Path connectedness and entropy density of the space of ergodic hyperbolic measures
Path connectedness and entropy density of the space of ergodic hyperbolic measures
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遍历双曲测度空间的路径连通性和熵密度
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Y. Pesin
中科院分区：
文献类型：
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作者：
A. Gorodetski;Y. Pesin
我们证明，如果该类中的任何两个双曲周期点是同宿相关的，则孤立同宿类支持的给定索引的双曲遍历测度空间是路径连通且熵密集的。作为推论，我们得出该空间的闭合也是路径连通的。在本文中，我们考虑紧流形 C 1+α 微分同态的周期点的同宿类，并讨论其上支持的不变测度空间的两个性质，并配备了双曲遍历测度子空间的周�-拓扑-连通性和熵密度。对后一个空间连通性的研究是由西格蒙德在一篇短文中发起的（27）。他在传递拓扑马尔可夫位移的情况下以及公理 A 微分同胚的推论中建立了该空间的路径连通性。西格蒙德的想法是首先证明任何两个周期测度（即周期点上的不变原子测度）可以通过遍历测度的连续路径连接，其次如果两个周期测度之一位于另一个周期测度的一个小邻域内，则可以选择整个路径位于该邻域内。为了执行第一步，西格蒙德证明任何周期性测度都可以通过马尔可夫测度来近似，并且任何两个马尔可夫测度都可以通过马尔可夫测度的路径连接。我们在定理 1.1 的证明中使用了西格蒙德的想法。西格蒙德定理的另一种方法是证明传递拓扑马尔可夫移位的遍历测度在所有不变测度的空间中是密集的。由于后一个空间是单纯形，且遍历测度是其极值点，因此这意味着该空间是波尔森单纯形（在同胚方面是唯一的）。现在期望的结果来自于（23）中给出的波尔森单纯形的完整描述（另见（14））。我们用这种方法来证明我们的定理 1.2。自从西格蒙德的工作以来，人们对双曲遍历测度空间的连通性研究的兴趣已经不知何故失去了， 1 直到最近它才重新受到关注。在（15）中，Gogolev 和 Tahzibi 受非双曲不变测度存在性研究的启发，提出了遍历空间是否存在的问题。
We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are ho- moclinically related. As a corollary we obtain that the closure of this space is also path connected. In this paper we consider homoclinic classes of periodic points for C 1+α dif- feomorphisms of compact manifolds and we discuss two properties of the space of invariant measures supported on them and equipped with the week � -topology - connectedness and entropy density of the subspace of hyperbolic ergodic measures. The study of connectedness of the latter space was initiated by Sigmund in a short article (27). He established path connectedness of this space in the case of tran- sitive topological Markov shifts and as a corollary, of Axiom A diffeomorphisms. Sigmund's idea was to show first that any two periodic measures (i.e., invariant atomic measures on periodic points) can be connected by a continuous path of er- godic measures and second that if one of the two periodic measures lies in a small neighborhood of another one, then the whole path can be chosen to lie in this neighborhood. In order to carry out the first step Sigmund shows that any peri- odic measure can be approximated by Markov measure and that any two Markov measures can be connected by a path of Markov measures. We use Sigmund's idea in our proof of Theorem 1.1. A different approach to Sigmund's theorem is to show that ergodic measures on a transitive topological Markov shift are dense in the space of all invariant measures. Since the latter space is a simplex and ergodic measures are its extremal points, it means that this space is the Poulsen simplex (which is unique up to a homeomorphism). The desired result now follows from a complete description of the Poulsen simplex given in (23) (see also (14)). We use this approach to prove our Theorem 1.2. Since Sigmund's work the interest to the study of connectedness of the space of hyperbolic ergodic measures has somehow been lost, 1 and only recently it has regain attention. In (15) Gogolev and Tahzibi, motivated by their study of existence of non-hyperbolic invariant measures, raised a question of whether the space of ergodic