Non-hyperbolic ergodic measures with the full support and positive entropy

The aim of this note is to give an alternative proof for the following result originally proved by Bonatti, Díaz and Kwietniak. For every n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document} there exists a compact manifold without boundary M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {M}}$$\end{document} of dimension n and a non-empty open set U⊂Diff(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U\subset \text {Diff}({\mathbf {M}})$$\end{document} such that for every f∈U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in U$$\end{document} there exists a non-hyperbolic measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} invariant for f with positive entropy and full support. We also investigate the connection between the Feldman-Katok convergence of measures and the Kuratowski convergence of their supports.


Introduction
Thanks to the work of Abraham and Smale it is well known that hyperbolic diffeomorphisms are not dense in the space of all diffeomorphisms of a given manifold [1]. Even more is true: diffeomorphisms that do not admit any non-hyperbolic measure are not dense in this space as well. Let us recall that a measure is non-hyperbolic if it has at least one zero Lyapunov exponent (we recall the definition of the Lyapunov exponents in the preliminaries). This notion was introduced by Pesin [ [2,9].
In 2005 Gorodetski, Ilyashenko, Kleptsyn and Nalsky introduced royal measures 1 and used them to construct non-hyperbolic diffeomorphisms on the skew product of Bernoulli shift with a circle as a fibre. They also conjectured that there exists a non-empty open subset U of the set of all diffeomorphisms of the 3-dimensional torus such that every f ∈ U admits at least one non-hyperbolic measure. Note that Lapunov exponents are sensitive under perturbation so one could suspect that the opposite is true. The conjecture was however proved by Kleptsyn and Nalsky in [19] and the idea of their work was somehow inspired by [15].
Royal measures were also under consideration of other authors, see for instance [6,8,10,11,19,21,28]. It was shown in [21] that the entropy of every royal measure vanishes. The main tool used for that was the Feldman-Katok convergence of measures. It allowed the authors to reprove also some already known properties of royal measures originally proved in [15] and in [6] in the more abstract setting. They did not mention however the connection between the Feldman-Katok convergence and the Kuratowski convergence of measures' supports. In this note we prove that under some additional assumptions the Feldman-Katok convergence implies the Kuratowski convergence of supports. This result can be applied to royal measures. We also give an example demonstrating that the sole Feldman-Katok convergence is not enough to conclude that the support of the limit measure is also a limit in the sense of Kuratowski. Alternative to the construction of non-hyperbolic measures from [15] was the one introduced by Bochi, Bonatti and Díaz. In the series of papers [3][4][5] they strengthened the results of Kleptsyn and Nalsky showing the following: The above result ( [4, Theorem 9]) applies in fact to a wide class of manifolds, including any manifold carrying a transitive Anosov flow, in particular to the n-dimensional torus for every n ≥ 3.
In [4, p. 3] authors conjectured that one may strengthen their results so that the resulting non-hyperbolic measure would have both: positive entropy and full support. That was shown in [7]. In this note we give an alternative, shorter proof of this fact. We demonstrate that actually the construction from [4] already leads to a measure with positive entropy.

The Kuratowski limit
In this section we outline Kuratowski limit, based on [20,Volume I]. For a sequence of non-empty, compact sets A 1 , A 2 , . . . ⊂ X we define the lower Kuratowski limit Li top n→∞ A n and the upper Kuratowski limit Ls top n→∞ A n of such a sequence by the conditions: Naturally, for every sequence of non-empty, compact sets A 1 , A 2 , . . . ⊂ X we have: We say that Such a limit is then denoted by The Kuratowski convergence is equivalent to the convergence with respect to the Hausdorff metric (defined on the family of closed and non-empty subsets of X ) if X is a compact metric space.

Invariant measures
Let M T (X ) be a set of all Borel probabilistic measures on X that are invariant under the map T acting on the space X and M e T (X ) ⊂ M T (X ) be the set of measures that are ergodic as well. We consider the weak * topology on M T (X ), making it a compact and a metrizable space. A sequence (μ n ) ∞ n=1 converges to μ with respect to this topology if and only if for all continuous functions ϕ : X → R a sequence ϕ dμ n converges to ϕ dμ in R. It is known that if X is a compact metric space, then this topology is given by the Prokhorov metric D P : where B ε = {y ∈ X : dist(y, B) < ε} is an ε-hull of B. Moreover, the following portmanteau theorem holds (the portmanteau theorem says in fact more, but we cite only the part we will use in this paper).

