Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
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DOI：
10.1007/bf02684777
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发表时间：
1980-12
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通讯作者：
A. Katok
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文献类型：
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作者：
A. Katok
介绍。本文结合s-轨迹和Lyapunov特征指数两种不同的技术，研究了紧流形上的微分同态的一些动力学性质。这两种方法是分别为不同的目的而开发的。鲁弗斯·鲍恩（Rufus Bowen） [i], b[2]？[3])和(i D-V-Anosov（[4]；证明见[5]，[6]）是基于这样的观察：假设某些双曲条件，观察到的几乎在某些微分同构中发生的动力学现象通常在该微分同构中发生。利用这一方法，Bowen证明了关于公理a微分同态和流的周期轨道的渐近增长和极限分布、平衡态的唯一性和遍历性等一系列深刻的结果（[2],[3],[7]）。第二种方法是由Ja开发的。B. Pesin[8]用于研究具有等效黎曼体积不变测度的光滑动力系统的遍历性质（如遍历性、熵、k性质、伯努利性质）（[9],[10],[ii]）。这一系列论文中使用的许多想法都出现在布林和佩辛的早期工作中。佩辛的大部分论证都是在没有对不变测度进行特殊假设的情况下成立的（D. Ruelle在2010年也观察到了这一事实）。本节包含了在李雅普诺夫意义上的轨道正则附近的微分同态的行为描述（关于正则的定义，见[8]，no. 6）。3);[9]，§3；[13])和不变缩展开流形的构造和性质，
Introduction. i. In this paper I study some dynamical properties of diffeomorphisms on compact manifolds by combining two different techniques, s-trajectories and the Lyapunov characteristic exponents. These two approaches were developed separately and for different purposes. The technique of s-trajectories introduced by Rufus Bowen ([i],[2]?[3]) an (i D-V-Anosov ([4]; for proofs see [5],[6]) is based on the observation that assuming some hyperbolicity conditions, dynamical phenomena which are observed to almost occur for some diffeomorphism usually do occur for that diffeomorphism. Using this approach Bowen proved a number of profound results concerning the asymptotic growth and the limit distribution of periodic orbits for Axiom A diffeomorphisms and flows, uniqueness and the ergodic properties of equilibrium states and so on ([2],[3],[7]).The second approach was developed by Ja. B. Pesin [8] for the study of ergodic properties (such as ergodicity, entropy, K-property, Bernoulli property) of smooth dynamical systems with an invariant measure equivalent to a Riemannian volume ([9],[10],[ii]). Many of the ideas used in this cycle of papers had occurred in the earlier work of Brin and Pesin [12]. A large part of Pesin's arguments works without special assumptions about the invariant measure (D. Ruelle has also observed this fact in [21]). This section contains the description of the behavior of a diffeomorphism near a trajectory regular in the Lyapunov sense (for definitions of regularity, see [8], no.(o. 3);[9], § 3;[13]) and the construction and the properties of invariant contracting and expanding manifolds,