essential tools which are notavailable here: a) the space of admissible sequences is Σ , b) the fibers are circles, andc) synchronisation techniques using that the measures of maximal entropy project to aBernoulli measure on Σ .To complete this section, we point out some further properties that immediately fol-low from the above results (see for instance the methods in [33, 22] that apply here ipsislitteris ), hence stated without proof. Corollary 2.24 (Ergodic arcwise connectedness) . Assume (H1)–(H2+). Then each of thesets M erg ,< (Γ) and M erg ,> (Γ) is arcwise connected. Moreover, M erg , (Γ) is nonemptyif and only if the space M erg (Γ) is arcwise connected. Bifurcation settings and homoclinic scenarios.
Returning to the idea that the map F : Γ → Γ may serve as a plug in a semi-local analysis of the dynamics, we will see inSection 11 how this plug acts and interacts with other pieces of dynamics of the globalmap ˜ F . We consider a family of maps ˜ f and ˜ f ,t (for simplicity we assume that ˜ f doesnot depend on t ) satisfying our hypotheses and study the corresponding globally definedone-parameter family of skew-products ˜ F t . For each parameter t we define the maximalinvariant set Γ ( t ) similarly as in (1.2) and the space of admissible sequences Σ ( t ) as in(2.2). This family has two distinguished parameters t h < t c , corresponding to a heterodi-mensional cycle associated to P and Q and to the collision of a pair of homoclinic classes,respectively. When t varies from t h to t c the maps ˜ F t “unfolds completely” the heterodi-mensional cycle, in the sense that the topological entropy of ˜ F t | Γ ( t ) goes from zero at t h tofull log 2 entropy (the maximal possible entropy) at t c .We prove that the space of admissible sequences Σ ( t ) converges to Σ as t → t c inHausdorff distance. To each family ( ˜ F t ) t ∈ [ t h ,t c ] there is naturally associated a constant C ( t ) . If C ( t c ) < ∞ then there is an explosion of the set Σ ( t ) at the collision parameter t = t c . Moreover, if C ( t c ) < then there is also an explosion of the space of invari-ant measures on Σ ( t ) and a jump of the topological entropy of Σ ( t ) (and hence of Γ ( t ) ),see Propositions 11.6 and 11.7. We provide an interpretation for those explosions and ex-amples illustrating the different dynamical scenarios that may occur. We also discuss the“twinning" and “merging" of hyperbolic and nonhyperbolic ergodic measures correspond-ing to Theorem 2.13 for ˜ F t c , see Proposition 11.8.Regarding the dynamics at the collision parameter t = t c , we recall that collisions ofhomoclinic classes were studied from the merely topological point of view in [29, 28]. Wealso observe an IFS somewhat similar to the one associated to the maps ˜ f , ˜ f ,t c consideredhere also appears in [30, 1, 2], where different questions were being asked.In Section 12 we discuss the role of our concavity hypothesis and observe the possibleappearance of further homoclinic classes when this hypothesis fails. Organization of the paper.
Sections 3 and 5 deal with two underlying key ingre-dients: the symbolic space Σ and its associate IFS and homoclinic relations and classes,respectively. Theorem 2.1 is proved in Section 4 which is dedicated to the coded natureof Σ . In Section 6 we prove some auxiliary results about concave maps. In Section 7we prove Theorem 2.6 dealing with the decomposition of the non-wandering set into ho-moclinic classes. Section 8 is dedicated to the study of “decompositions" of the space ofmeasures. We prove Theorem 2.13 about the structure of the space of measures in Sec-tion 8.4 and Theorem 2.15 about disintegration of measures in Section 8.5. Theorem 2.19about approximation of nonhyperbolic ergodic measures by hyperbolic ones is proved inSection 9. We will prove Corollary 2.23 at the end of Section 9.5. Theorem 2.11 about thePoulsen structure of M (Σ) is proved in Section 10. In Section 11 we explore bifurcationscenarios. In Section 12 we discuss homoclinic classes and the importance of the concavityhypothesis. The paper closes with Appendix A about the Wasserstein distance.3. U NDERLYING STRUCTURES : SYMBOLIC SPACE AND THE
IFSIn this section we assume (H1). In Section 3.1 hypothesis (H2) is not required and wewill additionally assume (H2) only in Section 3.2. Hypothesis (H2+) is not required.In the following, we consider the shift space Σ def = { , } Z equipped with the metric(3.1) d ( ξ, η ) def = e − n ( ξ,η ) , where n ( ξ, η ) def = sup {| (cid:96) | : ξ i = η i for i = − (cid:96), . . . , (cid:96) } . We use the notation ξ = ( ξ i ) i ∈ Z = ( ξ − .ξ + ) ∈ Σ , where ξ + = ( ξ ξ . . . ) ∈ Σ +2 def = { , } N and ξ − = ( . . . ξ − ξ − ) ∈ Σ − def = { , } − N . We equip Σ × R with the metric d (( ξ, x ) , ( η, y )) def = max { d ( ξ, η ) , | x − y |} . Consider the projections π : Σ × [0 , → Σ , π ( ξ, x ) def = ξ, and (cid:37) : Σ × [0 , → [0 , , (cid:37) ( ξ, x ) def = x. Admissible compositions.
Given n ≥ , call τ = ( τ . . . τ n ) ∈ { , } n a word and | τ | def = n its length ; a subword of τ is a word of the form ( τ i . . . τ k ) with ≤ i ≤ k ≤ n .Given words ( ξ − m . . . ξ − ) and ( τ . . . τ n − ) , we denote the corresponding cylinders by [ ξ − m . . . ξ − . ] def = { η : η k = ξ k , k = − m, . . . , − } , [ τ . . . τ n − ] def = { η : η k = τ k , k = 0 , . . . , n − } . Given a point x ∈ [0 , , a word ( ξ . . . ξ n − ) ∈ { , } n is forward admissible for x iffor every k = 0 , . . . , n − the map f [ ξ ... ξ k ] def = f ξ k ◦ · · · ◦ f ξ is well defined at x . We denote by I [ ξ ... ξ n − ] the set of points x ∈ [0 , for which ( ξ . . . ξ n − ) is admissible. Given ξ ∈ Σ and n ≥ such that x ∈ I [ ξ ... ξ n − ] , sometimeswe will also adopt the notation f nξ def = f [ ξ ... ξ n − ] . Analogously we adopt the notations I [ ξ − m ... ξ − . ] and f − mξ def = f [ ξ − m ... ξ − . ] def = f − ξ − m ◦ · · · ◦ f − ξ − . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 17
Remark 3.1.
Monotonicity of the maps f , f implies that each of the sets I [ ξ ... ξ n ] and I [ ξ − m ... ξ − . ] , when nonempty, is a (possibly degenerate) interval and of the form [ a [ ξ ... ξ n ] , and [0 , b [ ξ − m ... ξ . ] ] , respectively, where(3.2) f [ ξ ... ξ n ] ( a [ ξ ... ξ n ] ) = 0 and f [ ξ − m ... ξ − . ] ( b [ ξ − m ... ξ . ] ) = 1 . We say that ξ + ∈ Σ +2 is admissible for x if ( ξ . . . ξ n ) is forward admissible for x for every n ≥ , analogously for ξ − ∈ Σ − . We say that a bi-infinite sequence ξ ∈ Σ is admissible for x if ξ + and ξ − are both are admissible for x . Denote by I ξ + the set of points x such that ξ + is admissible for x , analogously for I ξ − . Note that, given ξ = ( ξ − .ξ + ) ∈ Σ , the families of intervals { I [ ξ ... ξ n − ] } n ≥ and { I [ ξ − m ... ξ − . ] } m ≥ both are nested. Wehave (cid:92) n ≥ I [ ξ ... ξ n − ] = I ξ + , (cid:92) m ≥ I [ ξ − m ... ξ − . ] = I ξ − , and I ξ def = I ξ − ∩ I ξ + . By writing ( ξ, x ) we assume that ξ + and ξ − are admissible for x , hence x ∈ I ξ . Remark 3.2.
By the previous comments, we have I ξ + = [ x ξ + , and I ξ − = [0 , x ξ − ] forsome x ξ ± ∈ [0 , (provided these intervals are nonempty). Therefore, if I ξ (cid:54) = ∅ then I ξ = [ x ξ + , x ξ − ] (and x ξ + ≤ x ξ − ). Remark 3.3.
Any word which is forward admissible for is of the form k and any wordwhich is backward admissible for is of the form (cid:96) . Remark 3.4.
Note that ξ n = 1 if and only if a [ ξ ... ξ n − ] < a [ ξ ... ξ n ] ,ξ − m = 1 if and only if b [ ξ − m ... ξ . ] < b [ ξ − m +1 ... ξ . ] . Given x ∈ [0 , and n ≥ , let Σ + ( n, x ) def = { ( ξ . . . ξ n − ) : admissible for x } , analogously Σ − ( n, x ) . We denote by Σ( x ) ⊂ Σ the set of (infinite) sequences which areadmissible for x and by Σ + ( x ) and Σ − ( x ) the corresponding sets of admissible one-sidedsequences. Note that the set Σ defined in (2.2) coincides with the set of all admissiblesequences Σ = (cid:91) x ∈ [0 , Σ( x ) . With the definition (1.2), we have
Γ = { ( ξ, x ) ∈ Σ × [0 ,
1] : x ∈ [0 , and ξ ∈ Σ( x ) } . Remark 3.5.
Clearly, it follows from (H1) that Σ + ( x ) ⊂ Σ + (1) and Σ − ( x ) ⊂ Σ − (0) .Hence, it holds Σ ⊂ { ( ξ − .ξ + ) : ξ − ∈ Σ − (0) and ξ + ∈ Σ + (1) } , and this inclusion is in general strict. Moreover, for every x Σ( x ) = { ( ξ − .ξ + ) : ξ − ∈ Σ − ( x ) , ξ + ∈ Σ + ( x ) } . Remark 3.6. If ξ = ( ξ . . . ξ n − ) Z is an admissible periodic sequence, then f [ ξ ... ξ n − ] hasone fixed point with nonpositive Lyapunov exponent and one with nonnegative Lyapunovexponent. Note that these points may coincide and then this point is parabolic. Remark 3.7.
Given ξ + ∈ Σ + (1) such that ξ n = 1 , for the new sequence obtained insertingone in the n th position, we have ( ξ . . . ξ n − ξ n +1 . . . ) ∈ Σ + (1) . Hence, inductively, ( ξ . . . ξ n − k ξ n +1 . . . ) ∈ Σ + (1) , for every k ≥ .To see why this is so, just note that f is defined on [0 , , hence f [ ξ ... ξ n − (1) iswell defined, and f [ ξ ... ξ n − ξ n ] (1) > f [ ξ ... ξ n − ξ n ] (1) . The above now follows from themonotonicity of f and f .Similarly, given ξ + ∈ Σ + (1) such that ξ n = 1 , the sequence obtained exchanging ξ n = 1 for ξ n = 0 and keeping all remaining terms also belongs to Σ + (1) .Recall that by (H1) the point d ∈ (0 , is defined by f ( d ) = 0 . Remark 3.8.
For k ≥ sufficiently large, the periodic sequence (10 k ) Z belongs to Σ .Indeed, for ε > small and k ≥ sufficiently large, we have ( f k ◦ f )([ d − ε, ⊂ [ d, .Hence, there is x ∈ [ d, such that x = ( f k ◦ f )( x ) and thus the periodic sequence ξ = (10 k ) Z is admissible for x . Remark 3.9.
Combining Remarks 3.7 and 3.8 and also applying Remark 3.4, the followingis now immediate. For k ≥ sufficiently large, we have ξ = (0 k Z ∈ Σ and a [0 k ≤ a [0 k k ... k ≤ . . . < x (0 k N = x ξ + . On the other hand, for every sequence of positive integers ( (cid:96) n ) n ≥ satisfying (cid:96) n ≤ k forevery n ≥ we have a [0 k k ... k ≤ a [0 (cid:96) (cid:96) ... (cid:96)n , whenever I [0 (cid:96) (cid:96) ... (cid:96)n (cid:54) = ∅ . Remark 3.10. [Consecutive ’s in admissible sequences] Define k ≥ to be the integersuch that f k − (1) ∈ [ d, and f k (1) < d . Note that this number is well defined since f ( x ) < x for all x ∈ [ d, .Consider a word ( ξ . . . ξ n ) which is forward admissible for x . The definition of k implies that it has at most k consecutive ’s. Moreover, ( ξ . . . ξ n is also forwardadmissible for x (just observe that the domain of f is the whole interval [0 , ).The forward orbit of a point x ∈ [0 , by the IFS is defined as O + ( x ) def = (cid:91) n ≥ (cid:91) ( ξ ...ξ n − ) ∈ Σ + ( n,x ) f [ ξ ... ξ n − ] ( x ) . The backward orbit of x , O − ( x ) , is defined analogously. The orbit of x by the IFS is O ( x ) def = (cid:91) ξ ∈ Σ( x ) (cid:16) (cid:91) n ≥ f [ ξ ... ξ n − ] ( x ) ∪ (cid:91) m ≥ f [ ξ − m ... ξ − . ] ( x ) (cid:17) . In view of Remark 3.5, O ( x ) = O − ( x ) ∪ O + ( x ) . Lemma 3.11.
The set O − (0) is dense in [0 , if and only if for every x, y ∈ [0 , , x (cid:54) = y ,we have Σ + ( x ) (cid:54) = Σ + ( y ) . Similarly, the set O + (1) is dense in [0 , if and only if for every x, y ∈ [0 , , x (cid:54) = y , we have Σ − ( x ) (cid:54) = Σ − ( y ) .Proof. Note first that for any two points x, y ∈ [0 , , x < y , the set Σ + ( x ) differs from Σ + ( y ) if and only if Σ + ( n, x ) (cid:54) = Σ + ( n, y ) for some n . Assume Σ + ( n, x ) (cid:54) = Σ + ( n, y ) for some n and assume that n is the smallest one. Then there exists a word ( ξ . . . ξ n − ) forward admissible for y but not for x , while the word ( ξ . . . ξ n − ) is forward admissiblefor both x and y . Hence, by Remark 3.4, we have ξ n − = 1 and f [ ξ ... ξ n ] ( x ) < d ≤ f [ ξ ... ξ n ] ( y ) . Thus, f [ ξ ...ξ n . ] (0) ∈ ( x, y ) . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 19
In the other direction, if O − (0) ∈ ( x, y ) , similarly there is ( ξ . . . ξ n − ) ∈ Σ − ( n, such that x < f [ ξ n − ...ξ . ] (0) < y . This implies that ( ξ . . . ξ n − ) ∈ Σ + ( n, y ) \ Σ + ( n, x ) .The second part of the lemma for forward orbits is obtained in an analogous way. (cid:3) Hyperbolic and parabolic periodic points.
In this section, we will assume (H1)–(H2).
Lemma 3.12.
Let ( ξ . . . ξ n ) be a word such that I [ ξ ... ξ n − ] (cid:54) = ∅ and g = f [ ξ ... ξ n − ] .There are the following possibilities: (1) If g has some fixed point then ξ = ( ξ . . . ξ n − ) Z ∈ Σ and there are two cases: (1a) g has exactly two fixed points p +[ ξ ... ξ n − ] < p − [ ξ ... ξ n − ] and they are repellingand contracting, respectively. In this case, I ξ = [ p +[ ξ ... ξ n − ] , p − [ ξ ... ξ n − ] ] . (1b) g has exactly one fixed point p [ ξ ... ξ n − ] and it is parabolic. In this case, I ξ = { p [ ξ ... ξ n − ] } . (2) If g has no fixed point then ( ξ . . . ξ n − ) Z (cid:54)∈ Σ .Proof. Let p be a fixed point of g . Then (( ξ . . . ξ n − ) Z , p ) ∈ Γ and the first assertionfollows. By monotonicity and f ( x ) < x (hypothesis (H1)) it follows that ( ξ . . . ξ n − ) contains at least one and hence, since f (cid:48) is strictly decreasing and f (cid:48) is nonincreasing(hypothesis (H2)), g has strictly decreasing derivative. This immediately implies the twopossibilities (1a) and (1b) claimed in the lemma.Case (2) is an immediate consequence of the monotonicity of the maps f , f and of thefact that the graph of g is below the diagonal. (cid:3)
4. C
ODED SYSTEMS
In this section we only assume (H1), hypotheses (H2)–(H2+) are not required.The goal of this section is to prove Theorem 2.1.Let us first recall some standard definitions, see for example [41] for details. We onlyconsider two-sided sequence spaces. Given a subset S ⊂ Σ , define W n ( S ) def = { τ : | τ | = n, τ = ( ξ k +1 . . . ξ k + n ) for some ξ ∈ S and some k ∈ Z } the set of all allowed words of length n in S and let W ( S ) def = (cid:91) n ≥ W n ( S ) , where W ( S ) = ∅ by convention. A subshift is a σ -invariant set in Σ . A subset S ⊂ Σ is a subshift of finite type ( SFT ) if it is specified by finitely many “forbidden" words, all offinite length, that is, if there exists a finite family F ⊂ W (Σ ) so that S = Σ F def = { ξ ∈ Σ : W ( { ξ } ) ∩ F = ∅ } . Equivalently, there exist n ≥ and a finite family of words of equal length n , F (cid:48) ⊂W n (Σ ) , such that S = Σ F (cid:48) . It follows that any SFT is σ -invariant and that Σ F = { ξ ∈ Σ : ( ξ k +1 . . . ξ k + n ) (cid:54)∈ F for all k ∈ Z } . Let us introduce the concept of coded systems, though we will skip the original def-inition (see, for example, [41, Chapter 13.5] and references therein) and instead use thecharacterization by Krieger in [38]. By [38], a transitive subshift S ⊂ Σ is coded if andonly if there is an increasing family of irreducible SFTs whose union is dense in S . Recall the definition of the compact and σ -invariant set Σ ⊂ Σ in (2.2). Consider thesets W het (Σ) def = { τ : τ ∈ W (Σ) , f [ τ ] (1) = 0 } , W (Σ) def = { k : k ≥ } , and let W cod (Σ) def = W (Σ) \ ( W het (Σ) ∪ W (Σ)) . Note that with this notation, the set Σ het defined in (2.3) is precisely Σ het = { − N τ N : τ ∈ W het (Σ) } = { σ k ( ξ ) : ξ = (0 − N .τ N ) , τ ∈ W het (Σ) , k ∈ Z } . Let us now prepare the proof of Theorem 2.1. Recall notations in Section 3.
Proposition 4.1.
For any two disjoint SFTs S i ⊂ Σ , i = 1 , , not containing Z , thereexists a transitive SFT S ⊂ Σ such that S ∪ S ⊂ S and Z (cid:54)∈ S .Proof. As by assumption we have Z (cid:54)∈ ( S ∪ S ) and since the sets S i are compact, thereis N ≥ such that [0 N ] ∩ ( S ∪ S ) = ∅ . By Remark 3.8, without loss of generality, wecan assume that N also satisfies that (0 N − Z ∈ Σ .By the choice of N , N is a “forbidden word" in S i , i = 1 , , and, in particular,every sequence in S i must be of the type ξ = ξ − .ξ + , with ξ + = (0 (cid:96) m (cid:96) m . . . ) suchthat (cid:96) k ≤ N − and ≤ m k ≤ N for some N ≥ for all k ∈ Z (recall Remark3.10). Considering the points in (3.2) and letting a k = a [0 (cid:96) m ... (cid:96)n mk ] , we get a nestedsequence of intervals [ a k , such that a k ≤ x ξ + and a k monotonically converges to x ξ + as k → ∞ . Note that for every k ≥ we have a [0 (cid:96) (cid:96) ... (cid:96)k ≤ a [0 (cid:96) m (cid:96) m ... (cid:96)k mk ] . Moreover, by Remark 3.9, we have a [0 N − ... N − ≤ a [0 (cid:96) ... (cid:96)k . This allows us to conclude that for every ξ ∈ S i , i = 1 , , we have < a def = x (0 N − N ≤ x ξ + The argument for x ξ − is analogous, and we let b def = x (0 N − − N , where (0 N − − N =( . . . N − N − . Hence, for i = 1 , (4.1) < a = x (0 N − N ≤ min ξ ∈ S i x ξ + ≤ max ξ ∈ S i x ξ − ≤ x (0 N − − N = b < . Claim 4.2.