Empirical measures
Given n ∈ N and a sequence x = (x i ) i∈N ⊂ X we define the empirical measure m(x, n) as follows: where δ z denotes the Dirac measure supported at z. If we fix a map T : X → X , then for x ∈ X we put and we sometimes omit the subscript T if it can be derived from the context. A point x ∈ X is said to be generic for μ ∈ M T (X ) if (m(x, n)) n∈N converges to μ with respect to the weak * topology. We denote byω(x) the set of all weak * accumulation points of the sequence (m(x, n)) n∈N (if x is generic for μ thenω(x) = {μ}, but in generalω(x) can contain more than one element).

Hyperbolic measures
Let M be a smooth m-dimensional manifold. For a diffeomorphism T : M → M and T -invariant ergodic measure μ, there exists ⊂ M as well as χ 1 μ ≤ . . . ≤ χ m μ ∈ R, such that μ( ) = 1 and for all x ∈ and 0 = v ∈ T x M the following is true: Numbers χ i μ are the Lyapunov exponents and if they are all non-zero we call μ hyperbolic. If χ i μ = 0 for some i ∈ {1, . . . , m} then the measure μ is non-hyperbolic.

Theorem 2
We are given a sequence of T -periodic orbits ( n ) n∈N such that | n | increases with n, and the ergodic measure μ n supported on n . If there exist sequences of real positive numbers (γ n ) ∞ n=1 and (κ n ) ∞ n=1 that satisfy (1) for each n the orbit n+1 is a (γ n , κ n )-good approximation of n , The sequence of T -periodic orbits fulfilling the assumptions of Theorem 2 is called a GIKN sequence. Royal measures are (by the definition) weak * limits of GIKNsequences.

The symbolic dynamics
Throughout this paper A is a finite set with the discrete topology. We define a shift space over an alphabet A as A ∞ equipped with the product topology together with the shift map σ given by the formula We denote by A n the set of the words of length-n over A. Let A * = n≥1 A n and let |u| be the word-length of u ∈ A * . Each word u ∈ A * defines its cylinder set [u] ⊂ A ∞ consisting of all the sequences in A ∞ which are prefixed by u. The cylindric sets form a open-closed basis of the topology for the shift space A ∞ . For A = a 1 · · · a n ∈ A * and k ∈ N let A k := a 1 · · · a n a 1 · · · a n · · · a 1 · · · a n , k times A ∞ := a 1 · · · a n a 1 · · · a n a 1 · · · a n a 1 · · · a n · · · Let also p(A) ∈ M e σ (A ∞ ) be a periodic measure generated by a periodic sequence A ∞ .

Entropy
Let P be a finite and a measurable division of X and μ ∈ M T (X ). The entropy P with respect to μ and T is denoted by h(X , T , μ, P) = h(μ, P) and the entropy μ with respect to T -by h(X , T , μ) = h(μ), that is h(μ) = sup P h(μ, P). In [12] the reader will find more information about the entropy.

Thef pseudometric
In this section we introduce thef pseudometric that was defined by Feldman [14] and independently by Katok [17].
where k is such that for some If A = a 1 · · · a n , A = a 1 · · · a n and m ≤ |A|, |A |, then we definē The following theorem comes from [13].

The Feldman-Katok pseudometric8
Let (X , ρ) be a compact metric space Thef δ -pseudometric between x and z is given bȳ Finally, we define the Feldman-Katok pseudometric on X ∞ as follows The above formula induces a pseudometric on X (which we also call the Feldman-Katok pseudometric and denote by¯ ) in the following way: We definef n,δ (x, z) andf δ (x, z) in the obvious way. It is not known if the pseudometric¯ induces somehow a (pseudo)metric on the simplex of invariant measures M T (X ). It gives us however the notion of convergence as explained below. We say that z We cite the below lemmas for the future reference. The details and other properties of the Feldman-Katok topologies can be found in [21].