Every τ ∈ W ( S ) ∪ W ( S ) is forward admissible for b and backward admis-sible for a and satisfies < a ≤ f [ τ ] ( b ) < . Proof.
By the definition of admissibility, given ξ ∈ S i , i = 1 , , we have that ξ is admis-sible for x if and only if x ξ + ≤ x ≤ x ξ − . Hence, by the above, a ≤ x ξ + ≤ x ≤ x ξ − ≤ b and, in particular, for every n ∈ Z we have a ≤ f nξ ( x ) ≤ b. Therefore, every τ ∈ W ( S i ) is forward admissible for b . The proof of backward admissi-bility is analogous. (cid:3) The naive idea for the construction of the SFT S is to choose some appropriate N andto consider all concatenations of words of lengths at least N which come from the sub-shifts S i , i = 1 , , which are separated by words N . The precise definition is S = Σ F ,where a F is a certain finite set of forbidden words of length N . Instead of describing F , we define its complement in the set W N (Σ ) of allowed words of length N . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 21
We determine N > N as follows. Let k i ≥ be such that S i is a SFT generatedby a family of words of length k i , i = 1 , . Without loss of generality we can assumethat k = k ≥ N . Moreover, we can also assume that any pair of cylinders of length N in S i for i = 1 , , respectively, are disjoint. Now, as < a ≤ b < , we can choose N ≥ k (= k ) such that f [0 N ] ( a ) = f N ( a ) > b. Hence, for every x ∈ [ a, b ] we also have(4.2) f − N ( x ) < a ≤ b < f N ( x ) . The complement of the set F in the set W N (Σ ) is defined as follows:i) any word of length N allowed either in S or in S ,ii) any word (cid:96) v , with (cid:96) ∈ { , . . . , N } , | v | = 2 N − (cid:96) , and v ∈ W ( S ) ∪ W ( S ) ,iii) any word v (cid:96) , with (cid:96) ∈ { , . . . , N } , | v | = 2 N − (cid:96) , and v ∈ W ( S ) ∪ W ( S ) ,iv) any word of the form v N w , with | v | ≥ , | w | ≥ , | v | + | w | = N , and v, w ∈W ( S ) ∪ W ( S ) . Claim 4.3.
We have Z (cid:54)∈ S = ∅ and S ∪ S ⊂ S .Proof. By items ii)–iii), N is not allowed, as by the above N is not allowed neitherin S nor in S . It is also not allowed by i) nor by iv) (the latter – because one of thewords v, w has length at least N / > N ). Hence, we have [0 N ] ∩ S = ∅ , getting thefirst claim. By item i), we immediately get W N ( S i ) ⊂ W N ( S ) , i = 1 , , and hence S ∪ S ⊂ S . (cid:3) Claim 4.4. S is transitive.Proof. It is enough to check that for every pair of words v, w ∈ W ( S ) there exists aword η ∈ W (Σ ) such that vηw ∈ W ( S ) . Without loss of generality, we can assume | v | , | w | > N . There are three possible cases:(1) if v ends with and w begins with , then take η = 0 N ,(2) if v ends with and w begins with (cid:96) for some (cid:96) ∈ { , . . . , N − } , (cid:96) beingmaximal with this property, then take η = 0 N − (cid:96) , analogously for the reversedcase,(3) if v ends with (cid:96) and w begins with m for some (cid:96), m ∈ { , . . . , N − } , (cid:96) and m being maximal with these properties, – if (cid:96) + m < N , then take η = 0 N − (cid:96) − m , – if (cid:96) + m ≥ N , then take η = ∅ .Let us see that indeed in case (1) the word vηw is allowed in S . We can write v = v (cid:48)(cid:48) v (cid:48) and w = w (cid:48) w (cid:48)(cid:48) where | v (cid:48) | + | w (cid:48) | = N . Then it is enough to apply item (iv). Cases (2)and (3) are analogous. This proves the claim. (cid:3) What remains to prove is that S ⊂ Σ , which is an immediate consequence of thefollowing claim. Claim 4.5.
Every ξ ∈ S is admissible for some point in (0 , .Proof. Note that by the above definition of S , for ξ ∈ S we have either ξ ∈ S or ξ ∈ S or ξ = ( . . . τ k τ k . . . ) with τ n being subwords allowed in either S or in S . Withoutloss of generality, it is enough to assume that ξ = ( . . . τ − k .τ k τ k . . . ) and to showthat ξ is admissible for b .We start by checking ξ + is admissible for b . By Claim 4.2 the word τ is forwardadmissible for b and we have a ≤ f [ τ ] ( b ) < . Clearly, k is forward admissible for f [ τ ] ( b ) and by (4.2) we have f [ τ k ] ( b ) > b . As by Claim 4.2 the word τ is forwardadmissible for b and hence for f [ τ k ] ( b ) , we have that τ k τ is forward admissible for b defined in (4.1). Now we proceed by induction to show that ξ + is admissible for b .To check backward admissibility, first recall that by (4.2) we have f − k ( b ) < a . As byClaim 4.2 the word τ − is backward admissible for a , we have that τ − is also backwardadmissible for f − k ( b ) . We now argue inductively as before. (cid:3) The proof of the proposition is now complete. (cid:3)
Proof of Theorem 2.1.
We start by analyzing the “heteroclinic part" Σ het of Σ . Lemma 4.6.
Every ξ ∈ Σ het is an isolated point in Σ .Proof. It suffices to show that every τ ∈ W het (Σ) we have [ τ ] ∩ Σ = { − N .τ N } and, inparticular, τ has a unique continuation to a bi-infinite admissible sequence. By the choiceof τ , f [ τ ] (1) = 0 and therefore is only forward admissible at and hence τ can only becontinued to a backward admissible sequence by − N . Analogously, again by f [ τ ] (1) = 0 and also Remark 3.3, τ can only be continued to a forward admissible sequence by N .This proves the lemma. (cid:3) As every point in Σ het is isolated and non-periodic, it is wandering. In particular, thereis no invariant measure supported on this set. Further, Σ het is countable and hence itstopological entropy is zero. This proves the claimed properties of Σ het in the theorem.The facts that Σ cod is compact and σ -invariant follow from the above derived propertiesof Σ het . What remains to show is that it is coded. We start by the following lemma. Lemma 4.7.
For every SFT S ⊂ Σ and every word τ satisfying [ τ ] ∩ S = ∅ , τ (cid:54) = (0 . . . ,and f [ τ ] (1) ∈ (0 , , there is a(n infinite) SFT S (cid:48) satisfying S (cid:48) ∩ S = ∅ , Z (cid:54)∈ S (cid:48) , and [ τ ] ∩ S (cid:48) (cid:54) = ∅ .Proof. Since f [ τ ] (1) ∈ (0 , , there is a ∈ (0 , for which τ is forward admissible. Let b = f [ τ ] ( a ) . We can also assume that b > . Choose k ≥ such that f k ( b ) > a . Now it isenough to consider the SFT S (cid:48) generated by the family of words { τ k , τ k +1 } . (cid:3) We now inductively construct an increasing countable family { S k } of transitive SFTssuch that (cid:83) k S k = Σ cod . First observe that the set W cod (Σ) is countable and let { τ ( k ) } besome enumeration of it. Now let S = ∅ and for k = 1 , , . . . apply the following iterativeprocedure: • If [ τ ( k ) ] ∩ S k − (cid:54) = ∅ then S k def = S k − , • otherwise, if [ τ ( k ) ] ∩ S k − = ∅ , then – first apply Lemma 4.7 to S k − to obtain a SFT S (cid:48) k such that S (cid:48) k ∩ S k − = ∅ , Z (cid:54)∈ S (cid:48) k , and [ τ ( k ) ] ∩ S (cid:48) k (cid:54) = ∅ , – thereafter apply Proposition 4.1 to S k − and S (cid:48) k to obtain a transitive SFT S k such that S k ⊃ S k − ∪ S (cid:48) k and Z (cid:54)∈ S k . Observe that we have S k ⊃ S k − , Z (cid:54)∈ S k , and S k ∩ [ τ ( k ) ] (cid:54) = ∅ . This provides an increasing family of transitive SFTs { S k } k satisfying (cid:91) k ≥ S k ⊃ Σ cod \ { Z } = Σ cod . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 23
Moreover, since points from Σ het have arbitrarily long subsequences of zeros they do notbelong to any S k . Moreover, as by Lemma 4.6 the points in Σ het are isolated, they cannotbelong to the closure of (cid:83) S k . Thus, (cid:83) k S k = Σ cod .Finally, to see that σ is topologically mixing on Σ cod , consider two forward admissiblewords ( ξ . . . ξ n ) and ( η . . . η m ) and points x and y for which these words are admissible,respectively. Note that for every k ≥ sufficiently large it holds f [ ξ ... ξ n k ] ( x ) > y .Hence, the composed word ( ξ . . . ξ n k η . . . η m ) is admissible at y . This immediatelyimplies the mixing property.This completes the proof of Theorem 2.1. (cid:3)
5. U
NDERLYING STRUCTURES : H
OMOCLINIC CLASSES
In this entire section we only assume (H1)–(H2), hypothesis (H2+) is not required.We establish the notion of a homoclinic class of a hyperbolic periodic point of the skew-product F induced by the map ˜ F defined in (1.1), translating it from the differentiablesetting. The analogous analysis can be done for ˜ F but will be skipped. We see that there areonly two classes: one containing contracting orbits and the other one containing expandingones, see Propositions 5.1. These classes may intersect. Moreover, the “boundary of the set Γ " has two graph-like parts: one contained in H ( P, F ) and the other one in H ( Q, F ) , seeProposition 5.7. We also introduce homoclinic relations for parabolic periodic points, seeSection 5.3, and see that they are related simultaneously to periodic points of both typesof hyperbolicity and, in particular, to P and Q , see Proposition 5.11. This section onlydiscusses the topological structure of homoclinic classes. The study of their hyperbolicand ergodic properties is postponed.5.1. Homoclinic relations and classes.
In the differentiable setting, the homoclinic classof a hyperbolic periodic point is the closure of the transverse intersections of the stable andthe unstable invariant manifolds of its orbit. These homoclinic classes are transitive setswith a dense subset of periodic orbits. In our setting, the definition of a homoclinic class issimilar, the only difference is that transversality is not involved (note that here we can onlyspeak of invariant sets and cannot invoque any differentiable structure for these sets). Wewill follow closely the presentation in [18, Sections 2 and 3] where a similar discussion isdone for an specific class of skew-product maps (falling in the concave class studied here)and skip some details, see this reference for further details.In what follows, we denote by O ( X ) the F -orbit of a point X . Consider a periodicpoint R = (( ξ . . . ξ n − ) Z , r ) of F , note that f [ ξ ... ξ n − ] ( r ) = r . Recall that its orbitis hyperbolic if f (cid:48) [ ξ ... ξ n − ] ( r ) (cid:54) = 1 (by hypothesis, this derivative is positive). This orbitis contracting if f (cid:48) [ ξ ...ξ n − ] ( r ) ∈ (0 , , otherwise it is expanding . When the derivativeis equal to one the orbit is called parabolic . We define the stable set of R , W s ( R, F ) ,as the set of points X such that F i ( X ) → R as i → ∞ . The stable set of the orbitof R , W s ( O ( R ) , F ) , is the union of the stable sets of the points in O ( R ) . The unstablesets of R and O ( R ) are defined by W u ( R, F ) = W s ( R, F − ) and W u ( O ( R ) , F ) = W s ( O ( R ) , F − ) .Given now a hyperbolic periodic point R and its orbit O ( R ) ⊂ Γ we consider its homo-clinic points X ∈ W s ( O ( R ) , F ) ∩ W u ( O ( R ) , F ) and its homoclinic class H ( R, F ) def = { W s ( O ( R ) , F ) ∩ W u ( O ( R ) , F ) } . Note again that the “transversality" of the homoclinic intersections is not required. Notingthat H ( R, F ) is F -invariant and that Γ is a locally maximal invariant set it follows that H ( R, F ) ⊂ Γ , for every hyperbolic periodic point R with O ( R ) ⊂ Γ . The homoclinic class H ( R, F ) can be alternatively defined as follows. First, we say that apair of hyperbolic periodic points R and R of the same type of hyperbolicity are homo-clinically related if the un-/stable invariant sets of their orbits intersect cyclically, W s ( O ( R ) , F ) ∩ W u ( O ( R ) , F ) (cid:54) = ∅ (cid:54) = W u ( O ( R ) , F ) ∩ W s ( O ( R ) , F ) . Note that transversality is not required, but it is required that the two orbits have the sametype or hyperbolicity . It follows that being homoclinically related defines an equivalencerelation among hyperbolic periodic points of the same type of hyperbolicity. This is duethe fact that the fiber dynamics has no critical points and hence the intersections betweenthe invariant sets of O ( R ) and O ( R ) behave as transverse ones and have well definedcontinuations. In Section 5.3, we will extend homoclinic relation to include also parabolicperiodic points and will provide the proof that this new relation is an equivalence relation.Then H ( R, F ) = closure { R (cid:48) : R (cid:48) is homoclinically related to R } . As in the differentiable setting, H ( R, F ) is a transitive set. Note that, in general, twohomoclinic classes of periodic points may fail to be disjoint (see Remark 12.1 item (1c)).We will see that in our setting hyperbolic periodic points of the same type are homo-clinically related. Hence there exist only two homoclinic classes (related to P and Q ,respectively). Proposition 5.1.
Every hyperbolic periodic point R ∈ Γ of expanding (contracting) typeis homoclinically related to Q (to P ). In particular, two points R and R of expanding(contracting) type are homoclinically related and their common homoclinic class coincideswith the one of Q (of P ). Proof of Proposition 5.1.
To continue our discussion, we state a simple lemma abouthomoclinic relations which is just a reformulation of [18, Corollary 3.1] for the fixed points P and Q of F . For completeness and to illustrate the dynamics, we will sketch its proof. Lemma 5.2 (Characterisation of homoclinic points) . Consider a point X = ( ξ, x ) ∈ Γ .The point X is a homoclinic point of Q if and only if (5.1) ξ = (0 − N ξ − (cid:96) . . . ξ − .ξ . . . ξ k N ) , where (5.2) f [ ξ ... ξ k ] ( x ) = 0 and f [ ξ − (cid:96) ... ξ − . ] ( x ) ∈ [0 , . The point X is a homoclinic point of P if and only if ξ = (0 − N ξ − (cid:96) . . . ξ − .ξ . . . ξ k N ) where f [ ξ − m ... ξ − . ] ( x ) = 1 and f [ ξ ... ξ n ] ( x ) ∈ (0 , . Indeed, we may have periodic orbits with different type of hyperbolicity whose invariant sets intersectcyclically, this leads to a heterodimensional cycle involving these orbits, see Remark 5.5
KEW-PRODUCTS WITH CONCAVE FIBER MAPS 25
Proof.
We only prove the first part. Suppose that X = ( ξ, x ) is a homoclinic point of Q .This immediately implies that ξ = (0 − N ξ − (cid:96) . . . ξ − .ξ . . . ξ k N ) . Then the conditions f [ ξ ... ξ k ] ( x ) ∈ W s (0 , f ) = { } and f [ ξ − (cid:96) ... ξ − . ] ( x ) ∈ W u (0 , f ) ∩ [0 ,
1] = [0 , prove one implication. To prove the converse one note that (5.2) implies that f [ ξ ... ξ k n ] ( x ) = 0 , for every n ≥ and lim n →∞ f [0 − n ξ − (cid:96) ... ξ − . ] ( x ) = 0 , which together with (5.1) implies that X ∈ W s ( Q, F ) ∩ W u ( Q, F ) , ending the proof. (cid:3) For the next remarks, consider two hyperbolic periodic points R and R , where(5.3) R = (( ξ . . . ξ n − ) Z , r ) , R = (( η . . . η m − ) Z , r ) . Remark 5.3 (Homoclinic relations) . Assume that the points in (5.3) are of expanding type.They are homoclinically related if and only if there are points of the form X = (cid:0) (( ξ . . . ξ n − ) − N τ − (cid:96) . . . τ − .τ . . . τ (cid:96) ( η . . . η m − ) N ) , x (cid:1) , and Y = (cid:0) (( η . . . η m − ) − N ρ − (cid:96) . . . ρ − .ρ . . . ρ (cid:96) ( ξ . . . ξ n − ) N ) , y (cid:1) , such that f [ τ ... τ (cid:96) ] ( x ) ∈ { r } = W sloc ( r , f [ η ... η m − ] ) f [ τ − (cid:96) ... τ − . ] ( x ) ∈ W uloc ( r , f [ ξ ... ξ n − ] ) (5.4)and f [ ρ ... ρ (cid:96) ] ( y ) ∈ { r } = W sloc ( r , f [ ξ ... ξ n − ] ) f [ ρ − (cid:96) ... ρ − . ] ( y ) ∈ W uloc ( r , f [ η ... η m − ] ) . (5.5)Note that X ∈ W s ( R , F ) ∩ W u ( R , F ) and Y ∈ W s ( R , F ) ∩ W u ( R , F ) .There is a similar version for homoclinic relations of contracting points. Remark 5.4.
An immediate consequence of Lemma 5.2 and Remark 5.3 is that the homo-clinic classes H ( P, F ) and H ( Q, F ) are both non-trivial and hence F has infinitely manyhyperbolic periodic points (homoclinically related either to P or Q ). Indeed, as f ( d ) = 0 it follows that ((0 − N . N ) , d ) ∈ W s ( Q, F ) ∩ W u ( Q, F ) ⊂ H ( Q, F ) and (0 − N . N , f (1)) ∈ W s ( P, F ) ∩ W u ( P, F ) ⊂ H ( P, F ) . Hence, both homoclinic classes are infinite sets and hence they contain infinitely manyhyperbolic periodic points. We need to understand the homoclinic relations among them.We know that some of them are related to P and some to Q . The point of Proposition 5.1is that these are the only two possibilities.For completeness, we state the corresponding result for Remark 5.3 for periodic points(5.3) of different type of hyperbolicity. Remark 5.5 (Heterodimensional cycles) . Assume now that R and R in (5.3) are ofcontracting and expanding type, respectively. Note that in this case W uloc ( r , f [ ξ ... ξ n − ] ) = { r } and W sloc ( r , f [ η ... η m − ] ) = { r } , while W sloc ( r , f [ ξ ... ξ n − ] ) and W uloc ( r , f [ η ... η m − ] ) are open intervals. Assume that thereare points X = (cid:0) (( ξ . . . ξ n − ) − N .τ . . . τ (cid:96) ( η . . . η m − ) N ) , r (cid:1) , and Y = (cid:0) (( η . . . η m − ) − N .ρ . . . ρ (cid:96) ( ξ . . . ξ n − ) N ) , y (cid:1) , such that • f [ τ ... τ (cid:96) ] ( r ) ∈ { r } = W sloc ( r , f [ η ... η m − ] ) , • y ∈ W uloc ( r , f [ η ... η m − ] ) , • f [ ρ ... ρ (cid:96) ] ( y ) ∈ W sloc ( r , f [ ξ ... ξ n − ] ) .Then X ∈ W s ( R , F ) ∩ W u ( R , F ) and Y ∈ W s ( R , F ) ∩ W u ( R , F ) . In this case,following the terminology in the differentiable case, we say that R and R form a het-erodimensional cycle. Proof of Proposition 5.1.
We only consider the expanding case, the other one is analo-gous. Write R = (( ξ . . . ξ n − ) Z , r ) . Invoking Lemma 3.12, the fact that r is an expand-ing fixed point of f [ ξ ...ξ n − ] implies that f [ ξ ...ξ n − ] has exactly two periodic points in I [ ξ ...ξ n − ] = [ a, (with a = a [ ξ ...ξ n − ] and hence f [ ξ ... ξ n − ] ( a ) = 0 , using the notationin Remark 3.1): the point r = p +[ ξ ... ξ n − ] and a point r (cid:48) = p − [ ξ ... ξ n − ] with r < r (cid:48) and f (cid:48) [ ξ ...ξ n − ] ( r (cid:48) ) < (using the notation in Lemma 3.12). We immediately have [ a, r (cid:48) ) ⊂ W u ( r, f [ ξ ...ξ n − ] ) and [0 , ⊂ W u (0 , f ) . Consider the points X = ((( ξ . . . ξ n − ) − N . ( ξ . . . ξ n − )0 N ) , a ) and Y = ((0 − N . ( ξ . . . ξ n − ) N ) , r ) . Hence we have that X ∈ W u ( R, F ) and from r ∈ [0 , ⊂ W u (0 , f ) we get Y ∈ W u ( Q, F ) . Similarly, f [ ξ ...ξ n − ] ( a ) = 0 implies that X ∈ W s ( Q, F ) . Finally, Y ∈ W s ( R, F ) is obvious. Therefore, the invariant sets of Q and O ( R ) intersect cyclically andhence the points Q and R are homoclinically related. (cid:3) Corollary 5.6.