Non-hyperbolic measures with the full support and positive entropy
To prove Theorem 7 (which together with the theorem of Bochi, Bonatti and Díaz implies our main result -Theorem 9 concerning the existence of non-hyperbolic measures with the full support and positive entropy) we need the below technical lemma. Informally, it says the following. Assume that two weak * convergent sequences of periodic measures on the shift space are given. If the periodic orbits generating these sequences are Feldman-Katok close to each other, then using them one can build generic points for the limit measures for which the Feldman-Katok pseudometric is small.
Then there exist s , w ∈ A ∞ such that s is generic for μ, w is generic for ν and f (s , w ) ≤ δ.
Proof For every n ∈ N pick M n ∈ N such that for all m ≥ M n one has min{|s (m) |, |w (m) |} > 2 n . Since the cylinder sets are clopen it follows from the portmanteau theorem that for every n ∈ N there is k n ∈ N such that for all k ≥ k n and A ∈ A n the following inequalities hold:

|p(s (k) )([A]) − μ([A])| < 1/2 n and |p(w (k) )([A]) − ν([A])| <
Let (l n ) n∈N ⊂ N satisfy for every n ∈ N the following condition: We will show that the sequences are generic for μ and ν, respectively. Since cylinder sets form a basis for the product topology on A ∞ and are clopen it is enough to prove that for every A ∈ A * one has Fix A ∈ A * and ε > 0. Let N ∈ N be such that ε > (N + 2)/2 N and |A| ≤ N . Pick n ≥ N . Let c(A, n) denote the frequency with ihich A occurs as a subword of s (k n ) . Then Since |s (k n ) | > 2 N and |A| ≤ N it follows from (1) that Define Fix p ≥ P 0 and put In other words prefix of s of the length p is of the form where t is (possibly, empty) prefix of s (k R+1 ) (different from s (k R+1 ) ). Since p ≥ P 0 , one has R ≥ N . Consequently, it follows from (3) that The above inequalities imply that the frequency with which A occurs in the block is not smaller than μ([A]) − N /2 N and not larger than μ([A]) + 1/2 N . What is more, the condition (2) says that the length of this block is at least 2 N /(2 N + 2) times more than the length of the prefix of s of the length p. Therefore Since ε is arbitrary we have that | m(s , p)( To finish the proof it is enough to notice thatf (s , w ) ≤ δ. This is however obvious since for sufficiently large n the match π of n-prefixes of s and w satisfying |D(π )| ≥ n(1 − δ) can be obtained by taking a concatenation of the optimal matches of words s (k i ) and w (k i ) from these prefixes.  ∈ (0, h(μ)) and pick δ > 0 for this ε as in Theorem 3. Assume also that: (1) ψ| K (+) > α and ψ| K (−) < −α, (2) M 1 divides M n for every n ∈ N, (3) for every n ∈ N there exists a set I n ⊂ {0, 1, · · · M n /M 1 } such that |s (n) | = |I n |, the inequality |I n | > (1 − δ)M n /M 1 is satisfied and for every j ∈ I n one has where the function n : {0, 1, · · · , |I n | − 1} → N is given by (4) p(s (n) ) → μ as n → ∞.
Let ω be an accumulation point with respect to the weak * topology of the sequence of empirical measures ⎛ Proof Note that ω is a T -invariant measure concentrated on the orbit's closure of the point x. For every n ∈ N and 0 ≤ j < M n /M 1 put l Since ω is a finite measure and the interval (−α, α) is uncountable, there is a real number β such that −α < β < α and Passing to a subsequence if necessary we can assume that ω is a limit (not only an accumulation point) of the sequence ⎛ (p(w (n) )) n∈N →ω and (m(w (n) )) n∈N →ν for someν,ω ∈ M σ ({+, −} ∞ ). We will show that h(ω, P) > 0. To this end note that the sequences (s (n) ) n∈N and (w (n) ) n∈N satisfy the assumptions of Lemma 6 for δ. Therefore we can use Theorem 3 to conclude that h(ν) > h(μ) − ε and consequently h(ω, Remark 8 Bochi, Bonatti and Díaz proved in [5] that for every n ≥ 3 there exists a compact manifold without boundary M of dimension n such that for every δ > 0 one can find a non-empty open set U ⊂ Diff(M) such that for every map T ∈ U one of the Lapunov exponents with respect to T is given by the integral of some continuous function ψ T : M → R and there are non-empty compact sets K (+), K (−) ⊂ M, a number α > 0 and an increasing sequence (M n ) n∈N ⊂ N satisfying the following properties: there are sets I n ⊂ {0, 1, · · · M n /M 1 } for n ∈ N such that: (ii) if we pick a sequence s ∈ {+, −} ∞ in such a way that |s (n) | = |I n | for every n ∈ N and s (m) is a prefix of s (n) for all m < n, then we can find a point x ∈ X such that for every j ∈ I n we have where the function n : {0, 1, · · · , |I n | − 1} → N is given by the formula In addition, for the above point x the following holds. For every measure ω ∈ω(x) and for ω-almost every point z ∈ M the conditions below are satisfied: ii. the orbit of z is dense in M..
We will now prove the main theorem of this section. We get h(ω) > (h(μ) − ε)/M 1 > 0. Note that ω ∈ω(x) and hence it follows from Remark 8 that for ω-almost every point z ∈ M the following conditions are satisfied: (ii) the orbit of z is dense in M. Let ω be a measure from the ergodic decomposition of ω with a positive entropy (such a measure exists since h(ω) > 0). Let Z ⊂ M be the set of points for which both conditions (i) and (ii) are satisfied. It follows from the ergodic decomposition theorem that ω (Z ) = 1. The Birkhoff ergodic theorem together with the condition (i) imply that M ψ T dω = 0, and so ω is a non-hyperbolic measure. Moreover, from the condition (ii) we get that ω is fully supported.