Consider two basic sets Γ , Γ ⊂ Γ of the same type of hyperbolicity. Thenthere is a horseshoe Γ ⊂ Γ containing Γ and Γ . We conclude this subsection justifying the comments in Remark 2.14. Recall the defi-nition of Σ cod in (2.4). Proposition 5.7.
Given ξ = ( ξ − .ξ + ) ∈ Σ cod , with I ξ = [ x ξ + , x ξ − ] we have ( ξ, x ξ − ) ∈ H ( P, F ) and ( ξ, x ξ + ) ∈ H ( Q, F ) . Note that in the above result we may have x ξ + = x ξ − . In such a case we have that H ( P, F ) ∩ H ( Q, F ) (cid:54) = ∅ (in that case both classes are nonhyperbolic). Proof of Proposition 5.7.
We prove the proposition for the point ( ξ, x ξ + ) , the proof for ( ξ, x ξ − ) is similar considering negative iterates. First let [ a k ,
1] = I [ ξ ... ξ k ] , where a k def = a [ ξ ... ξ k ] and recall (see Remark 3.1) that(5.6) f [ ξ ... ξ k ] ( a k ) = 0 and that ( a k ) k converges monotonically to x ξ + . Also recall that a k ≤ x ξ + . It is enough toshow the following. Claim.
There are infinitely many pairs ( m, k ) , both arguments unbounded, such that (5.7) f [ ξ − m ... ξ − . ] ( a k ) ∈ [0 , . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 27
Assuming the above claim, there is a sequence ( m (cid:96) , k (cid:96) ) (cid:96) , m (cid:96) → ∞ , k (cid:96) → ∞ , satisfying(5.7). Lemma 5.2 implies that A (cid:96) def = (cid:0) (0 − N ξ − m (cid:96) . . . ξ − .ξ . . . ξ k (cid:96) N ) , a k (cid:96) (cid:1) . is a homoclinic point of Q and hence belongs to H ( Q, F ) . As A (cid:96) → ( ξ, x ξ + ) , the propo-sition will follow. As (5.6) is always valid, it only remains to prove the claim. Proof of Claim.
There are the following cases according to the number of s in ξ . Case 1) ξ + has infinitely many s: We list the positions of s by i k . Note that ξ − isadmissible for x ξ + , hence for every point in [0 , x ξ + ] , and in particular for a i k . Note that inthis case we have a i k < x ξ + . We claim that f [ ξ − m ... ξ − . ] ( a i k ) ∈ [0 , for all m, k ≥ . Otherwise condition a i k < x ξ + together with monotonicity would imply < f [ ξ − m ... ξ − . ] ( x ξ + ) , in contradiction with the admissibility for x ξ + . Case 2) ξ + has finitely many s: Let (cid:96) ≥ such that ξ (cid:96) = 1 and ξ k = 0 for all k > (cid:96) . ByRemark 3.4 this implies that(5.8) a k = x ξ + for every k > (cid:96). We consider two subcases according to the number of s in ξ − : Case 2.a) ξ − has infinitely many s: By Remark 3.10, the number of consecutive ’s isbounded. List by j k the positions such that ξ − j k = 1 and ξ − j k − = 0 . By the above, ( j k ) k defines an increasing sequence. For each k there are two possibilities: • f [ ξ − jk ... ξ − . ] ( x ξ + ) = f [ ξ − jk ... ξ − . ] ( a k ) ∈ [0 , , • f [ ξ − jk ... ξ − . ] ( x ξ + ) = 1 and f [ ξ − jk − ξ − jk ... ξ − . ] ( x ξ + ) = f [0 ξ − jk ... ξ − . ] ( x ξ + ) = f [0 ξ − jk ... ξ − . ] ( a k ) ∈ [0 , . This yields (5.7) considering the sequences ( j k , k ) k and ( j k + 1 , k ) k , respectively. Case 2.b) ξ − has finitely many s: Let n ≥ such that ξ − n = 1 and ξ − m = 0 for all m > n . Note that we cannot have f [ ξ − n ... ξ − ] ( x ξ + ) = 1 . Indeed, by (5.8) we would have f [ ξ − n ... ξ − ξ ...ξ (cid:96) (1) = f [ ξ ... ξ (cid:96) ◦ f [ ξ − n ... ξ − ] (1)= f [ ξ ... ξ (cid:96) ( x ξ + ) = f [ ξ ... ξ (cid:96) ( a (cid:96) +1 ) = 0 . which would imply (recalling the definition of Σ het in (2.3)) ξ = ( ξ − .ξ + ) = (0 − N ξ − n . . . ξ − .ξ . . . ξ (cid:96) N ) ∈ Σ het and hence ξ (cid:54)∈ Σ cod , contradiction. Thus, we have f [ ξ − n ... ξ − . ] ( x ξ + ) ∈ [0 , and hence for every r ≥ we obtain f [ ξ − n − r ...ξ − n ... ξ − . ] ( x ξ + ) = f [0 r ξ − n ... ξ − . ] ( x ξ + ) = f [0 r ξ − n ... ξ − . ] ( a k ) ∈ [0 , This then yields (5.7) taking the sequence ( k, k ) k .This proves the claim. (cid:3) The proof of the proposition is now complete. (cid:3)
Homoclinic relations for parabolic periodic points.
We now consider homoclinicrelations including parabolic points. Note that in our concave setting a parabolic periodicpoint behaves as an attracting periodic point (to its right) and a repelling periodic point(to its left). In this way, to each such a point we will associate two “homoclinic classes”,one taking into account its contracting nature and the other one its expanding one. Let usprovide the details.Consider the set
Per ≤ ⊂ Γ of periodic points which are either parabolic or of contract-ing typ. Define Per ≥ analogously. We say that a pair of points A = (( ξ . . . ξ m − ) Z , a ) and B = (( η . . . η n − ) Z , b ) in Per ≤ are homoclinically ≤ related if: • either we have O ( A ) = O ( B ) , • or we have O ( A ) (cid:54) = O ( B ) and there are words ( α . . . α k ) and ( β . . . β (cid:96) ) such that f [ β ...β (cid:96) ] ( a ) ∈ int (cid:0) W sloc ( b, f [ η ...η n − ] ) (cid:1) and f [ α ...α k ] ( b ) ∈ int (cid:0) W sloc ( a, f [ ξ ...ξ m − ] ) (cid:1) . Analogously, we define being homoclinically ≥ related on the set Per ≥ . Remark 5.8 ((Parabolic) homoclinic relations) . Comparing the above definition with Re-mark 5.3, we note that there are some subtle differences considering the interior of thestable sets. Let us observe that if in the above definition A is parabolic, then int (cid:0) W † loc ( a, f [ ξ ...ξ m − ] ) (cid:1) = (cid:0) W † loc ( a, f [ ξ ...ξ m − ] ) (cid:1) \ { a } , † ∈ { s , u } . If A is of contracting type, then int (cid:0) W sloc ( a, f [ ξ ...ξ m − ] ) (cid:1) = W sloc ( a, f [ ξ ...ξ m − ] ) . Finally, if A is of expanding type, then int (cid:0) W uloc ( a, f [ ξ ...ξ m − ] ) (cid:1) = W uloc ( a, f [ ξ ...ξ m − ] ) . Hence, if A and B are both of contracting type (expanding type), we recover the usualhomoclinic relation (recall the characterization in (5.4), (5.5)). Lemma 5.9.
To be homoclinically ≤ (homoclinically ≥ ) related is an equivalence relationon Per ≤ (on Per ≥ ).Proof. We only consider the case ≤ . The only fact which remains to prove is transi-tivity. Let us consider the case A = (( ξ . . . ξ m − ) Z , a ) , B = (( η . . . η n − ) Z , b ) , C =(( τ . . . τ r − ) Z , c ) ∈ Per ≤ , where A and B are related and B and C are related. Bydefinition, there are words ( β . . . β (cid:96) ) and ( γ . . . γ i ) satisfying f [ β ...β (cid:96) ] ( a ) ∈ int (cid:0) W s ( b, f [ η ...η n − ] ) (cid:1) and f [ γ ...γ i ] ( b ) ∈ int (cid:0) W sloc ( c, f [ τ ...τ r − ] ) (cid:1) . Hence, for every x sufficiently close to b we have f [ γ ...γ i ] ( x ) ∈ int (cid:0) W sloc ( b, f [ τ ...τ r − ] ) (cid:1) . Since f [ β ...β (cid:96) ( η ...η n − ) j ] ( a ) → b as j → ∞ , for every j big enough we have that f [ β ...β (cid:96) ( η ...η n − ) j γ ...γ i ] ( a ) ∈ int (cid:0) W sloc ( c, f [ τ ...τ r − ] ) (cid:1) . This implies the first condition in the relation. The other one is completely analogous. (cid:3)
Remark 5.10.
Note that in the above definition for ≤ ( ≥ ) we needed to consider theinterior of the stable (unstable) manifolds. Without this hypothesis such a relation mayfail to be transitive. Note that for parabolic points the un-/stable manifolds are half-openintervals and intersections may occur in the boundary and this may cause non-transitivity. KEW-PRODUCTS WITH CONCAVE FIBER MAPS 29
Let H ≤ ( A, F ) be the closure of the set of points in Per ≤ which are homoclinically ≤ related to A and note that if A is of contracting type then H ≤ ( A, F ) = H ( A, F ) . Analo-gously for H ≥ ( A, F ) . If A is a parabolic point then the sets H ≤ ( A, F ) and H ≥ ( A, F ) necessarily intersect through the orbit of A , but they may be different. Proposition 5.11.
Every parabolic periodic point S ∈ Γ is homoclinically ≤ related to P and homoclinically ≥ related Q . Hence H ≤ ( S, F ) = H ( P, F ) and H ≥ ( S, F ) = H ( Q, F ) . In particular, if F has a parabolic periodic point then H ( P, F ) ∩ H ( Q, F ) (cid:54) = ∅ .Proof. We only show that a parabolic periodic point S = (( ξ . . . ξ m − ) Z , s ) is homo-clinically ≤ related to P . By Remark 3.1, we have I [ ξ ...ξ n − ] = [ a, , where a = a [ ξ ...ξ n − ] . By concavity, we have that ∈ W s ( f [ ξ ...ξ n − ] , s ) and thus f [ ξ ...ξ n − ] (1) ∈ int( W s ( f [ ξ ...ξ n − ] , s )) . As s ∈ (0 , ⊂ int( W s ( f , , it follows that S are P arehomoclinically ≤ related. As H ≤ ( P, F ) = H ( P, F ) we are done. (cid:3)
6. C
ONCAVE ONE - DIMENSIONAL MAPS
In this section we collect some auxiliary results. Throughout, we assume (H1)–(H2+)and let M be a constant as in (H2+). Similar arguments, in particular those in Lemma 6.3,can also be found in [2]. Recall the notation f nξ = f [ ξ ... ξ n − ] .6.1. Distortion control.
We prove two distortion results.
Lemma 6.1 (Controlled distortion) . For every ξ ∈ Σ , every x, y ∈ I ξ , x < y , and every n ≥ we have M − ≤ log( f nξ ) (cid:48) ( x ) − log( f nξ ) (cid:48) ( y ) (cid:80) n − k =0 ( f kξ ( y ) − f kξ ( x )) ≤ M. Proof.
By hypothesis (H2+), log( f nξ ) (cid:48) ( x ) − log( f nξ ) (cid:48) ( y ) = n − (cid:88) k =0 (cid:0) log f (cid:48) ξ k ( f kξ ( x )) − log f (cid:48) ξ k ( f kξ ( y )) (cid:1) ≥ M − n − (cid:88) k =0 ( f kξ ( y ) − f kξ ( x )) . The lower bound is analogous. (cid:3)
Lemma 6.2.
There are positive increasing functions C , C : (0 , → R with C i ( x ) → as x → such that for every interval I = [ x, y ] ⊂ [0 , satisfying | I | ≥ a we have | f i ( I ) || I | e C ( a ) ≤ f (cid:48) i ( x ) ≤ | f i ( I ) || I | e C ( a ) , i = 0 , . Proof.
For the first inequality, using (H2+), we have | f i ( I ) | = (cid:90) I f (cid:48) i ( z ) dz ≤ f (cid:48) i ( x ) (cid:90) I e − M − ( z − x ) dz = f (cid:48) i ( x ) 1 − e − M − | I | M − . Thus, if | I | ≥ a then f (cid:48) i ( x ) ≥ | f i ( I ) || I | | I | M − − e − M − | I | ≥ | f i ( I ) || I | e C ( | I | ) , where C ( a ) def = log( aM − / (1 − e − aM − )) has the claimed properties. The other inequality follows analogously, taking C ( a ) def = log( aM/ (1 − e − aM )) . (cid:3) Rescaling of moving intervals.
The following scheme will be used several times.Consider ξ ∈ Σ such that I ξ + is not a singleton and points x , x , x ∈ I ξ + satisfying x < x < x . Let I def = [ x , x ] . For n ≥ consider the “moving intervals" I n def = f nξ ( I ) = [ x n , x n ] , where x n def = f nξ ( x ) , x n def = f nξ ( x ) . Given k , for (cid:96) > k consider the rescaled map g (cid:96)k : I k → [0 , ∞ ] defined by(6.1) g (cid:96)k ( x ) def = | I k || I (cid:96) | ( f ξ (cid:96) − ◦ . . . ◦ f ξ k )( x ) . Observe that ( g (cid:96)k ) (cid:48) ( x ) = ( g (cid:96)(cid:96) − ) (cid:48) ( f (cid:96) − ξ ( x )) · . . . · ( g k +1 k ) (cid:48) ( f kξ ( x )) . Hence, for every x ∈ I we have(6.2) ( g n ) (cid:48) ( x ) = | I || I n | ( f nξ ) (cid:48) ( x ) . Further, observe that | g k +1 k ( I k ) | = | I k | . Note that concavity also implies(6.3) ( g k +1 k ) (cid:48) ( x k ) ≥ . Hence, arguing inductively, for every n ≥ we have(6.4) | g n ( I ) | = | I | . Lemma 6.3. If (cid:80) ∞ k =0 ( x k − x k ) = ∞ , then for every choice of x ∈ ( x , x ) we have lim n →∞ ( x n − x n ) = 0 . Proof.
Note that the sequence ( x n − x n ) n is not necessarily monotone, so convergence isnot immediate. We start by proving a slightly stronger fact. Claim. d def = lim n →∞ x n − x n x n − x n = 0 . As < | x n − x n | ≤ , the lemma is then an immediate consequence of the above claim. Proof of the claim.
Note that concavity of the maps implies that the sequence in the claimis monotonically decreasing, hence its limit exists. By contradiction, suppose that d > .By monotonicity of the derivatives of f , f and the choice of x , by (6.2) we have | ( g n ) (cid:48) ( x ) | = | I || I n | ( f nξ ) (cid:48) ( x ) = x − x x n − x n ( f nξ ) (cid:48) ( x ) ≥ x − x x n − x n x n − x n x − x > d. (6.5)Fix some y ∈ ( x , x ) and let α ∈ (0 , such that x = αy + (1 − α ) x . By concavity of g n , letting y n = f nξ ( y ) , we get | I || I n | ( αy n + (1 − α ) x n ) = αg n ( y ) + (1 − α ) g n ( x ) ≤ g n ( αy + (1 − α ) x ) = | I || I n | x n KEW-PRODUCTS WITH CONCAVE FIBER MAPS 31 and hence αy n + (1 − α ) x n ≤ x n , which implies x n − x n ≥ x n − y n ≥ (1 − α )( x n − y n ) ≥ (1 − α )( x n − x n ) . Hence x n − y n x n − x n ≥ (1 − α ) x n − x n x n − x n ≥ (1 − α ) d > . Further, again by monotonicity of the derivatives, (6.2), and by Lemma 6.1 together withthe above we obtain max z ,z ∈ [ y,x ] ( g n ) (cid:48) ( z )( g n ) (cid:48) ( z ) = ( g n ) (cid:48) ( y )( g n ) (cid:48) ( x ) = ( f nξ ) (cid:48) ( y )( f nξ ) (cid:48) ( x ) ≥ exp (cid:16) M − n − (cid:88) k =0 ( x k − y k ) (cid:17) ≥ exp (cid:16) M − (1 − α ) d n − (cid:88) k =0 ( x k − x k ) (cid:17) . By hypothesis, the latter diverges as n → ∞ . By (6.5), we obtain ( g n ) (cid:48) ( y ) → ∞ . Byconcavity, min z ∈ [ x ,y ] ( g n ) (cid:48) ( z ) → ∞ as n → ∞ . But this implies that g n ( y ) − g n ( x ) → ∞ , which contradicts (6.4), proving the claim. (cid:3) This proves the lemma. (cid:3)
Distance to fixed points.
The following lemma will be instrumental in the proof ofTheorem 2.6.
Lemma 6.4.
There exists a function h : (0 , ∞ ) → (0 , ∞ ) with lim t → h ( t ) = 0 satisfyingthe following property. Let ( ω . . . ω n ) ∈ { , } n , n ≥ , be a word such that the map g = f [ ω ... ω n ] has some fixed point in [0 , . If | g ( x ) − x | < ε for some x ∈ [ a [ ω ... ω n ] , ,then the distance of x to the closest fixed point of g is not larger than h ( ε ) .Proof. Let z be the (unique) point satisfying g (cid:48) ( z ) = 1 (that is, the maximum point for t (cid:55)→ g ( t ) − t ). Denote by x ± the fixed point(s) of g , x + ≤ z ≤ x − . Note that we mayhave x + = x − = z .We only study the case x ≤ z , the case x > z is analogous. Claim.
For every y ≤ z − √ ε we have g (cid:48) ( y ) ≥ e M − √ ε .Proof. By (H2+), we have log( f (cid:48) ω ( y ) /f (cid:48) ω ( z )) ≥ M − √ ε . Hence g (cid:48) ( y ) = f (cid:48) ω ( y ) · . . . · f (cid:48) ω n ( f [ ω ... ω n − ] ( y )) > f (cid:48) ω ( y ) f (cid:48) ω ( z ) · f (cid:48) ω ( z ) · . . . · f (cid:48) ω n ( f [ ω ... ω n − ] ( z )) ≥ e M − ε / g (cid:48) ( z ) = e M − ε / , proving the claim. (cid:3) Define h ( ε ) def = ( e M − √ ε − − ε + √ ε. Given ε > such that | g ( x ) − x | < ε , denote y = z − √ ε. There are two cases:
Case y < x + ≤ z : If y ≤ x ≤ z , then clearly we have | x − x + | ≤ h ( ε ) . Assume now x < y . By hypothesis and since x ≤ x + , we have ε > | g ( x ) − x | = − ( g ( x ) − x ) . Since g ( y ) − y < , by concavity we have ε > − ( g ( x ) − x ) = ( g ( x + ) − x + ) − ( g ( y ) − y ) + ( g ( y ) − y ) − ( g ( x ) − x ) > g − id) (cid:48) ( y )( y − x ) ≥ ( e M − √ ε − y − x ) , where for the latter we used the above claim. Thus, by the above, we obtain | x − x + | ≤ | x − y | + | y − x + | ≤ ε ( e M − √ ε − − + √ ε ≤ h ( ε ) . Case x + ≤ y : If y ≤ x ≤ z , by hypothesis and as x ≥ x + , we get ε > | g ( x ) − x | = g ( x ) − x. By concavity, applying the above claim, we have ε > ( g ( x ) − x ) − ( g ( y ) − y ) + ( g ( y ) − y ) − ( g ( x + ) − x + ) > g − id) (cid:48) ( y )( y − x + ) ≥ ( e M − √ ε − y − x + ) , which implies, as above, | x − x + | ≤ | x − y | + | y + x + | ≤ h ( ε ) . Finally, if x < y , we have ε > | g ( x ) − x | = | ( g ( x ) − x ) − ( g ( x + ) − x + ) | . Letting now w = max { x + , x } , using again the claim, we have ε > | ( g ( x ) − x ) − ( g ( x + ) − x + ) | ≥ ( g − id) (cid:48) ( w ) | x − x + | ≥ ( e M − √ ε − | x − x + | , obtaining | x − x + | < h ( ε ) . This proves the lemma. (cid:3)
7. D
ENSITY OF PERIODIC POINTS – P
ROOF OF T HEOREM
Approximation by (hyperbolic) periodic points.
First observe that, as our maps arelocal diffeomorphisms, every periodic point has positive derivative which might be equalto (parabolic) or different from (hyperbolic, either of contracting or of expanding type).The next lemma deals with the approximation of parabolic periodic points – in case suchpoints do exist – by hyperbolic ones. Lemma 7.1.
Assume (H1)–(H2). Every parabolic periodic point is accumulated by hyper-bolic periodic points of either type of hyperbolicity. Moreover, every parabolic periodicmeasure is weak ∗ accumulated by hyperbolic periodic measures of either type of hyper-bolicity.Proof. We only prove the lemma for periodic points of contracting type, the other case ofexpanding type is similar and hence omitted.Let X = F n ( X ) = (( ω . . . ω n − ) Z , x ) be a parabolic periodic point. Abbreviate theword ( ω . . . ω n − ) simply by ω . Hence f [ ω ] ( x ) = x and ( f [ ω ] ) (cid:48) ( x ) = 1 and therefore ω (cid:54) = 0 n and x (cid:54) = 0 . Notice also that ( f [ ω ] ) (cid:48) < in ( x, . Thus, f [ ω ] (( x, ⊂ ( x, . As x (cid:54) = 0 , there is k ≥ such that ( f k ) (cid:48) ( x ) < . Hence, we have ( f k ) (cid:48) < in [ x, . Note also that f k (( x, ⊂ ( x, . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 33
Noting that is in the basin of attraction of x with respect to f [ ω ] , given ε > suffi-ciently small, for every (cid:96) ≥ sufficiently large we have f [ ω (cid:96) ] (1) ∈ ( x, x + ε ) . Observethat f [ ω (cid:96) k ω (cid:96) ] (1) ≤ f [ ω (cid:96) ] (1) < x + ε. By the above, we have f [ ω (cid:96) k ω (cid:96) ] (( x, ⊂ ( x, x + ε ) and hence there is a periodic point p ( (cid:96) ) for f [ ω (cid:96) k ω (cid:96) ] in ( x, x + ε ) . Moreover ( f [ ω (cid:96) k ω (cid:96) ] ) (cid:48) < on ( x, . Since the periodicsequences η ( (cid:96) ) def = ( ω (cid:96) k ω (cid:96) ) Z = (( ω (cid:96) k ω (cid:96) ) − N . ( ω (cid:96) k ω (cid:96) ) N ) , satisfy η ( (cid:96) ) → ( ω − N .ω N ) = ω Z as (cid:96) → ∞ , the corresponding F -periodic points Y (cid:96) =( η ( (cid:96) ) , p ( (cid:96) ) ) converge to X = ( ω Z , x ) . Moreover, each Y (cid:96) is hyperbolic of contracting type.The claim about weak ∗ approximation of the parabolic measure by hyperbolic periodicones is immediate by construction. (cid:3) Proof of Theorem 2.6.
Throughout this section, we assume (H1)–(H2+). By Propo-sition 5.1, we have closure { A ∈ Γ : A hyperbolic and periodic } = H ( P, F ) ∪ H ( Q, F ) . Hence, the first claim in the theorem is then a consequence of the following lemma.
Lemma 7.2.
We have
Ω(Γ , F ) = closure { A ∈ Γ : A hyperbolic and periodic } .Proof. First recall that, by Theorem 2.1, every point in Γ het is isolated in Γ and non-periodic, hence it is wandering. Hence, to prove the lemma, it is enough to see that everynonwandering point in Γ cod is accumulated by hyperbolic periodic points. By Lemma7.1, every parabolic periodic point is accumulated by hyperbolic periodic ones. Hence itremains to show approximation by (either hyperbolic or parabolic) periodic points.Let X = ( ξ, x ) ∈ Γ cod be a nonwandering point, X (cid:54)∈ { P, Q } . For k ∈ Z denote x k = f kξ ( x ) . As the set of nonwandering points
Ω(Γ , F ) is F -invariant, for every k ∈ Z the point X k = F k ( X ) = ( σ k ( ξ ) , x k ) is also nonwandering. There are the following cases:1) there exists a smallest m ≥ so that x m = 0 ,2) there exists a smallest m ≥ so that x m = 1 .3) for every m ≥ we have x m ∈ (0 , . Case 1):
Given n ≥ , observe that f [ ξ − n ... ξ m ] ( x − n ) = 0 , hence the only forwardadmissible continuation at 0 is N , that is, we have ( ξ − n . . . ξ m ) = ( ξ − n . . . ξ m m − m ) for all m > m . Thus, for every m > m we have f [ ξ − n ... ξ m ] ( x − n ) = 0 . At the sametime, given ε > small, f [ ξ − n ... ξ m ] ( x − n + ε ) → as m → ∞ . Thus, we can find m ( ε ) > max { , n − m } such that for every m > m ( ε ) it holds f [ ξ − n ... ξ m ] ( x − n + ε ) > x − n + ε, and hence this map has a fixed point in the interval [ x − n , x − n + ε ] . Note that each suchpoint corresponds to a F -periodic point of period m , Y = (( η . . . η m − ) Z , y ) , where ( η . . . η m − ) = ( ξ − n . . . ξ − ξ . . . ξ n − ξ n . . . ξ m ) and y ∈ [ x − n , x − n + ε ] . Hence, F n ( Y ) ∈ [ ξ − n . . . ξ − .ξ . . . ξ n − ] × f [ ξ − n ... ξ − . ] ([ x − n , x − n + ε ])= [ ξ − n . . . ξ − ξ . . . ξ n − ] × [ x , x + τ ( ε )] , where τ ( ε ) → as ε → . Passing first with ε to and then with n to ∞ , we see that X isaccumulated by F -periodic points. Case 2):
This case is analogous to Case 1 considering backward iterates.
Case 3):
Given n ≥ choose ε > sufficiently small such that x − n + ε < and thatthe word ( ξ − n . . . ξ n − ) is backward admissible for all t > x − n − ε . Indeed, such ε existsbecause f [ ξ − n ... ξ m ] ( x − n ) ∈ (0 , for every m = 1 , . . . , n . Consider the neighborhood U = [ ξ − n . . . ξ n − ] × [ x − n − ε, x − n + ε ] of the nonwandering point F − n ( X ) . There exists a point Y = ( η, y ) ∈ U and a number m > n such that F m ( Y ) = ( σ m ( η ) , y m ) ∈ U . Note that as m > n , we have(7.1) ( η . . . η n − ) = ( ξ − n . . . ξ n − ) . Note also that I [ η ... η n − ] ⊃ I [ η ... η m − ] .Let g = f [ η ... η m − ] . Observe that y, g ( y ) ∈ [ x − n − ε, x − n + ε ] implies | g ( y ) − y | ≤ ε .There are two subcases to consider. Case 3a) The map g has a fixed point: By Lemma 6.4, there is a fixed point of g whichis h (2 ε ) -close to y . Case 3b) The map g has no fixed point: As there is no fixed point for g , we have(7.2) g ( t ) < t for all t ∈ I [ η ... η m − ] . In particular, y m def = g ( y ) satisfies y m < y . Moreover, observe that the number of ad-missible concatenations of g is bounded from above by some number (cid:96) . Given any δ ∈ (0 , ε ) , we now find a fixed point within the interval [ y m , y + δ ) . Since y m ≥ x − n − ε ,by the choice of ε and with (7.1), we have y m ∈ I [ ξ − n ... ξ n − ] = I [ η ... η n − ] , that is, ( η . . . η n − ) is forward admissible at y m . Note that for every k ≥ sufficiently large wehave ˜ y k def = ( f k ◦ f [ η ... η n − ] )( y m ) > y , hence ˜ y k ∈ I [ η ... η m − ] and we can apply g to ˜ y k .Observing again (7.2) and recalling that [ y m , y ) is a fundamental domain for g , for every k , there is a unique number (cid:96) = (cid:96) ( k ) ∈ { , . . . , (cid:96) } such that z k def = h ( k ) ( y m ) ∈ [ y m , y ) , where h ( k ) def = g (cid:96) ◦ f k ◦ f [ η ... η n − ] . Note that, by construction, the derivative of h ( k ) in [ y m , tends to as k → ∞ (here weuse the fact that (cid:96) ≤ (cid:96) and that f is contracting at ). Hence, h ( k ) has a unique fixed point q k ∈ [ y m , . Consider now the sequence ( q k ) k . Taking a subsequence, we can assumethat it converges to some point q ∞ ∈ [ y m , y ] . In the case when q ∞ < y , then q k ∈ [ y m , y ) for every large enough k . If q ∞ = y , then q k → y and hence q k ∈ [ y m , y + δ ) for everylarge enough k . And again we can apply Lemma 6.4 to find a fixed point of h ( k ) which is h (3 ε ) -close to y m .In both cases, as in Case 1), it follows that X is accumulated by F -periodic points.The proof of the lemma is now complete. (cid:3) What remains to show are the properties related to hyperbolicity. For that we will useTheorem 2.19 whose proof is postponed but is independent of what comes next.
Lemma 7.3.
If the sets H ( P, F ) and H ( Q, F ) both are hyperbolic, then they are disjoint.Proof. Note that P ∈ H ( P, F ) implies that F is uniformly contracting on H ( P, F ) and Q ∈ H ( Q, F ) implies that F is uniformly contracting on H ( Q, F ) . Thus, both sets aredisjoint. (cid:3) To conclude the proof of the theorem, it is enough to prove the following lemma.
Lemma 7.4.
Assume that H ( P, F ) ∩ H ( Q, F ) = ∅ . Then the sets H ( P, F ) and H ( Q, F ) are both hyperbolic.Proof. The lemma is a consequence of the following claim.
KEW-PRODUCTS WITH CONCAVE FIBER MAPS 35
Claim.
Every measure supported on H ( P, F ) is hyperbolic (of contracting type) and everymeasure supported on H ( Q, F ) is hyperbolic (of expanding type). By the above claim, the set H ( P, F ) is compact F -invariant and hyperbolic of contract-ing type on a set of total probability . Thus, we can invoke [3, Corollary E] and obtainthat H ( P, F ) is hyperbolic of contracting type. Analogously, H ( Q, F ) is hyperbolic ofexpanding type. This ends the proof of the lemma. Proof of the claim.
First recall that, by Proposition 5.11, parabolic periodic points are si-multaneously in both classes H ( P, F ) and H ( Q, F ) . As, by hypothesis, these classes aredisjoint, there are no such points. Also observe that, by Proposition 5.1, every periodicpoint of expanding type is in H ( Q, F ) . As a consequence, all periodic points in H ( P, F ) are hyperbolic of contracting type. Analogously, all periodic points in H ( Q, F ) are hyper-bolic of expanding type.It remains to see that all ergodic measures are hyperbolic. By contradiction, assumethat there is some ergodic nonhyperbolic measure. Then, by Theorem 2.19 such measureis simultaneously weak ∗ accumulated by hyperbolic periodic measures of contracting type(hence supported on H ( P, F ) ) and expanding type (hence on H ( Q, F ) ). The latter con-tradicts the disjointness of the homoclinic classes. (cid:3) This proves the lemma. (cid:3)
This completes the proof of Theorem 2.6. (cid:3)
8. P
ARTITION OF THE SPACES OF ERGODIC MEASURES – P
ROOFS OF T HEOREMS
AND Σ lifts to at least one (Section8.1) and at most two (Section 8.2) ergodic measures in Γ . In the latter case we call those twin-measures and study their distance (Section 8.3). Thereafter, we will conclude theproofs of Theorems 2.13 and 2.15. We close this section discussing the frequencies of ’sand ’s, which will be used in the bifurcation analysis in Section 11.3.Unless otherwise stated, in this section we will assume hypotheses (H1)–(H2+). Recallthat hypothesis comes with a constant M .8.1. Lifting measures.
Observe that in Theorem 2.13, given ν ∈ M erg (Σ) , the existenceof ergodic measures µ ∈ M erg (Γ) projecting to ν is a simple fact which does not dependon (H2+). The main point of Theorem 2.13 is the fact that either there is precisely onehyperbolic measure of each type projecting to ν or there is only one nonhyperbolic one;our proof requires (H2+). Indeed, we have the following lemma. Lemma 8.1.
Assume (H1)–(H2). Given any ν ∈ M erg (Σ) , there exist measures µ ∈ M erg , ≤ (Γ) and µ ∈ M erg , ≥ (Γ) such that π ∗ µ = ν = π ∗ µ .Proof. In the case when ν is the Dirac measure supported on Z then the lemma followsimmediately taking µ and µ being the Dirac measures supported on P and Q , respec-tively.Assume now that ν ∈ M erg (Σ) is different from such a Dirac measure. Consider a ν -generic point ξ = ξ − .ξ + . Observe that ξ + (cid:54) = 0 N . Hence the forward admissible intervalis I ξ + = [ a ξ + , , where a ξ + > . Take any ξ -admissible point x . Denote x n = f nξ ( x ) .Let y def = lim inf n x n , and observe that hence y > . Recall that a set A is of total probability if µ ( A ) = 0 for every F -invariant probability measure µ . Claim.
There exists a sequence ( n k ) k such that x n k ≥ x n .Proof. If y = 1 then y = lim n x n and the claim follows.Assume now y ∈ (0 , . Recall that the graph of f is below the diagonal. Let ε > small so that f ( x ) < x − ε for every x ∈ [ d, and f ( x ) > y + ε for every | x − y | < ε (which is possible because f ( y ) > y ). Choose n ≥ so that x n < y + ε . Consider asubsequence ( n k ) k such that | x n k − y | < ε for every k . There are two cases: First, if x n k ≥ x n for infinitely many k ’s, then we are done taking the sequence ( x n k ) k . Otherwise, wecan assume that x n k < x n for every k . Then we must have ξ n k +1 = 0 eventually. Indeed,otherwise the sequence ( x n k ) k satisfies x n k < x n and, by our choice of ε , x n k +1 = f ( x n k ) < x n k − ε < x n − ε < y, contradicting the definition of y . Hence, by the choice of ε , in this case it follows x n k +1 = f ( x n k ) > y + ε > x n and we are done taking the sequence ( x n k +1 ) k . (cid:3) For the following, for simplicity, assume that n = 0 .Consider now for each k the periodic measure ν k supported on η ( k ) = ( ξ . . . ξ n k − ) Z .By the above claim, we have f [ ξ ... ξ nk − ] ([ x , ⊂ [ x n k , ⊂ [ x , . Hence, η ( k ) is forward admissible at x . By Remark 3.6, there exists a point y k = f n k ξ ( y k ) = f n k η ( k ) ( y k ) with nonnegative Lyapunov exponent. Consider now the periodic measure µ + k supportedon the periodic orbit of ( η ( k ) , y k ) with nonnegative fiber Lyapunov exponent. By construc-tion, ν k = π ∗ µ + k . Analogously, using the nonpositive exponent-periodic points providedby Remark 3.6, we get an ergodic measure µ − k with nonpositive Lyapunov exponent andsatisfying ν k = π ∗ µ − k . Note that both measures µ ± k may coincide, in which case the ex-ponent is zero. Consider weak ∗ accumulation measures µ ± of µ ± k as k → ∞ . Observethat, by construction, π ∗ µ ± = ν . Clearly, µ + has a nonnegative ( µ − has a nonpositive)Lyapunov exponent. A priori , the measures µ ± are not ergodic, but each one will have some ergodic compo-nent having the claimed properties (recall Remark 2.12). This proves the lemma. (cid:3) Projection, disintegration, and twin-measures.Lemma 8.2.
For every ν ∈ M erg (Σ) , there are at least one and at most two ergodicmeasures µ , µ ∈ M erg (Γ) with π ∗ ( µ ) = ν = π ∗ ( µ ) .Proof. Given ν ∈ M erg (Σ) , let ξ be some ν -generic sequence. Given x ∈ I ξ , let X =( ξ, x ) and consider a weak ∗ accumulation measure µ = lim n (cid:96) →∞ n (cid:96) ( δ X + δ F ( X ) + . . . + δ F n(cid:96) − ( X ) ) , which is an F -invariant probability measure. By Remark 2.12, every ergodic component µ (cid:48) of µ satisfies ν = π ∗ µ (cid:48) . This proves that there is at least one ergodic F -invariant measureprojecting to ν .To show that there are no more than two ergodic measure projecting to ν , assume bycontradiction that there are (at least) three such measures, say µ i satisfying π ∗ µ i = ν , i = KEW-PRODUCTS WITH CONCAVE FIBER MAPS 37 , , . Observe that we can choose points ( ξ ( i ) , x i ) which are generic for µ i , i = 1 , , respectively, and satisfy ξ (1) = ξ (2) = ξ (3) def = ξ . Hence lim n →∞ n n − (cid:88) k =0 δ F k ( ξ,x i ) = µ i . Up to relabelling, we can assume that x < x < x . Denote x ni def = f nξ ( x i ) , i = 1 , , .By monotonicity, we have x n < x n < x n for every n . The following two cases can occur:(1) (cid:80) ∞ k =0 ( x k − x k ) = ∞ ,(2) (cid:80) ∞ k =0 ( x k − x k ) < ∞ .In Case (1), by Lemma 6.3, we have lim n ( x n − x n ) = 0 . Since ( ξ, x ) and ( ξ, x ) aregeneric points within the common fiber, it follows µ = µ .In Case (2), the sum of the series being finite immediately implies that lim n ( x n − x n ) =0 and thus lim n ( x n − x n ) = 0 . Hence, arguing as before, we get µ = µ and µ = µ and hence the three measures µ , µ , and µ coincide. (cid:3) Lemma 8.3.
Assume that ( µ n ) n ⊂ M (Γ) is a sequence satisfying • lim n π ∗ µ n = ν ∈ M erg (Σ) , • π − ∗ ν has just one element µ ∈ M erg (Γ) ,then lim n µ n = µ .Proof. Consider some subsequence ( n i ) i such that ( µ n i ) i weak ∗ converges to some mea-sure ˜ µ ∈ M (Γ) . Since ( π ∗ µ n ) n converges to ν , this is also true for its subsequence ( π ∗ µ n i ) i . By continuity of π ∗ , we obtain π ∗ ˜ µ = ν . As π − ∗ ν is just one element, we have ˜ µ = µ . Since the subsequence was arbitrary, we conclude weak ∗ convergence. (cid:3) “Distance" between twin-measures.Lemma 8.4. For every ν ∈ M erg (Σ) and every ergodic measures µ , µ satisfying π ∗ µ i = ν , i = 1 , , we have M − ≤ χ ( µ ) − χ ( µ ) − (cid:82) x dµ + (cid:82) x dµ ≤ M. Proof.
As in the proof of Lemma 8.2, we choose µ i -generic points ( ξ, x i ) , i = 1 , , in acommon fiber. Up to relabelling, we can assume x < x . Then, by ergodicity, we obtain lim n →∞ n log( f nξ ) (cid:48) ( x ) − n log( f nξ ) (cid:48) ( x ) n (cid:80) n − k =0 f kξ ( x ) − n (cid:80) n − k =0 f kξ ( x ) = χ ( µ ) − χ ( µ ) (cid:82) x dµ − (cid:82) x dµ . By applying Lemma 6.1, we conclude the proof. (cid:3)
Hyperbolicity of measures – Proof of Theorem 2.13.
Let ν ∈ M erg (Σ) . By Lemma8.2 there are at least one and at most two ergodic measures projecting to ν that are preciselyCases a) and b), respectively, claimed in the theorem: Case a).
There are two measures µ , µ ∈ M erg (Γ) such that π ∗ µ = ν = π ∗ µ . Asin the proof of Lemma 8.2, we choose µ i -generic points ( ξ, x i ) , i = 1 , , in a commonfiber. Up to relabelling, we can assume x < x . As µ (cid:54) = µ , their Wasserstein distance D def = W ( µ , µ ) is positive. Note that, by monotonicity of the fiber maps, we can invokeLemma A.1 and obtain D = (cid:90) x dµ ( ξ, x ) − (cid:90) x dµ ( ξ, x ) . Let I = [ x , x ] and I n def = f nξ ( I ) = [ x n , x n ] , where x n def = f nξ ( x ) , x n def = f nξ ( x ) . As we have chosen generic points within the same fiber, there is an infinite sequence ofpositive integers n such that | I n | ≥ D/ > . Denote by J = J ( D ) the infinite set ofsuch indices, J ( D ) def = (cid:110) n ≥ | I n | ≥ D (cid:111) . Claim 8.5.