Feldman-Katok convergence of measures vs kuratowski convergence of supports
The aim of this section is to describe the relationship between the Feldman-Katok convergence of measures and the Kuratowski convergence of their supports.

Theorem 10
Let (x n ) n∈N ⊂ X be a Cauchy sequence of periodic points with respect to¯ . For m, n ∈ N denote by p n the period of x n , by μ n the measure generated by x n and by π Proof We will prove that Ls top n→∞ supp μ n ⊂ supp μ which is enough as the inclusion supp μ ⊂ Li top n→∞ supp μ n follows from the weak * convergence of the sequence (μ n ) n∈N (see [22,Theorem 1.59]). Choose z ∈ Ls top n→∞ supp μ n .Let k n ∞ and (x n ) n∈N be such that for every n ∈ N one has x n ∈ supp μ k n and ρ(z, x n ) → 0 as n → ∞. Fix ε > 0. We will show that μ (B(z, ε)) > 0. To this end choose n such that (i) x n ∈ B(z, ε/3), (ii) sup m≥n¯ (x m , x n ) < ε/3 (this condition is satisfied for n large enough as it follows from Lemma 5 that¯ (x n , x m ) =¯ (x k n , x k m )). Let x n = T j (x k n ) for some 0 ≤ j < p k n . Denote Fix m ≥ n. Note that for every p ∈ N we can consider the match π ( p) k m ,k n as ( p,¯ (x m , x n ))-match π ( p) of the points x n and x m that satisfies the following: This together with (i) imply that Therefore: Consequently, Because ε is arbitrary, we get that z ∈ supp μ.

Remark 11
As a corollary of the above Theorem 10 we obtain a part of Theorem 2: the support of every royal measure equals the Kuratowski limit of supports of periodic measures used for the construction of that measure. The notion of the Feldman-Katok convergence allows to give an alternative proof for the whole Theorem 2, see [21] for the details.

Example 12
The Feldman-Katok convergence of measures does not imply the Kuratowski convergence of their supports. To see that consider a point x, whose trajectory is convergent to z with respect to the natural topology (we assume that x is not a fixed point). Let (x n ) n∈N be a sequence of periodic points such that for every n ∈ N the following holds: (i) the period of the point x n equals 2(n 2 + n), (ii) for every 0 ≤ j < 2n one has ρ T j (x n ), T j (x) < 1/n, (iii) for every 2n ≤ j ≤ 2n 2 + 2n one has ρ T j (x n ), z < 1/n. Then¯ (x n , z) ≤ 1/n and so the sequence (x n ) n∈N tends to z with respect to¯ . On the other hand, z is a generic point for the atomic measure supported at {z}, while the Kuratowski limit of supports of measures generated by x n contains the whole orbit of x.
It is easy to see that a dynamical system that admits the above described situation can be constructed. For example (see Fig. 1), let We equip X with the maximum metric (induced from R 2 ). Let T : X → X be given by the formula We define x = (1, 0) and x n = (1, 1/2 n ) for n ∈ N. It obvious that such a system satisfies the requested conditions.
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