The set J ( D ) has density at least D/ , that is, lim inf n →∞ n card { k ∈ { , . . . , n − } : k ∈ J ( D ) } ≥ D . Proof.
Since ( ξ, x i ) , i = 1 , , are generic, we have D = W ( µ , µ ) = lim n →∞ n n − (cid:88) k =0 ( x k − x k ) = lim n →∞ n n − (cid:88) k =0 | I k | . Hence, for every n sufficiently large we have n n − (cid:88) k =0 | I k | ≥ D. Hence, writing J n def = J ( D ) ∩ [0 , n ] and denoting by card J n its cardinal, we can concludethat for every such nn D ≤ n − (cid:88) k =0 | I k | < card J n · n − card J n ) D . Therefore, n card J n > D, proving the claim. (cid:3) To estimate the exponent of µ , recalling that ( ξ, x ) is µ -generic (guaranteeing theexistence of the first limit below) and recalling the definition of the rescaled maps g (cid:96)k in(6.1) and using (6.2), we have(8.1) χ ( µ ) = lim n →∞ n log( f nξ ) (cid:48) ( x ) = lim n →∞ n log (cid:18) ( g n ) (cid:48) ( x ) | I n || I | (cid:19) . Claim 8.6.
Taking C ( D/ > as in Lemma 6.2, we have ( g k +1 k ) (cid:48) ( x k ) (cid:40) ≥ e C ( D/ if k ∈ J ( D ) , ≥ otherwise . Proof. If k ∈ J ( D ) , then using | I k | ≥ D/ and applying Lemma 6.2 to I k , we obtain e C ( D/ ≤ | I k || I k +1 | f (cid:48) ξ k +1 ( x k ) = ( g k +1 k ) (cid:48) ( x k ) . Note that for every k we have ( g k +1 k ) (cid:48) ( x k ) ≥ , recall (6.3). (cid:3) KEW-PRODUCTS WITH CONCAVE FIBER MAPS 39
Recall that ( g n ) (cid:48) ( x ) = ( g ) (cid:48) ( x )( g ) (cid:48) ( x ) · · · ( g nn − ) (cid:48) ( x n − ) . Thus, Claims 8.5 and8.6 together imply lim n →∞ n log( g n ) (cid:48) ( x ) ≥ lim n →∞ n log (cid:16) ( e C ( D/ ) card J n · n − card J n (cid:17) ≥ C ( D · D . With (8.1), we have χ ( µ ) = lim n →∞ n log (cid:18) ( g n ) (cid:48) ( x ) | I n || I | (cid:19) ≥ lim n →∞ ,n ∈ J n log (cid:18) ( g n ) (cid:48) ( x ) D/ | I | (cid:19) ≥ D · C ( D . Now we take κ ( D ) def = D/ · C ( D/ .The analogous argument for the inverse fiber maps implies that χ ( µ ) ≤ − κ ( D ) < where κ ( D ) = D/ C ( D/ , and hence completing the proof of Case a). Case b).
There is only one ergodic measure µ such that π ∗ µ = ν . This implies ν (cid:54) = δ Z and hence any ν -generic sequence ξ = ξ − .ξ + is such that ξ + contains infinitely many ’s.To prove the claim, arguing by contradiction, assume that χ ( µ ) (cid:54) = 0 , say χ ( µ ) > (thecase χ ( µ ) > is analogous). Given ε ∈ (0 , χ ( µ )) , there is a (forward) generic point ( ξ, x ) so that there exists n ≥ such that for every n ≥ n we have(8.2) ( f nξ ) (cid:48) ( x ) ≥ e nε . Note that by the previous comment, we can assume x ∈ (0 , . Take any y ∈ ( x, . Noticethat y ∈ I ξ + (see Remark 3.2). Considering the sequence of orbital measures µ n uniformlydistributed on { ( ξ, y ) , F ( ξ, y ) , . . . , F n − ( ξ, y ) } , there is some subsequence which weak ∗ converges to some F -invariant measure µ (cid:48) . Notice that π ∗ µ (cid:48) = π ∗ µ . Hence, by hypothesis, µ (cid:48) = µ . In particular, it is unnecessary to consider a subsequence and thus ( ξ, y ) is in fact µ -forward generic. Thus, taking the limit of Birkhoff sums of the function ( ξ, s ) (cid:55)→ s , weobtain lim n →∞ n n − (cid:88) k =0 ( f kξ ( y ) − f kξ ( x )) = 0 . For sufficiently large n , by using (8.2) we obtain n − (cid:88) k =0 ( f kξ ( y ) − f kξ ( x )) ≤ n ε M .
Thus, with Lemma 6.1 max u,w ∈ [ x,y ] | log( f nξ ) (cid:48) ( u ) − log( f nξ ) (cid:48) ( w ) | ≤ M n − (cid:88) k =0 ( f kξ ( y ) − f kξ ( x )) ≤ M n ε M = n ε . Therefore, for every z ∈ [ x, y ] , with (8.2) we obtain ( f nξ ) (cid:48) ( z ) > e nε/ . Hence, | f nξ ([ x, y ]) | → ∞ as n → ∞ , which is a contradiction.This completes the proof of Case b) and hence the proof of Theorem 2.13. (cid:3) Remark 8.7.
Notice that in the proof of Theorem 2.13 Case b), the µ -generic point is ofthe form ( ξ, x ) with x (cid:54) = 1 . This allowed as to take y ∈ ( x, . Note also that the onlymeasure having a generic point of the form ( ξ, is the Dirac measure at P = (0 Z , .This will no longer be true in the setting of Section 11 studying bifurcation scenarios. Thisis precisely the place where another measure will “appear", see Proposition 11.8 Case c). Proof of Theorem 2.15.
Let µ ∈ M erg (Γ) and ν = π ∗ µ . Observe that, by Remark2.12, ν is ergodic. By invariance of the disjoint subsets Σ sing and Σ spine in (2.7), ν issupported on one of them only.If ν (Σ sing ) = 1 then I ξ = { x ξ } is a singleton ν -almost everywhere. Hence, by disinte-gration, µ ξ = δ x ξ ν -almost everywhere. In particular, there is no other measure projectingto ν . Hence, by Theorem 2.13 b), µ is nonhyperbolic.Otherwise, ν (Σ spine ) = 1 and I ξ = [ x ξ + , x ξ − ] , where x ξ + < x ξ − ν -almost every-where. Consider the measures µ ± def = (cid:90) Σ δ x ξ ± dν ( ξ ) . It is clear that both are F -invariant. Since x ξ + < x ξ − almost everywhere, we have µ + (cid:54) = µ − . What remains to see is that these measures are ergodic and of the claimed type ofhyperbolicity. Observe that for every µ (cid:48) ∈ M (Γ) satisfying π ∗ µ (cid:48) = ν we have (cid:90) x dµ + ( ξ, x ) = (cid:90) x ξ + dν ( ξ ) ≤ (cid:90) x dν ( ξ ) ≤ (cid:90) x dµ (cid:48) ( ξ, x ) ≤ (cid:90) x ξ − dν ( ξ ) = (cid:90) x dµ − ( ξ, x ) . Hence, the measures µ ± are extremal points in the subspace { µ (cid:48) : µ (cid:48) ∈ M (Γ) , π ∗ µ (cid:48) = ν } .By Remark 2.12, any ergodic component of µ ± also projects to ν . Thus µ ± cannot havea nontrivial ergodic decomposition. Hence µ ± both are F -ergodic. By Theorem 2.13 a), µ ± are hyperbolic with opposite type of hyperbolicity and there are no further ergodicmeasures projecting to ν . Hence, µ ∈ { µ + , µ − } . This proves Theorem 2.15. (cid:3) Frequencies.
We conclude with some consequences of Theorem 2.13. They willbe used in Section 11 when analyzing explosion of entropy and of the space of ergodicmeasure in bifurcation scenarios.
Lemma 8.8.
Assume (H1)–(H2). For every ν ∈ M erg (Σ) we have ν ([0]) log f (cid:48) (1) + ν ([1]) log f (cid:48) (1) ≤ . Proof.
Let ν ∈ M erg (Σ) . Arguing by contradiction, suppose that the statement is false.Any ergodic measure µ projecting to ν satisfies χ ( µ ) = (cid:90) log f (cid:48) ξ ( x ) dµ ( ξ, x ) ≥ (cid:90) log f (cid:48) ξ (1) dµ ( ξ, x )= ν ([0]) log f (cid:48) (1) + ν ([1]) log f (cid:48) (1) > . But this contradicts Lemma 8.1 which guarantees the existence of an ergodic measureprojecting to ν with nonpositive Lyapunov exponent. (cid:3) Given ξ ∈ Σ and a ∈ { , } , for natural numbers n < m we define freq mn ( ξ, a ) def = 1 m − n + 1 card { k ∈ { n, . . . , m } : ξ k = a } . Let(8.3) freq( ξ, a ) def = lim sup n →−∞ ,m →∞ freq mn ( ξ, a ) and define freq( ξ, a ) analogously taking lim inf instead of lim sup . Observe that thosefunctions are measurable and σ -invariant. Hence, for every ν ∈ M erg (Σ) for ν -almostevery ξ we have freq( ξ, a ) = freq( ξ, a ) = ν ([ a ]) . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 41
Corollary 8.9.
Assume (H1)–(H2). For every ξ ∈ Σ we have freq( ξ,
0) log f (cid:48) (1) + freq( ξ,
1) log f (cid:48) (1) ≤ . Proof.
Given ξ , consider subsequences ( n i ) i and ( m i ) i such that lim i →∞ freq m i n i ( ξ,
0) = freq( ξ, and consider the probability measures m i − n i + 1 m i (cid:88) k = n i δ σ k ( ξ ) . Then any weak ∗ accumulation measure ν of those measures is σ -invariant and satisfies ν ([0]) = freq( ξ, . Observe that ν ([1]) = freq( ξ,
1) = 1 − freq( ξ, . Now it suffices to consider the ergodic decomposition of ν and apply Lemma 8.8. (cid:3)
9. A
CCUMULATIONS OF ERGODIC MEASURES IN M (Γ) : P ROOF OF T HEOREM ∗ approxima-tion, the remainder of this section discusses approximation also in entropy. One essentialstep in the proof are so-called skeletons, defined and discussed in Section 9.2.9.1. Periodic approximation of ergodic measures.Proposition 9.1 (Density of hyperbolic periodic measures) . Assume (H1)–(H2+). Everymeasure in M erg (Γ) is weak ∗ accumulated by hyperbolic periodic measures in M erg (Γ) . For the above result in the case when the measure is hyperbolic see Remark 2.21. Wewill provide a proof, building upon the concavity hypotheses, which has intrinsic interest.It applies to any (also nonhyperbolic) ergodic measure µ (cid:54)∈ { δ P , δ Q } . The simple cases µ ∈ { δ P , δ Q } we check separately in Lemma 9.3. Corollary 9.2.
Assume (H1)–(H2). Every measure in M erg (Σ) is weak ∗ accumulated byperiodic measures in M erg (Σ) . Assuming (H1)–(H2+), the above corollary is a consequence of Proposition 9.1 togetherwith the fact that for every ν ∈ M erg (Σ) there exists µ ∈ M erg (Γ) with π ∗ µ = ν , seeLemma 8.2. The general case, assuming only (H1)–(H2), we prove at the end of thissection. Proof of Proposition 9.1.
We first deal with the simplest case.
Lemma 9.3.
The measure δ Q (the measure δ P ) is weak ∗ accumulated by periodic mea-sures.Proof. Note that the sequence ξ ( n ) = (0 n Z is admissible for any n sufficiently large (seeRemark 3.8). Moreover, we can apply Lemma 3.12 to obtain two fixed points p + n < p − n for the map f [0 n . Observe that for every z ∈ (0 , f (1)) there exists n such that forevery n ≥ n we have ( f ◦ f n )( z ) > z and hence p + n ≤ z ≤ p − n . This implies that lim n →∞ p + n = 0 . Moreover, we have q + n def = f n ( p + n ) = f − ( p + n ) → d = f − (0) . As the periodic orbit of the point ((0 n Z , p + n ) (projected to [0 , ) consists of the points q + n , f − ( q + n ) , . . . , f − n ( q + n ) , almost all of them stay close to . More precisely, given any z ∈ (0 , , there exists n = n ( z ) such that for every n ≥ n we have f − n ( d ) < z .As z > can be chosen arbitrarily close to , in this way we construct a sequence Q n =( ξ ( n ) , p + n ) of periodic points whose F -invariant probability measures supported on its orbitconverge weak ∗ to δ Q .The analogous arguments apply to δ P considering F − . (cid:3) We collect some preparatory results. Recall that (cid:37) ( ξ, x ) = x is the canonical projectionto the second coordinate. Remark 9.4.
For every µ ∈ M erg (Γ) , µ (cid:54)∈ { δ P , δ Q } , there exists a ∈ (0 , such that (cid:37) ∗ µ ( J ) > for both intervals J = (0 , a ) and J = ( a, . Lemma 9.5.
For every µ ∈ M erg (Γ) , µ (cid:54)∈ { δ P , δ Q } , there exist µ -generic points ( ξ, x ) such that there exists infinitely many times n ≥ with f nξ ( x ) > x .Proof. By Remark 9.4, there exists a ∈ (0 , such that (cid:37) ∗ µ ( J ) > for J = (0 , a ) and J = ( a, . Hence, there exists a µ -generic point R = ( ξ, x ) in Σ × (0 , a ) . Then,by Poincaré recurrence, the orbit of R by F has infinitely many return times n ≥ to Σ × ( a, and therefore satisfies f nξ ( x ) > a > x , proving the lemma. (cid:3) Lemma 9.6.
Let µ ∈ M erg (Γ) and ( ξ, x ) be any µ -generic point such that there areinfinitely many times n i ≥ with f n i ξ ( x ) > x . Then for every i ≥ the sequence ξ ( n i ) = ( ξ . . . ξ n i − ) Z is admissible and there exist repelling and contracting points p +[ ξ ... ξ ni − ] = f [ ξ ... ξ ni − ] ( p +[ ξ ... ξ ni − ] ) < x < p − [ ξ ... ξ ni − ] = f [ ξ ... ξ ni − ] ( p − [ ξ ... ξ ni − ] ) . Let P ± n i def = (( ξ . . . ξ n i − ) Z , p ± [ ξ ... ξ ni − ] ) and consider the sequence of ergodic measures ( µ ± n i ) n i supported on their orbits µ ± n i = 1 n i ( δ P ± n + δ F ( P ± n ) + . . . + δ F n − ( P ± n ) ) . Then we have that lim i →∞ π ∗ µ ± n i = ν def = π ∗ µ. Moreover, any weak ∗ accumulation point µ + of ( µ + n i ) i and any weak ∗ accumulation point µ − of ( µ − n i ) i satisfies (9.1) χ ( µ + ) ≥ ≥ χ ( µ − ) and χ ( µ + ) ≥ χ ( µ ) ≥ χ ( µ − ) . Finally, there are ergodic components ˜ µ ± of µ ± , respectively, such that π ∗ ˜ µ ± = ν and χ (˜ µ + ) ≥ ≥ χ (˜ µ − ) and χ (˜ µ + ) ≥ χ ( µ ) ≥ χ (˜ µ − ) . Proof.
Let [ a i , be the admissible interval for the map f [ ξ ... ξ ni − ] . We have f [ ξ ... ξ ni − ] ( a i ) =0 and f [ ξ ... ξ ni − ] ( x ) > x . Hence this map has at least one fixed point, which cannotbe parabolic as its graph crosses the diagonal. Hence Lemma 3.12 case (1a) proves thefirst claim providing the repelling point p +[ ξ ... ξ ni − ] and the contracting point p − [ ξ ... ξ ni − ] .Moreover, the choices of the times n i prove the inequalities for x .Let ν = π ∗ µ . Observe that ξ is ν -generic and that the sequence of ergodic measures ( ν n i ) i supported on the periodic sequences ξ ( n i ) converges in the weak ∗ topology to ν .Consider the measures µ ± n i in the statement of the lemma. Observe that π ∗ µ + n i = ν + n i KEW-PRODUCTS WITH CONCAVE FIBER MAPS 43 converges in the weak ∗ topology to ν . By the first part of the lemma, P + n i is of expandingtype and P − n i of contracting type. This implies the first part of (9.1). The second part of(9.1) follows from concavity and the relative position of x .Finally, by Remark 2.12, ν is ergodic and any ergodic component ˜ µ + of µ + also satisfies π ∗ ˜ µ + = ν . By the ergodic decomposition of µ + , there is one component with nonnegativeexponent and exponent not smaller than χ ( µ ) . The argument for µ − is analogous. (cid:3) Lemma 9.7.
Assume (H1)–(H2+). For every µ ∈ M erg , (Γ) there is the weak ∗ limit of asequence of periodic measures in M erg ,> (Γ) . Analogously, there is a sequence of periodicmeasures in M erg ,< (Γ) converging weak ∗ to µ .Proof. Given µ ∈ M erg , (Γ) , clearly µ (cid:54)∈ { δ P , δ Q } . Let ν = π ∗ µ ∈ M erg (Σ) . Byapplying Lemma 9.5, we have that the hypotheses of Lemma 9.6 are satisfied and hencethere is a sequence ( µ ± n i ) i of periodic measures such that π ∗ µ ± n i = ν ± n i converges to ν . ByTheorem 2.13 b), µ is the only F -invariant (ergodic) measure projecting to ν . Hence, byLemma 8.3 the sequences ( µ ± n i ) i both weak ∗ converge to µ . (cid:3) Lemma 9.8.
Every µ ∈ M erg ,> (Γ) is the weak ∗ accumulation point of a sequence ofperiodic measures ( µ n ) n ⊂ M erg ,> (Γ) . Analogously for measures in M erg ,< (Γ) .Proof. The case µ = δ Q is just Lemma 9.3. For µ (cid:54) = δ Q , by Lemma 9.5 the hypothesesof Lemma 9.6 are satisfied and hence there is a sequence of periodic measures ( µ + n i ) i converging to some F -invariant measure ˜ µ . Clearly χ (˜ µ ) ≥ and π ∗ ˜ µ = π ∗ µ = ν . ByLemma 9.6, indeed we have χ (˜ µ ) ≥ χ ( µ ) > . We claim that ˜ µ = µ .If ˜ µ is ergodic, then by Theorem 2.13 a) we have ˜ µ = µ . Otherwise, if ˜ µ is not ergodic,by Theorem 2.13 a), then there is exactly one further measure µ (cid:48) ∈ M erg ,< (Γ) alsoprojecting to ν such that ˜ µ = αµ + (1 − α ) µ (cid:48) for some α ∈ (0 , and hence χ (˜ µ ) < χ ( µ ) ,a contradiction. (cid:3) The proposition now follows from Lemmas 9.3, 9.7, and 9.8. (cid:3)
Proof of Corollary 9.2 assuming only (H1)–(H2).
Recall again that for every ν ∈ M erg (Σ) there exists µ ∈ M erg (Γ) with π ∗ µ = ν , see Lemma 8.2. Replacing Lemma 9.7 by Theo-rem 2.22 (which only requires (H1)–(H2)), we conclude the proof. (cid:3) Skeletons.
In this section we first collect some ingredients necessary to prove The-orem 2.22. They are somewhat similar to the ones in [21, Section 3.1], though herewe essentially rely on concavity arguments and neither on minimality nor one expand-ing/contracting itineraries. The focus in this section is on nonhyperbolic measures, only.In this section we adopt the following notation. Given k ≥ and ξ ∈ Σ , consider thefollowing probability measure put on the orbit of ξ (9.2) A k ξ def = 1 k ( δ ξ + δ σ ( ξ ) + . . . + δ σ k − ( ξ ) ) . The analogous notation we use for one-sided sequences.
Definition 9.9 (Skeleton ∗ property) . By a measure µ ∈ M erg , (Γ) having the skeleton ∗ property we mean that for every ε ∈ (0 , h ( µ )) there exist numbers a ∈ (0 , , C > , and n ≥ such that for every n ≥ n there is k ∈ { n (1 − ε ) , . . . , n } and there is a finiteset X = X ( h, ε, a, C ) = { X i } of points X i = ( ξ ( i ) , x i ) (a Skeleton relative to
F, µ, ε ) sothat: (i) the set X has cardinality card X ≥
Cnε e n (1 − ε )( h − ε ) , (ii) the words ( ξ ( i )0 . . . ξ ( i ) k − ) are all different,(iii) − kε < log ( f kξ ( i ) ) (cid:48) ( x i ) < kε, (iv) x i ∈ (0 , a ) and f kξ ( i ) ( x i ) ∈ ( a, ,(v) for ν = π ∗ µ the Wasserstein distance W in M (Σ) satisfies(9.3) W (cid:16) A k ξ ( i ) , ν (cid:17) < ε. Remark 9.10 (Relation with skeletons in other contexts) . All previously used definitionsof so-called skeletons (see [21, Section 4] and [24, Section 5.2] have the following essentialingredients: (i) cardinality governed by entropy, (ii) spanning property, (iii) finite-timeLyapunov exponents close to zero, (iv) connecting times, and (v) proximity in the weak ∗ topology. In our context, item (iv) is clearly different as we are not using minimalityproperties but obtain connecting times from concavity. Moreover, item (v) is stated interms of the Wasserstein distance. Proposition 9.11.
Assuming (H1), every µ ∈ M erg , (Γ) has the skeleton ∗ property.Proof. The proof will use some preliminary standard arguments that we borrow from [21,Section 3.1]. First note that, as by hypothesis we have µ (cid:54)∈ { δ P , δ Q } , by Remark 9.4 thereis a ∈ (0 , such that (cid:37) ∗ µ ( J ) > for J = (0 , a ) and J = ( a, , where (cid:37) is the projectionto the second coordinate. Let A def = Σ × (0 , a ) and B def = Σ × ( a, . By the above, bothsets have positive measure µ . Choose ε > satisfying ε < min { µ ( A ) / , µ ( B ) / } . Lemma 9.12.
There are a set A ⊂ Γ satisfying µ ( A ) > − ε and a number N ≥ such that for every n ≥ N and every X ∈ A we have F (cid:96) ( X ) ∈ Σ × ( a, for some (cid:96) ∈ { n (1 − ε ) − , . . . , n } .Proof. By ergodicity, recurrence, and Egorov’s theorem, there are a set A ⊂ Γ satisfying µ ( A ) > − ε and a number N (cid:48) ≥ such that for every n ≥ N (cid:48) and every X ∈ A wehave (cid:12)(cid:12)(cid:12)(cid:12) n card { (cid:96) ∈ { , . . . , n − } : F k ( X ) ∈ B } − µ ( B ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . Now let N ≥ N (cid:48) such that for every n with n (1 − ε ) ≥ N we have(9.4) nε ( µ ( B ) − ε ) > . Hence, by the above, for every X ∈ A and every n with n (1 − ε ) ≥ N card { (cid:96) ∈ { n (1 − ε ) , . . . , n } : F (cid:96) ( X ) ∈ B } = card { (cid:96) ∈ [0 , n ] : F (cid:96) ( X ) ∈ B } − card { (cid:96) ∈ [0 , n (1 − ε ) −
1] : F (cid:96) ( X ) ∈ B }≥ ( µ ( B ) − ε ) n − ( n (1 − ε ) − µ ( B ) + ε ) ≥ nε ( µ ( B ) − ε ) > , where in the last inequality we used (9.4). This proves the lemma. (cid:3) Lemma 9.13.
There are a set A ⊂ Γ satisfying µ ( A ) > − ε and N ≥ such that forevery n ≥ N and every X = ( ξ, x ) ∈ A and for every (cid:96) ∈ { n (1 − ε ) , . . . , n } it holds W ( A (cid:96) ξ, π ∗ µ ) < ε . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 45
Proof.
First, by Remark 2.12, π ∗ µ is ergodic. We now consider a countable dense set ofcontinuous functions and to each of them apply Birkhoff’s ergodic theorem to π ∗ µ . Usingagain Egorov’s theorem, we conclude the proof. (cid:3) Lemma 9.14.
There are A ⊂ Γ satisfying µ ( A ) > − ε and N ≥ such that for every X = ( ξ, x ) ∈ A and every (cid:96) ≥ N we have − (cid:96)ε < log ( f (cid:96)ξ ) (cid:48) ( x ) < (cid:96)ε. Proof.
The lemma follows from Birkhoff’s theorem using χ ( µ ) = (cid:90) log f (cid:48) ξ ( x ) dµ ( ξ, x ) = 0 and then applying Egorov’s theorem. (cid:3) Lemma 9.15.
There are A ⊂ Γ satisfying µ ( A ) > − ε and N ≥ such that for every n ≥ N and every X = ( ξ, x ) ∈ A it holds e − n ( h ( µ )+ ε ) ≤ ν ([ ξ . . . ξ n − ]) ≤ e n ( h ( µ ) − ε ) . Proof.
Note that by Remarks 2.12 and 2.5, ν def = π ∗ µ is ergodic and h ( ν ) = h ( µ ) . Thelemma now follows from Brin-Katok’s theorem [13] applied to µ and ν and then applyingEgorov’s theorem. (cid:3) By construction, the set A (cid:48) def = A ∩ A ∩ A ∩ A ∩ A satisfies µ ( A (cid:48) ) > µ ( A ) − ε > .Now let n def = max { N , N , N , N } . Take n ≥ n and for every (cid:96) ∈ { n (1 − ε ) , . . . , n } denote by A (cid:48) ( (cid:96) ) ⊂ A (cid:48) the set of points X ∈ A (cid:48) such that F (cid:96) ( X ) ∈ B . Let k be the index of a set A (cid:48) ( k ) with maximal measure.Hence, by the pigeonhole principle µ ( A (cid:48) ( k )) ≥ µ ( A (cid:48) ) nε + 1 > . Let S (cid:48) def = π ( A (cid:48) ( k )) and observe ν ( S (cid:48) ) ≥ µ ( A (cid:48) ( k )) > . Choose any point X =( ξ , x ) ∈ A (cid:48) ( k ) . Let S def = S (cid:48) \ [ ξ . . . ξ k − ] . We continue inductively: choose any ξ (cid:96) ∈ S (cid:96) − , let S (cid:96) def = S (cid:96) − \ [ ξ (cid:96) . . . ξ (cid:96)k − ] and X (cid:96) = ( ξ (cid:96) , x (cid:96) ) and repeat. As by Lemma 9.15we have ν ( S (cid:96) ) ≥ ν ( S (cid:48) ) − (cid:96)e − k ( h ( µ ) − ε ) , we can repeat this process at least m times, where m ≥ ν ( S (cid:48) ) e k ( h ( µ ) − ε ) ≥ µ ( A (cid:48) ( k )) e k ( h ( µ ) − ε ) ≥ µ ( A (cid:48) ) nε + 1 e k ( h ( µ ) − ε ) ≥ µ ( A ) − εnε + 1 e n (1 − ε )( h ( µ ) − ε ) Taking X def = { X i } Mi =1 , we obtain item (i) taking C > appropriately. Since, by construc-tion, the cylinders [ ξ ( i )0 . . . ξ ( i ) k − ] are pairwise disjoint, proving item (ii). As k ≥ N , item(iii) now follows from Lemma 9.14. Analogously, we have item (v) from Lemma 9.13.Finally, item (iv) follows from the very definition of X . (cid:3) Weak ∗ and entropy approximation of ergodic measures: Proof of Theorem 2.22. For any hyperbolic measure µ ∈ M erg ,< (Γ) ∪ M erg ,> (Γ) the weak ∗ and entropy ap-proximation of µ is a well known consequence of [15, 31]. Thus, in what follows, we willassume µ ∈ M erg , (Γ) and ν = π ∗ µ . Denote h = h ( µ ) and let ε ∈ (0 , h ) .The steps of this proof are the following. First, we choose an appropriate skeleton with n sufficiently large and k ∈ { n (1 − ε ) , . . . , n } . Based on that, we construct a certainSFT S + contained in Σ + which associates a IFS of contracting interval maps. Thereafter,we estimate the topological entropy of this SFT. And we also prove that every measuresupported on S + is close to ν + in the correspondingly defined Wasserstein distance. Here ν + is the measure ν projected to Σ + . Finally, we argue that the convergence (in entropyand Wasserstein distance) on Σ + implies the one on Σ , proving the proposition.
1. Choice of skeletons.
By Proposition 9.11, there are numbers a ∈ (0 , , C > , and n ≥ providing the claimed skeletons X = X ( h, ε, a, C ) = { X i } i of finitely manypoints X i = ( ξ i , x i ) . Below, we will choose n ≥ n sufficiently large. To that end, fix L ∈ (0 , | log f (cid:48) (1) | ) and consider(9.5) Z def = { z ∈ [0 ,
1] : f (cid:48) ( z ) ≤ e − L } , which is a nontrivial closed interval containing . Let (cid:96) ≥ be the smallest integer suchthat f (cid:96) ( a ) ∈ Z . Now choose(9.6) n > max (cid:26) n , (cid:96) (1 − ε ) ε · log f (cid:48) (0) (cid:27) so that(9.7) n + (cid:96) + 2 nε/L log (cid:18) Cnε e n (1 − ε )( h − ε ) (cid:19) ≥ h − ε. Let(9.8) ε (cid:48) def = max k ∈{ n (1 − ε ) ,...,n } k + (cid:96) + 2 kε/L (cid:18) (1 + kε ) + (cid:18) (cid:96) + 2 kεL (cid:19)(cid:19) . Note that ε (cid:48) = O ( ε ) as n → ∞ .Now take k ∈ { n (1 − ε ) , . . . , n } and the skeleton X = { X i } i , X i = ( ξ ( i ) , x i ) ∈ Γ , asprovided by Proposition 9.11. By item (i), it holds(9.9) card X ≥
Cnε e n (1 − ε )( h − ε ) . Let(9.10) s def = (cid:96) + (cid:96) (cid:48) , where (cid:96) (cid:48) def = (cid:24) kεL (cid:25) . For notational simplicity, let us assume that(9.11) (cid:96) (cid:48) = 2 kεL .
By property (iv) of the skeleton, for every i we have x i ∈ (0 , a ) and x (cid:48) i def = f kξ ( i ) ( x i ) = f [ ξ ( i )0 ... ξ ( i ) k − ] ( x i ) ∈ ( a, KEW-PRODUCTS WITH CONCAVE FIBER MAPS 47 and hence x i < a < x (cid:48) i . Note that applying any concatenation of f to any point x ∈ (0 , moves this point further to the right towards , and hence(9.12) f s ◦ f [ ξ ( i )0 ... ξ ( i ) k − ] ( a ) > f [ ξ ( i )0 ... ξ ( i ) k − ] ( a ) > f [ ξ ( i )0 ... ξ ( i ) k − ] ( x i ) > a > x i . Observe that, in particular, by the choice of (cid:96) , we have(9.13) ( f (cid:96) ◦ f [ ξ ( i )0 ... ξ ( i ) k − ] )( x i ) ∈ Z.
2. Construction of an admissible SFT.
We now introduce a certain SFT. Instead of de-scribing its set of forbidden words of length r = k + s (with s as in (9.10)), we define itscomplement. First, given the set { X i } i of points X i = ( ξ ( i ) , x i ) from the skeleton, define W (cid:48) def = { ( ξ ( i )0 . . . ξ ( i ) k − ) } i and prolong each word in W (cid:48) to an allowed word of length r def = k + s in the following way(9.14) W def = { ( η ( i )0 . . . η ( i ) r − ) = ( ξ ( i )0 . . . ξ ( i ) k − s ) : ( ξ ( i )0 . . . ξ ( i ) k − ) ∈ W (cid:48) } . Define(9.15) S + def = r − (cid:91) j =0 ( σ + ) j ( W N ) , where W N def = { ω + = ( η ( i ) η ( i ) . . . ) : η ( i m ) ∈ W , m ≥ } ⊂ Σ +2 is the one-sided subshift of all one-sided infinite concatenations of such words of lenght r .The next result asserts that it is a subshift of forward admissible (one-sided) sequences. Lemma 9.16. W N ⊂ Σ + .Proof. By (9.12), ( η ( i )0 . . . η ( i ) r − ) is forward admissible for a , that is, [ a, ⊂ I [ η ( i )0 ... η ( i ) r − ] ,and(9.16) f [ η ( i )0 ... η ( i ) r − ] ([ a, ⊂ ( a, . Thus, any pair of maps of the type f [ η ( i )0 ... η ( i ) r − ] , ( η ( i )0 . . . η ( i ) r − ) ∈ W , can be concatenated.Hence any one-sided infinite concatenations of words in W is forward admissible. (cid:3)
3. A contracting IFS.
The following is immediate from (9.16).
Lemma 9.17.
For every ( η ( i )0 . . . η ( i ) r − ) ∈ W there exists p − i = p − [ η ( i )0 ...η ( i ) r − ] satisfying p − i = f [ η ( i )0 ... η ( i ) r − ] ( p − i ) ∈ ( a, . For every i , define(9.17) g i def = f [ η ( i )0 ... η ( i ) r − ] and let p − def = min i p − i > a. The above ingredients will define the announced IFS.
Lemma 9.18.
For every i , it holds g i ([ p − , ⊂ [ p − , .Proof. By (9.12), for all i it holds a < g i ( a ) . This together with p − i = g i ( p − i ) implies thatthe graph of g i on ( a, p − i ) is above the diagonal. As a < p − ≤ p − i , we have p − ≤ g i ( p − ) for all i , proving the lemma. (cid:3) By Lemma 9.18, we can consider the IFS generated by the restriction of the maps g i tothe interval [ p − , . Lemma 9.19 (Uniform contractions) . The IFS { g i } i is contracting on [ p − , .Proof. By (9.13), for every i , we have z i def = ( f (cid:96) ◦ f [ ξ ( i )0 ... ξ ( i ) k − ] )( x i ) ∈ Z . Therefore, by(9.5) and (9.11), we have(9.18) ( f (cid:96) (cid:48) ) (cid:48) ( z i ) ≤ e − L(cid:96) (cid:48) = e − kε . Hence, ( g i ) (cid:48) ( x i ) = ( f (cid:96) (cid:48) ◦ f (cid:96) ◦ f [ ξ ( i )0 ... ξ ( i ) k − ] ) (cid:48) ( x i ) ≤ ( f [ ξ ( i )0 ... ξ ( i ) k − ] ) (cid:48) ( x i ) · (max f (cid:48) ) (cid:96) · ( f (cid:96) (cid:48) ) (cid:48) ( z i ) (by item (iii) of the skeleton, (9.18)) ≤ e kε · ( f (cid:48) (0)) (cid:96) · e − L kε/L = e − kε ( f (cid:48) (0)) (cid:96) (using n (1 − ε ) ≤ k ) ≤ e − n (1 − ε ) ε ( f (cid:48) (0)) (cid:96) (by (9.6)) < . Recall again that, by (9.12) and (9.17), x i < a < p − . Hence, concavity of the fiber mapsimplies that the above upper bounds also hold for any point in the interval [ p − , . (cid:3) Lemma 9.20.
For every ω + = ( η ( i ) η ( i ) . . . ) ∈ W N , η ( i m ) ∈ W for every m ≥ , theset z ( ω + ) def = (cid:92) j ≥ z j ( ω + ) , where z j ( ω + ) def = j (cid:92) m =1 ( g i ◦ . . . ◦ g i m )([ p − , , is just one point.Proof. By Lemma 9.19, the IFS is uniformly contracting. Note that z j ( ω + ) is a nestedsequence of compact intervals and hence z ( ω ) is either a point or a compact interval. Bycontraction, it is just one point. (cid:3) Lemma 9.21. S + ⊂ Σ + .Proof. By Lemma 9.16, every ω + ∈ W N is forward admissible and z ( ω + ) defined inLemma 9.20 is an admissible point. It follows that for every i = 0 , . . . , r − the map f iω + is well defined at z ( ω + ) and, in particular, ( σ + ) i ( ω + ) is forward admissible. (cid:3) Denote by σ + the one-sided shift on Σ +2 . Note that S + is a σ + -invariant subshift whichis a SFT. Clearly, the natural extension of σ + : S + → S + is the SFT σ : S → S , where S = r − (cid:91) i =0 σ i ( W Z ) are all bi-infinite concatenations of words in W . It is clear that, by construction, σ : S → S is topologically transitive. At the end of the proof, we will use the following fact. Lemma 9.22. S ⊂ Σ .Proof. Let ξ = ( . . . ξ − .ξ ξ . . . ) ∈ S . By definition, for every j ≥ its “one-sidedinfinite remainder" ( ξ − j . . . ξ − ξ + ) belongs to S + . By Lemma 9.21, it is forward admis-sible. Thus, observing that the two-sided sequence η ( j ) def = (0 − N .ξ − j . . . ξ − ξ + ) is (for-ward and backward) admissible, we can conclude that the shifted sequence σ − j ( η ( j ) ) = KEW-PRODUCTS WITH CONCAVE FIBER MAPS 49 ( . . . ξ − j . . . ξ − .ξ . . . ξ j . . . ) is (forward and backward) admissible. Therefore, we have C j ∩ Σ (cid:54) = ∅ , where C j def = [ ξ − j . . . ξ − .ξ . . . ξ j ] . Observe that { C j } j is a compact decreasing sequence of compact nonempty sets eachintersecting the compact set Σ . Hence, also (cid:84) j C j intersects Σ and therefore ξ ∈ Σ . (cid:3)
4. Estimate of entropy.Lemma 9.23. h top ( σ + , S + ) ≥ h ( µ ) − ε .Proof. Take k ≤ n from the skeleton X , for n as in (9.6), and s as in (9.10). Recall that,by our choice (9.14), we have card W = card X . Then h top ( σ + , S + ) = 1 k + s h top (( σ + ) k + s , S + ) = 1 k + s log card W (by (9.10) and (9.9)) ≥ k + (cid:96) + 2 kε/L log (cid:18) Cnε e n (1 − ε )( h − ε ) (cid:19) (using that k ≤ n ) ≥ n + (cid:96) + 2 nε/L log (cid:18) Cnε e n (1 − ε )( h − ε ) (cid:19) (by (9.7)) ≥ h − ε = h ( µ ) − ε, proving the lemma. (cid:3)
5. Weak ∗ approximation in M (Σ + ) . In the space Σ +2 we adopt the definition of thedistance d defined in (3.1) in the analogous way and consider the Wasserstein distance W on M (Σ +2 ) . Consider the natural projection π + : Σ → Σ +2 to the space of one-sidedsequences and let ν + def = ( π + ) ∗ ν . Recall the notation A k for the finite averages of shiftedmeasures in (9.2). Lemma 9.24.
For every ξ + ∈ Σ +2 and every ζ + ∈ [ ξ . . . ξ k − ] + ⊂ Σ +2 we have W ( A k ζ + , A k ξ + ) < k . Proof.
We have W ( δ σ (cid:96) ( ζ + ) , δ σ (cid:96) ( ξ + ) ) ≤ e − ( k − (cid:96) ) . For every m ≥ k and every ζ + ∈ [ ξ . . . ξ k − ξ k . . . ξ m − ] + we hence have W ( A k ζ + , A k ξ + ) ≤ k k − (cid:88) (cid:96) =0 e − ( k − (cid:96) ) , which implies the claim of the lemma. (cid:3) Lemma 9.25.
For every ˜ ν ∈ M ( S + ) it holds W (˜ ν, ν + ) ≤ ε (cid:48) , where ε (cid:48) is as in (9.8) .Proof. Every sequences ξ ( i ) in the skeleton X = { X i } of points X i = ( ξ ( i ) , x i ) } satisfies W ( A k ( ξ ( i ) ) + , ν + ) = W (( π + ) ∗ A k ( ξ ( i ) ) , ( π + ) ∗ ν ) by item (v) of the skeleton property, see (9.3) ≤ W ( A k ( ξ ( i ) ) , ν ) < ε. (9.19)Recall the choice of words ( ξ ( i )0 . . . ξ ( i ) k − . . . in (9.14) which define S + . Lemma 9.24and (9.19) then imply that for every η + ∈ [ ξ ( i )0 . . . ξ ( i ) k − ] we have(9.20) W ( η + , ν + ) ≤ W ( η + , A k ( ξ ( i ) ) + ) + W ( A k ( ξ ( i ) ) + , ν + ) ≤ k + ε. Recall that, by [47, Corollary to Theorem 4], every invariant measure has a genericpoint. Let now ˜ ν ∈ M erg ( S + ) and take a ˜ ν -generic point ζ + ∈ S + . Recall that ( σ + ) j ( ζ + ) is also ˜ ν -generic for every j ≥ . By construction of S + in (9.15), we can assume thatthere is some index i so that ζ + ∈ [ ξ ( i )0 . . . ξ ( i ) k − . . . . Observe that, by construction of S + , for every m ≥ we then also have(9.21) ( σ + ) mr ( ζ + ) ∈ [ ξ ( i m )0 . . . ξ ( i m ) k − . . . for some i m in the index set of the skeleton. Recall that r = k + s = k + (cid:96) + (cid:96) (cid:48) . Considerthe decomposition(9.22) A r ζ + = 1 k + (cid:96) + (cid:96) (cid:48) (cid:0) k A k ζ + + ( (cid:96) + (cid:96) (cid:48) ) R ( k, ζ + ) (cid:1) , for some probability measures R ( k, ζ + ) .By Lemma A.1, the Wasserstein distance satisfies(9.23) W ( sν + (1 − s ) ν , ν + ) = sW ( ν , ν + ) + (1 − s ) W ( ν , ν + ) for arbitrary ν , ν probability measures and s ∈ [0 , . Hence, with (9.22) we conclude W ( A r ζ + , ν + ) = 1 k + (cid:96) + (cid:96) (cid:48) (cid:0) k · W ( A k ζ + , ν + ) + ( (cid:96) + (cid:96) (cid:48) ) W ( R ( k, ζ + ) , ν + ) (cid:1) ≤ k + (cid:96) + (cid:96) (cid:48) (cid:0) k · W ( A k ζ + , ν + ) + ( (cid:96) + (cid:96) (cid:48) ) · (cid:1) (using (9.20)) ≤ k + (cid:96) + (cid:96) (cid:48) ((1 + kε ) + ( (cid:96) + (cid:96) (cid:48) )) (using (9.10)) = 1 k + (cid:96) + 2 kε/L ((1 + kε ) + ( (cid:96) + 2 kε/L )) ≤ ε (cid:48) , where ε (cid:48) was defined in (9.8).Using (9.21), we can repeat the above arguments replacing ζ + by η + = ( σ + ) mr ( ζ + ) and [ ξ ( i )0 . . . ξ ( i ) k − . . . by [ ξ ( i m )0 . . . ξ ( i m ) k − . . . and obtain W ( A r ( σ + ) mr ( ζ + ) , ν + ) for any m ≥ Hence, since A jr ζ + = 1 j (cid:16) A r ζ + + A r ( σ + ) r ( ζ + ) + . . . + A r ( σ + ) ( j − r ( ζ + ) (cid:17) and using again (9.23), we obtain W ( A jr ζ + , ν + ) ≤ j · jε (cid:48) . By the triangle inequality, W (˜ ν, ν + ) ≤ W (˜ ν, A jr ζ + ) + W ( A jr ζ + , ν + ) ≤ W (˜ ν, A jr ζ + ) + ε (cid:48) . As ζ + is generic, A jr ζ + → ˜ ν as r → ∞ in the weak ∗ topology, proving the lemma. (cid:3) Lemma 9.26.
There is a function ε (cid:55)→ δ ( ε ) , δ ( ε ) → as ε → , such that for every ν +1 , ν +2 ∈ M (Σ + ) satisfying W ( ν +1 , ν +2 ) < ε then their natural extensions ν , ν ∈ M (Σ) satisfy W ( ν , ν ) < δ .Proof. It suffices to observe that taking natural extension is a homeomorphism betweenthe two compact metric spaces M (Σ + ) and M (Σ) . (cid:3) KEW-PRODUCTS WITH CONCAVE FIBER MAPS 51
Now the following is an immediate consequence of Lemmas 9.25 and 9.26.
Lemma 9.27.
For every ˜ ν ∈ M ( S ) it holds W (˜ ν, ν ) ≤ δ ( ε (cid:48) ) , where δ ( · ) is as in Lemma9.26.
6. Construction of a basic set.
Recall z : W N → [0 , defined in Lemma 9.20 and notethat, letting z (( σ + ) i ( ω + )) def = f iω + ( z ( ω + )) , it extends to a map z : S + → [0 , . Let Υ + def = { ( ω + , z ( ω + )) : ω + ∈ S + } ⊂ Σ + × [0 , . Observe that Υ + is compact and F + -invariant, for F + : Σ + × [0 , → Σ + × [0 , definedby F + ( ξ + , x ) def = ( σ + ( ξ + ) , f ξ ( x )) . and take the natural extension Υ of Υ + relative to F + . Note that Υ is a compact F -invariant set is semi-conjugate to the two-sided transitive SFT σ : S → S . By construction,on Υ the map F r is hyperbolic of contracting type in the fiber direction. The set Υ is theclaimed basic set.
7. Weak ∗ and in entropy approximation in M (Σ) . The above construction depends on ε and on the hence chosen quantifiers. Given ε = 1 /n , let us now denote the correspondinglyconstructed (two-sided) SFT by S n and the corresponding basic set by Υ n . Clearly, any µ n ∈ M (Υ n ) projects to a measure ν n = π ∗ µ n ∈ M ( S n ) . By Lemma 9.27, it holds lim n →∞ ν n = ν in the weak ∗ topology in M (Σ) . Recalling that ν = π ∗ µ proves the firstassertion of the proposition.For the second assertion, note that h top ( F, Υ n ) = h top ( σ, S n ) . By Lemma 9.23, wehave L def = lim sup n h top ( σ, S n ) ≥ h ( µ ) = h ( ν ) . We claim that we have in fact equality. Arguing by contradiction, suppose that
L > h ( ν ) ,consider the sequence of measures ν n ∈ M ( S n ) of maximal entropy h ( ν n ) = h top ( σ, S n ) .By Lemma 9.25, ν = lim n ν n . Hence, by upper semi-continuity of the entropy map, wehave lim sup n h ( ν n ) ≤ h ( ν ) , a contradiction.This finishes the proof of Theorem 2.22. (cid:3) Proof of Theorem 2.19.
By Theorem 2.22, it remains to show that not only the pro-jections converge to the projection, but also the measures themselves converge to. Noteagain π ∗ µ n → π ∗ µ = ν in the weak ∗ topology.By our hypothesis, χ ( µ ) = 0 . As we assume (H2+), we can invoke Theorem 2.13 Caseb). Hence, µ is the only measure with π ∗ µ = ν . Thus, applying Lemma 8.3, the sequence µ n weak ∗ converges to µ . (cid:3) Weak ∗ and entropy approximation of linear combinations of hyperbolic ergodicmeasures with the same type of hyperbolicity. Let us first state some fundamental factabout SFTs.
Lemma 9.28.
Let ( X, T ) be a transitive SFT. Then every measure µ ∈ M ( X ) can beapproximated, weak ∗ and in entropy, by a sequence of ergodic measures. That is, M ( X ) is an entropy dense Poulsen simplex.Proof. That M ( X ) is a Poulsen simplex is well known [47]. The entropy denseness ap-pears first in [34, Lemma 3] for full shifts, the version for SFT is a simple modification andappears in [6, Proposition 3.3]. (cid:3) We will apply Lemma 9.28 to the following situation.
Remark 9.29.
Given Υ ⊂ Γ a basic set, then F | Υ is topologically conjugate to a topologi-cal Markov chain. In particular, this Markov chain is a SFT and it is topologically transitive(mixing) if F | Υ is topologically transitive (mixing) (see, for example, [36, Chapter 18.7]). Proposition 9.30.
Given any finite number of ergodic measures µ , . . . , µ k ∈ M erg , ≥ (Γ) and positive numbers λ , . . . , λ k satisfying (cid:80) ki =1 λ i = 1 , the measure µ (cid:48) = (cid:80) ki =1 λ i µ i isweak ∗ and in entropy approximated by ergodic measures in M erg ,> (Γ) .Proof. For every measure µ i ∈ M erg , ≥ (Γ) there exists a sequence of horseshoes (withuniform fiber expansion) (Γ ( i ) n ) n such that M (Γ ( i ) n ) converges weak ∗ to µ i and h top ( F, Γ ( i ) n ) converges to h ( µ i ) . Indeed, apply Remark 2.21 if µ i is hyperbolic and Theorem 2.19 oth-erwise. In particular, the measure µ (cid:48) is weak ∗ and in entropy approximated by measures µ (cid:48) n = (cid:80) ki =1 λ i µ (cid:48) i , where µ (cid:48) i ∈ M erg (Γ ( i ) n ) can be taken the measure of maximal entropyfor F in Γ ( i ) n . By Corollary 5.6, for every n there exists a horseshoe Γ (cid:48) n ⊃ (cid:83) i Γ ( i ) n . Inparticular, µ (cid:48) n ∈ M (Γ (cid:48) n ) . By Lemma 9.28, µ (cid:48) n can be weak ∗ and in entropy approximatedby ergodic measures in M erg (Γ (cid:48) n ) . Clearly, M erg (Γ (cid:48) n ) ⊂ M erg ,> (Γ) . (cid:3) Remark 9.31.
Given µ ∈ M (Γ) and U some neighborhood of µ in the weak ∗ topol-ogy, there exist µ , . . . , µ k ∈ M erg , ≥ (Γ) and positive numbers λ , . . . , λ k satisfying (cid:80) ki =1 λ i = 1 such that µ (cid:48) = (cid:80) ki =1 λ i µ i ∈ U (see, for example, [4, Lemma 2.1]). Proof of Corollary 2.23.
The Proposition 9.30 implies the assertion in Corollary 2.23 formeasures whose ergodic decomposition has only finitely many components. For the gen-eral case recall Remark 9.31 and note that for any µ (cid:48) = (cid:80) i λ i µ i we have h ( µ (cid:48) ) = (cid:80) i λ i h ( µ i ) . The corollary now follows from Proposition 9.30. (cid:3)
10. E
NTROPY - DENSE P OULSEN STRUCTURE OF M (Σ) : P ROOF OF T HEOREM ν = (cid:80) ki =1 λ i ν i for some ν , . . . , ν k ∈ M erg (Σ) and some positive numbers λ , . . . , λ k satisfying (cid:80) ki =1 λ i = 1 .For every ν i , by Lemma 8.1 there exists some measure µ i ∈ M erg , ≥ (Γ) with non-negative fiber Lyapunov exponent satisfying π ∗ µ i = ν i . For each i , by Theorem 2.22,there is a sequence of basic sets Υ ( i ) n ⊂ Γ such that their measure spaces project to measurespaces weak ∗ converging to ν i . Moreover, there is a sequence of measures µ ( i ) n such that π ∗ µ ( i ) n converge weak ∗ and in entropy to π ∗ µ i = ν i .By Corollary 5.6 for each n there exists a horseshoe Γ n ⊃ k (cid:91) i =1 Υ ( i ) n . By [6, Lemma 2.14], we can arbitrarily approximate weak ∗ and in entropy the measure (cid:80) ki =1 λ i µ ( i ) n by ergodic measures in M erg (Γ n ) , k (cid:88) i =1 λ i µ ( i ) n = we- lim m →∞ µ n,m , µ n,m ∈ M erg (Γ n ) , where we-lim denotes convergence in the weak ∗ topology and in entropy. Hence k (cid:88) i =1 λ i π ∗ µ ( i ) n = π ∗ k (cid:88) i =1 λ i µ ( i ) n = π ∗ (cid:16) we- lim m →∞ µ n,m (cid:17) = we- lim m →∞ π ∗ µ n,m , KEW-PRODUCTS WITH CONCAVE FIBER MAPS 53
Hence, by a diagonal argument, we find a sequence of ergodic measures ( µ n,m n ) n suchthat ν = (cid:88) i λ i ν i = lim n (cid:88) i λ i π ∗ µ ( i ) n = lim n (cid:16) we- lim m →∞ π ∗ µ n,m (cid:17) . This concludes the proof of the theorem. (cid:3)
11. B
IFURCATION EXIT SCENARIOS
In this section, we put our results in an extended ambient considering bifurcation sce-narios. The skew-products lead to globally defined one-parameter families that may have“explosions of entropy and of the space of ergodic measures" at some bifurcation associ-ated to a collision of a pair of homoclinic classes. This sort of question was studied in[29, 28], where the collision occurs through the orbit of a parabolic (saddle-node) point. Inour setting the collision may be of different nature. We now proceed to explain the details,not aiming for complete generality but for giving the general ideas.Ia) ˜ f ˜ f ,t h ˜ f ,t ˜ f ,t c Ib) ˜ f ˜ f ,t h ˜ f ,t ˜ f ,t c II) ˜ f ˜ f ,t h ˜ f ,t ˜ f ,t c F IGURE
4. The bifurcation scenariosReturning to the globally defined skew-product map ˜ F in (1.1), consider one whosefiber maps ˜ f and ˜ f satisfy conditions (H1)–(H2) in the interval [0 , . Our previous studyapplies to the set Γ defined in (1.2), which is indeed the locally maximal invariant set of ˜ F in the open set Σ × ( − ε, ε ) for some small ε > . Though the dynamics of ˜ F may have other “pieces" beyond Γ . To perform a bifurcation analysis, we embed ˜ F in aone-parameter family ˜ F t , t ∈ [ t h , t c ] , defined by means of fiber maps { ˜ f , ˜ f ,t } . Here theparameter t h corresponds to a heterodimensional cycle and t c to a “collision”. Concerningthe latter, there are two qualitatively different scenarios that will be studied below, compareFigure 4. Topologically, in Cases Ia) and Ib), there occurs a collision of a homoclinic classin Γ with one “coming from outside" that does not involve parabolic points. In Case II,there occurs an internal collision of the classes of P and Q along a parabolic orbit. In eithercase, the maximal invariant set that emerges at the bifurcation has full entropy log 2 . Thefurther analysis depends on the choice of the concave maps. On one hand, we may observean explosion of the space of admissible sequences (Proposition 11.6). On the other hand,in the ergodic level, we may observe an explosion in the space of measures and in entropy Giving an interpretation using the filtrations in Conley theory [14] for the study of chain recurrence classes,the set Γ is the maximal invariant set of a filtrating neighbourhood. This means that there are two compact setswith nonempty interiors M and M , with M ⊂ int( M ) and such that ˜ F ( M i ) ⊂ int( M i ) , i = 1 , . Thenone studies the dynamics in M \ M which is a level of the filtration. Further levels may also be considered. Insome cases, Γ may split into two separated parts, and hence there is another filtrating set separating them. Indeed,this occurs when the homoclinic classes H ( P, ˜ F ) and H ( Q, ˜ F ) both are hyperbolic (see Theorem 2.6). (Proposition 11.7). Finally, we see how the twin-structure of the measures in Theorem2.13 changes in Case I) (Proposition 11.8). In Sections 11.3.4 and 11.3.5 we will give aninterpretation of the Cases I) and II), respectively.11.1. Formal setting.
Consider real functions ˜ f , ˜ f whose restrictions to the interval [0 , , that again we denote by f , f , satisfy the following hypothesis that is slightly moregeneral than (H1) in the way that it allows f to have a fixed point.( ˜H1) f : [0 , → [0 , is a differentiable increasing map such that f is onto and sat-isfies f (0) = 0 , f (1) = 1 , f (cid:48) (0) > , f (cid:48) (1) ∈ (0 , , and f ( x ) > x for every x ∈ (0 , . Moreover, there is d ∈ [0 , such that f : [ d, → [0 , is a differen-tiable increasing map satisfying f ( x ) ≤ x for every x ∈ [ d, such that f ( d ) = 0 ,and f has at most one fixed point in [ d, .Unless stated otherwise, in this section, we fix a map ˜ f and consider a one-parameterfamily of maps { ˜ f ,t } t ∈ [ t h ,t c ] such that { f , f ,t } satisfy ( ˜H1) and (H2), with the corre-sponding (possibly degenerate) intervals [ d t , ⊂ [0 , , for every t ∈ [ t h , t c ] . We willinvoke (H2+) only in Proposition 11.8. We assume f ,t to be continuous in t in the C topology. Moreover, the family { f , f ,t } satisfies (H1) for every t ∈ ( t h , t c ) and at t = t h and t = t c complies the following “exit bifurcation scenarios" for a family of maps satis-fying ( ˜H1): • ( bifurcation parameter t = t h ) f ,t h (1) = 0 , • ( bifurcation parameter t = t c ) there exists a ∈ [0 , such that f ,t c ( a ) = a .Note that for every t ∈ ( t h , t c ] the interval [ d t , is nondegenerate. By assumption, forevery t ∈ ( t h , t c ) the family { f , f ,t } satisfies (H1)–(H2) and therefore the correspondingresults of previous sections hold.There are three cases of what can happen at the bifurcation point t = t c :I) (hyperbolic case) f (cid:48) ,t c ( a ) (cid:54) = 1 a) f (cid:48) ,t c ( a ) > (and hence a = 1) ,b) f (cid:48) ,t c ( a ) < (and hence a = 0 ),II) (parabolic case) f (cid:48) ,t c ( a ) = 1 (in this case there is no a priori restriction on thevalue a ∈ [0 , ).Cases Ia) and Ib) are actually identical up to the time reversal (compare also Remark 2.9),so we only consider Case Ia). Remark 11.1.
Notice that the induced IFS { f , f ,t c } on [0 , , corresponding to Case I,was also studied in [30], though the focus there was on the stationary measures (and alsoassuming contraction on average) which represent a special subclass of invariant measures.Closer to our approach is [1] studying the so-called mystery of the vanishing attractorwhere concavity properties similar to the ones in Section 6 are used. Example 11.2.
One simple example is the family f , ˜ f ,t = ˜ f + t, where f satisfies (H1)–(H2) and ˜ f is a differentiable increasing map of R such that ˜ f (cid:48) isnot increasing. Now consider the associated maps { f , f ,t } , where f = ˜ f | [0 , , f ,t = ˜ f ,t | [ ˜ f − ,t (0) , . One could also study a more general case when ˜ f also depends on the parameter t , hence changing itsfixed points. This analysis just would require straightforward modifications of the domains of the maps. KEW-PRODUCTS WITH CONCAVE FIBER MAPS 55
In this example, in either case we have t h = − ˜ f (1) . The value t c depends on the cases: • if ˜ f (cid:48) (1) > then a = 1 and t c = 1 − ˜ f (1) , • if ˜ f (cid:48) (1) ≤ and there exists c ∈ [0 , with ˜ f (cid:48) ( c ) = 1 then a = c and t c = c − ˜ f ( c ) , • if ˜ f (cid:48) (0) < then a = 0 and t c = − ˜ f (0) .The first and the last case correspond to Ia) and Ib), respectively, while the second casecorresponds to II).Let ˜ F t , t ∈ [ t h , t c ] , be the corresponding skew-product and denote by Γ ( t ) the analo-gously defined maximal invariant set of ˜ F t in Σ × ( − ε t , ε t ) for some small ε t > .Define F t def = ˜ F t | Γ ( t ) and let Σ ( t ) def = π (Γ ( t ) ) be the space of admissible sequences for F t . Remark 11.3.
Note that in Cases I) and II), we have Σ ( t c ) = { , } Z , because for t = t c all sequences are forward and backward admissible at a . In particular, Σ × { a } ⊂ Γ ( t c ) . Meanwhile, Σ ( t h ) is a countable set consisting of sequences with at most one appearanceof the symbol . Thus, the topological entropy of the bifurcating maps F t c and F t h are log 2 and 0, respectively. Remark 11.4 (Discussion of Figure 2) . Figure 2 depicts the following points A = ((0 − N . N ) , ∈ W s ( Q, ˜ F t h ) ∩ W u ( P, ˜ F t h ) ,B = ((0 − N . N ) , ∈ W s ( Q, ˜ F t c ) ∩ W u ( Q, ˜ F t c ) , B ∈ H ( Q, ˜ F t c ) ,C = ((0 − N . N ) , ˜ f ,t c (1)) ∈ W s ( P, ˜ F t c ) ∩ W u ( P, ˜ F t c ) , C ∈ H ( P, ˜ F t c ) ,D = ((0 − N . N ) ,
0) = W s ( Q, ˜ F t c ) ∩ W u ( P, ˜ F t c ) , D ∈ H ( Q, ˜ F t c ) ,E = ((0 − N . N ) , ∈ W s ( P, ˜ F t c ) ∩ W u ( P, ˜ F t c ) , E ∈ H ( P, ˜ F t c ) ,S = (1 Z , ,R = (1 Z , . The point R is a fixed point of ˜ F t c of expanding type and hence cannot be homoclini-cally related to the fixed points P of contracting type. Moreover, Σ × { } ⊂ H ( P, ˜ F t c ) ∩ H ( R, ˜ F t c ) and R and P are involved in a heterodimensional cycle. Analogous argumentsapply to S and Q and the set Σ × { } .Finally, the difference between the points C and E is that while both are homoclinicpoints of P , the latter is a contained in the intersection of the “strong stable” set and theunstable set of P . Analogously for the homoclinic points B and D of Q . Remark 11.5 (Nondecreasing complexity) . For the simple bifurcation family in Example(11.2), the family of compact sets { Σ ( t ) } t is nondecreasing in t . Hence, the topologicalentropy of ˜ F t on Γ ( t ) is nondecreasing and there is no “annihilation" of periodic pointsas t increases. This may not be the case in a general situation. As observed before, wepass from zero entropy (for t = t h ) to full entropy ( t = t c ). These features resemblesomewhat to the ones in the quadratic family of maps g λ ( x ) = λx (1 − x ) , λ ∈ [1 , , seefor example, [43]. In the quadratic family the creation of periodic points occurs throughsaddle-node and flip bifurcations as well as the creation of “homoclinic tangencies" (in thesense that the critical point is pre-periodic). In our case, periodic points are created either through saddle-node bifurcations or heterodimensional cycles (see Remark 5.5). In somesense, one may regard this family as a partially hyperbolic version of the quadratic familyshowing a nondecreasing transition from trivial dynamics to full chaos.11.2. Bifurcation at t h : heterodimensional cycles. By Remark 5.5, the map ˜ F t h has aheterodimensional cycle associated to P = (0 Z , and Q = (0 Z , .11.3. Bifurcation at t c : collisions of sets. Consider the function C : [ t h , t c ] → [0 , ∞ ] ,(11.1) C ( t ) def = | log f (cid:48) (1) | log f (cid:48) ,t (1) if f (cid:48) ,t (1) > , ∞ otherwise . By hypothesis, C is continuous.11.3.1. Space of admissible sequences.
Recall the definitions of the upper and lower fre-quencies freq and freq in (8.3).
Proposition 11.6.
Given C ≥ , let S C def = (cid:26) ξ ∈ Σ : freq( ξ, ξ, ≤ C (cid:27) . Then for every ε > there exists δ > such that for t ∈ ( t c − δ, t c ) we have Σ ( t ) ⊂ S C ( t c )+ ε . In particular, if C ( t c ) < ∞ then there is a jump in the space of admissible sequences.Moreover, independently of the value C ( t c ) , Σ ( t ) → Σ in the Hausdorff distance as t → t c .Proof. The first statement is an immediate consequence of Corollary 8.9. Indeed, the ratioof frequencies cannot be larger than C ( t ) and C ( t ) → C ( t c ) as t → t c .By Remark 11.3, we have Σ ( t c ) = Σ , which implies the claimed jump if C ( t c ) < ∞ .For the last assertion, observe that for any n ≥ there is ε n such that for every t ∈ ( t c − ε n , t c ) the word n is forward admissible in Σ ( t ) . Hence, every word of length n is forward admissible in Σ ( t ) , which means that Σ ( t ) intersects every (forward) n -th levelcylinder for t small enough. This, together with the fact that Σ ( t ) is shift-invariant, impliesthe convergence in Hausdorff distance. (cid:3) Spaces of measures and entropy.
Let H ( p ) def = − p log p − (1 − p ) log(1 − p ) , for p ∈ (0 , . Proposition 11.7.
For every t < t c and C ( t ) as in (11.1) we have that M (Σ ( t ) ) ⊂ (cid:110) ν ∈ M (Σ ) : ν ([1]) ν ([0]) ≤ C ( t ) (cid:111) is a closed proper subset of M (Σ ) . Moreover, for every t < t c we have (11.2) sup ν ∈ M erg (Σ ( t ) ) h ( ν ) = h top ( σ, Σ ( t ) ) ≤ H ( p t ) , where p t def = min (cid:110) , C ( t )1 + C ( t ) (cid:111) . In particular, if C ( t c ) < then the space M (Σ ) is not the weak ∗ limit of the subspaces M (Σ ( t ) ) as t → t c and there is a jump in entropy at t c , in the sense that H ( p t c ) = H ( C ( t c )1 + C ( t c ) ) < H ( 12 ) = log 2 = h top ( σ, Σ ) = h top ( σ, Σ ( t c ) ) . KEW-PRODUCTS WITH CONCAVE FIBER MAPS 57
Proof.
First observe that by Lemma 8.8 for any t < t c and every ν ∈ M (Σ ( t ) ) , we have ν ([1]) ν ([0]) ≤ | log f (cid:48) (1) | log f (cid:48) ,t (1) = C ( t ) . This implies the first statement.To prove (11.2), observe that A = { [0] , [1] } is a generating partition for Σ ( t ) . Hence,by the Kolmogorov-Sinai theorem, for any ergodic measure ν ∈ M erg (Σ ( t ) ) , letting p def = ν ([1]) and hence − p = ν ([0]) , we have h ( ν ) = h ( ν, A ) ≤ − p log p − (1 − p ) log(1 − p ) = H ( p ) . As p ∈ [0 , p t ] and is the maximum of H , equation (11.2) follows from the variationalprinciple for entropy, proving the proposition. (cid:3) Structure of the space of measures.
In this section we will put Theorem 2.13 inthe context of the bifurcation scenario.
Proposition 11.8.
Assume ( ˜H1)–(H2+). There exist continuous functions κ , κ : (0 , ∞ ) → (0 , ∞ ) which are increasing and satisfy lim D → κ i ( D ) = 0 , i = 1 , , such that, given any ν ∈ M erg (Σ ) , one of the following three cases occurs: a) There exist exactly two measures µ , µ ∈ M erg (Γ ( t c ) ) such that π ∗ µ = ν = π ∗ µ . In this case, both measures are hyperbolic and have fiber Lyapunov expo-nents with different signs. More precisely, if χ ( µ ) > > χ ( µ ) then for theWasserstein distance D def = W ( µ , µ ) between µ and µ , we have D = (cid:90) x dµ ( ξ, x ) − (cid:90) x dµ ( ξ, x ) > and − κ ( D ) < χ ( µ ) < − κ ( D ) < < κ ( D ) < χ ( µ ) < κ ( D ) . b) There exists exactly one measure µ ∈ M erg (Γ ( t c ) ) with π ∗ µ = ν and χ ( µ ) = 0 . c) There exists exactly one measure µ ∈ M erg (Γ ( t c ) ) with π ∗ µ = ν and χ ( µ ) > . Inthis case we are in Case Ia) and this measure is supported on Σ × { } . Below we see that case c) in the above proposition indeed occurs (see Remark 11.11).
Remark 11.9.
Given ξ = ( ξ . . . ξ n − ) Z ∈ Σ ( t c ) , consider the map g = f ξ ,t ◦ . . . ◦ f ξ n − ,t (writing f ,t = f ) defined on I ξ . There are the following possibilities according to thehyperbolic case Ia) and the parabolic case II):Ia) – either g has a unique fixed point that is parabolic, – or g has a pair of fixed points p +[ ξ ... ξ n − ] < p +[ ξ ... ξ n − ] that are repelling andcontracting, respectively.II) – either is the unique fixed point of g (which can be repelling or parabolic), – or g has a pair of fixed points p +[ ξ ... ξ n − ] < p +[ ξ ... ξ n − ] that are repelling andcontracting, respectively. Proof of Proposition 11.8.
The arguments of the proof of Theorem 2.13 essentially work.Case a) is as before. It remains to consider the cases where there is only one measure µ ∈ M erg (Γ ( t c ) ) projecting to ν . Let us first see that χ ( µ ) ≥ . By contradiction, if χ ( µ ) < then let ν = π ∗ µ . Observe that, by Corollary 9.2, ν is accumulated by periodicmeasures ν n supported on periodic orbits of sequences ξ ( n ) . By Remark 11.9, there is asequence of periodic points P n = ( ξ ( n ) , p n ) ∈ Γ ( t c ) that are either repelling or parabolic. Consider the sequence ( µ n ) n of periodic measures supported on those points. Taking, ifnecessary, a subsequence, we can assume that µ n → µ (cid:48) . By construction π ∗ ( µ (cid:48) ) = ν and χ ( µ (cid:48) ) ≥ . Thus µ (cid:48) (cid:54) = µ , a contradiction.To see that case c) does not occur in the parabolic case II), we argue as above. Indeed,arguing again by contradiction, by Remark 11.9, there is a sequence of periodic points Q n of contracting or parabolic type whose corresponding measures converge to some measure µ (cid:48)(cid:48) satisfying π ∗ µ (cid:48)(cid:48) = π ∗ µ and χ ( µ (cid:48)(cid:48) ) ≤ , which is a contradiction. (cid:3) Illustration of the hyperbolic case Ia).
We present an example of a skew-productwith two parts leading to a collision of homoclinic classes. The map ˜ F t in Σ × [0 , hasthe dynamics discussed in the previous sections. The dynamics of ˜ F t in Σ × [1 + δ t , ,for some small δ t > , is a “twisted twin copy" of the former one, see Figure 5. Inthis way, we get locally maximal invariant sets Γ ( t ) ⊂ Σ × ( δ t , δ t / and Υ ( t ) ⊂ Σ × (1 + δ t / , δ t ) , respectively, which have qualitatively “the same” dynamics wheninterchanging the roles of the maps f and f ,t . The sets Γ ( t ) and Υ ( t ) are disjoint for t ∈ ( t h , t c ) and collide for t = t c . Since Σ ( t c ) = Σ , this collision is big as the sets Γ ( t c ) and Υ ( t c ) intersect in the “topological horseshoe" Σ × { } whose hyperbolic-like naturedepends on the value C ( t c ) defined in (11.1). Assuming that C ( t c ) < , according to theresults above, we observe at the bifurcation t = t c an explosion of the symbolic space andof entropy. The example illustrates where a substantial part of the “additional” symbolicsequences, in particular periodic sequences, come from. Indeed, the hyperbolic periodicpoints do not simply appear “out of thin air" but come from outside, that is, from Υ ( t ) . ˜ f ˜ f ,t δ f f ,t c g − ,t c g − F IGURE
5. Bifurcation with collision of homoclinic classes along atopological horseshoe
Remark 11.10 (Entropy jump viewed from another side) . Recalling that there is no en-tropy in the fibers (see (2.5)), Proposition 11.7 implies that we have the correspondingjump in the topological entropy of F t at Γ ( t ) at t = t c .Observe that Υ ( t ) has a correspondingly defined IFS with fiber maps g ,t and g givenby g ,t = ˜ f − ,t and g = ˜ f − (on appropriately defined domains), compare Figure 5.Analogously, as in (11.1), we can define a function C (cid:48) ( t ) for the maps { g ,t , g } . Notethat for the collision parameter t = t c we have C ( t c ) = C (cid:48) ( t c ) − . Therefore, C ( t c ) < KEW-PRODUCTS WITH CONCAVE FIBER MAPS 59 implies C (cid:48) ( t c ) > and, as a consequence of Proposition 11.7, there is no entropy jump in Υ ( t c ) . Any entropy jump for Γ ( t ) in fact comes from entropy in Υ ( t ) . Remark 11.11 (Explosion of space of sequences and expanding measures without twins) . We also observe an explosion in the space of ergodic measures M erg (Γ ( t ) ) at t = t c . Thesimplest example is the Dirac mass δ R at R = (1 Z , recalling that by hypothesis Ia), χ ( δ R ) = log f (cid:48) ,t c (1) > . Indeed, for every periodic sequence ξ with large frequency of s we also have that themeasure uniformly distributed in the orbit of ( ξ, is of expanding type. Moreover, anyergodic measure of expanding type supported on Σ × { } has no twin. All these measuresprovide examples of expanding measures without twins, that is, to which Proposition 11.8Case c) apply. None of them can be obtained as weak ∗ limits of the sets of invariantmeasures M erg (Γ ( t ) ) as t → t c .Arguments similar to the ones in Remark 11.10 apply and imply that those “new" mea-sures come from M erg (Υ ( t ) ) .11.3.5. Illustration of the parabolic case II).
In what follows, we study the parabolic caserecalling the choice of the point a ∈ [0 , . In fact, we will assume a ∈ (0 , , when a ∈ { , } the statement also is true but the proof resembles methods of the previoussubsection. Proposition 11.12 (Convergence to full entropy) . We have lim t → t c h top ( σ, Σ ( t ) ) = log 2 = h top ( σ, Σ ( t c ) ) . Moreover, Σ ( t ) converges to Σ in the Hausdorff distance as t → t c .Proof. Choose k ∈ N . For every word ω ∈ Σ (cid:48) k := { , } k \ { k } there exists ε ω suchthat for t c − ε ω < t < t c we have f [ ω ] ,t ( a ) > a (adopting the corresponding notation). Asthere are only finitely many words of length k , we can find ε k > valid for every ω ∈ Σ (cid:48) k .Thus, for t ∈ ( t c − ε k , t c ) any concatenation of words from Σ (cid:48) k is allowed in Σ ( t ) . Thisallows us to estimate the entropy by h top ( σ, Σ ( t ) ) ≥ k log(2 k − , which immediately implies the first claim and also provides the convergence in Hausdorffdistance. (cid:3) Note that in the parabolic case, at t = t c we have C ( t c ) = ∞ and hence the secondclaim of Proposition 11.6 does not apply. Remark 11.13.
Arguing exactly as in the proof of Proposition 5.11, one sees that thehomoclinic classes H ( P, ˜ F t c ) and H ( Q, ˜ F t c ) both contain the parabolic fixed point A =(1 Z , a ) . Hence, under certain conditions implying that the homoclinic classes of P and of Q are hyperbolic for t < t c sufficiently close to t c , this leads to a collision of hyperbolichomoclinic classes.12. D ISCUSSION : H
OMOCLINIC SCENARIOS BEYOND CONCAVITY
Under conditions (H1)–(H2), Proposition 5.1 (together with Proposition 5.11 in thecase when there are parabolic points) allows us to consider H ( P, ˜ F ) and H ( Q, ˜ F ) as theonly two homoclinic classes of ˜ F in Σ × [0 , . According to the choice of ˜ F , there are two possibilities for the sets H ( P, ˜ F ) and H ( Q, ˜ F ) : either they are disjoint or they havenonempty intersection (in this latter case, the sets may be equal or not).The constructions in [18] provide an explicit two-parameter family (with parameters a and t ) of fiber maps f = g a (concave) and f = g ,t (affine) such that the correspondingskew-product map ˜ F a,t falls into one of the following cases, according to the choices ofthe parameters a and t :1. The sets H ( P, ˜ F a,t ) and H ( Q, ˜ F a,t ) are pairwise disjoint and hyperbolic and theirunion is the limit set of F a,t in Σ × [0 , ([18, Theorem 2.7 case (B)]).2. H ( P, ˜ F a,t ) ∩ H ( Q, ˜ F a,t ) is the orbit of a parabolic point of ˜ F a,t ([18, Theorem 2.7case (C.c)]).3. H ( P, ˜ F a,t ) = H ( Q, ˜ F a,t ) = Γ a,t ([18, Theorem 2.7 case (A)]).Regarding the above scenarios, the corresponding space of ergodic measures splits intotwo parts, corresponding to the measures of contracting and expanding type:1. These parts are disjoint.2. Their closures intersect in a measure supported on a parabolic periodic orbit.3. Their closures intersect in nonhyperbolic ergodic measures, some of them withpositive entropy (see [8]).In each of these cases, we can choose the fiber dynamics in a way that the each class islocally maximal. By Proposition 5.1, any pair of saddles of the same type of hyperbolicityare homoclinically related, hence we can apply [33] and conclude that the correspondingparts of the space of ergodic measures each are arcwise connected and have closures whichare a Poulsen simplex. Compare also with Corollary 2.24.We now discuss possible configurations of homoclinic classes for skew-products as in(1.1) assuming (H1) but not a priori ( H (that is, without the concavity assumptions).The following remark indicates that without the concavity assumption the scenery can bevast, with many possibilities for the interrelation between those classes (and hence for theresulting topological and ergodic properties). Remark 12.1 (Homoclinic scenarios when (H2) is not satisfied) . The map ˜ F may haveother hyperbolic periodic points R which may fail to be homoclinically related to P or Q . In this setting, it is fundamental to understand how these periodic points and theirhomoclinic classes are inserted in the dynamics of ˜ F . Indeed, the following (possiblynon-exhaustive) list of dynamical scenarios may occur:(1) H ( P, ˜ F ) and H ( Q, ˜ F ) are the only homoclinic classes of ˜ F in Σ × [0 , (that is,any other homoclinic class of ˜ F in Σ × [0 , is equal to one of these two classes)and these two classes are:(a) (transitivity) H ( P, ˜ F ) = H ( Q, ˜ F ) = Γ and hence F = ˜ F | Γ is nonhyperbolicand Γ is a transitive set, see [17].(b) (hyperbolicity) H ( P, ˜ F ) ∩ H ( Q, ˜ F ) = ∅ , each of them is hyperbolic, and thelimit set of ˜ F in Σ × [0 , is the union H ( P, ˜ F ) ∪ H ( Q, ˜ F ) , see [25].(c) (overlapping) H ( P, ˜ F ) (cid:54) = H ( Q, ˜ F ) but H ( P, ˜ F ) ∩ H ( Q, ˜ F ) (cid:54) = ∅ and hence F = ˜ F | Γ is nonhyperbolic, see [29, 28].(2) The homoclinic classes H ( P, ˜ F ) and H ( Q, ˜ F ) are disjoint and hyperbolic, butthere are other homoclinic classes (which are different as sets), see [27, Theorem(2)(i)]. In this case, there are periodic orbits (say expanding) O ( R ) and O ( R ) ∩ ( H ( P, ˜ F ) ∪ H ( Q, ˜ F )) = ∅ KEW-PRODUCTS WITH CONCAVE FIBER MAPS 61 such that π ( O ( R )) = π ( O ( R )) = O ( ω ) ∈ Σ = π (Γ) . Hence, the ergodic measure supported on the periodic orbit O ( ω ) has (at least)two lifts to different ergodic measures in M erg ,> (the ones supported on O ( R ) and O ( R ) ), thus failing Theorem 2.13. There is a similar construction replacingperiodic orbits by nontrivial basic sets.Note that, the scenarios in case (1) are compatible with our concavity assumptions (seefor instance [18]) while case (2) is not (see Theorem 2.6).A PPENDIX
A. W
ASSERSTEIN DISTANCE
Let us remind that for two probability measures µ , µ supported on a compact metricspace M we define their couplings as measures on M × M with marginals µ on the firstcoordinate and µ on the second. Denoting by Γ( µ , µ ) the space of all couplings of µ and µ , we can define a metric on M ( M ) (the Wasserstein distance ) by(A.1) W ( µ , µ ) def = inf γ ∈ Γ( µ ,µ ) (cid:90) M × M d ( x, y ) dγ ( x, y ) . As a special case of the duality theorem of Kantorovich and Rubinstein (see [35]), one cangive an equivalent definition as follows(A.2) W ( µ , µ ) = sup (cid:110) (cid:90) M f ( x ) dµ ( x ) − (cid:90) M f ( x ) dµ ( x ) : Lip( f ) ≤ (cid:111) . Here
Lip( f ) denotes the Lipschitz constant, and the supremum is taken over Lipschitzfunctions only. It is well known that the Wasserstein distance is a metric on the space P ( M ) of probability measures supported on M and that it induces the weak ∗ topology on P ( M ) .We will use the following lemma. Lemma A.1.
Assume that d is a metric on M ⊂ X × R satisfying d (( x , y ) , ( x , y )) ≥| x − x | , with equality if y = y . Assume also that µ and µ have a special coupling γ ∈ Γ( µ , µ ) such that γ ( { (( x , y ) , ( x , y )) : y = y , x ≥ x } ) = 1 . Then W ( µ , µ ) = (cid:90) M x dµ ( x, y ) − (cid:90) M x dµ ( x, y ) . Proof.
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