Erodic Properties for 'Almost Hyperbolic' Systems

摘要：本文研究了“几乎双曲”动力系统的遍历性质。这里的“几乎双曲”系统是指光滑的动力系统，除了有限的一组点外，它在任何地方都是双曲的。这些系统包括“几乎Anosov”微分同态、“几乎双曲”不变集、单位区间上具有不动点的分段展开式映射以及具有不动点的展开式映射的不动集。“几乎双曲”系统位于光滑动力系统空间中一致双曲系统集的边界上。从表面上的“几乎Anosov”微分同态得到的结果来看，这种系统的遍历性质可能与一致双曲系统的遍历性质大不相同，尽管拓扑性质相似。例如，这些系统可能允许也可能不允许SRB度量，即使它们允许，相关衰减也可能从指数变为多项式。本项目强调一般“几乎双曲”系统的SRB测度的存在性和性质、相关衰减率和小扰动下的随机稳定性。均匀双曲系统是60年代末以来光滑动力系统的主要研究对象。这些系统表现出多种类型的复杂动态行为，其行为通常被认为是混沌的。遍历理论关注的是对系统长期行为的理解。现在一致双曲系统的遍历性质已经很好地理解了，人们对非一致双曲系统的遍历性质很感兴趣。“几乎双曲”意味着双曲条件在有限的点集合上失效。这些系统位于一致双曲系统集的边界上，是最简单的非一致双曲系统。先前对某些特定系统（二维环面中的“几乎阿诺索夫”系统）获得的结果表明，这些系统可能（有时确实）表现出完全不同的长期行为。在这个项目中，我们将研究更一般的“几乎双曲”系统的遍历性质。我们对这类系统的轨道干扰、混合速率和小扰动下的随机稳定性特别感兴趣。

AbstractHuThis proposed research addresses ergodic properties of "almost hyperbolic" dynamical systems.Here "almost hyperbolic" systems means a smooth dynamical systems that is hyperbolic everywhere except for a finite set of points. These systems include "almost Anosov" diffeomorphisms, "almost hyperbolic" invariant sets, piecewise expanding maps on the unit interval with indifferent fixed points, and invariant sets of expanding maps with indifferent fixed points. "Almost hyperbolic" systems lie on the boundary of the set of uniformly hyperbolic systems in the space of smooth dynamical systems. By the results obtained from "almost Anosov" diffeomorphisms on the surface, the ergodic properties of such systems may be quite different from that of uniformly hyperbolic systems, though the topological properties are similar. For example, these systems may or may not admit SRB measures, and even when they do, correlation decay may change from exponential to polynomial.This project stresses existence and properties of SRB measures, rate of decay of correlations, and stochastic stability under small perturbations for general "almost hyperbolic" systems.Uniformly hyperbolic systems are the main research subjects in smooth dynamical systems since late 60's. These systems display many types of complex dynamic behavior, and whose behavoir are often regarded as chaotic. Ergodic theory concerns understanding the long-term behavior of systems. Now ergodic properties for uniformly hyperbolic systems are understood very well, and people are interested in such properties for nonuniformly hyperbolic systems. "Almost hyperbolic" means that hyperbolic conditions fail at a finite set of points. These systems lie on the boundary of the set of uniformly hyperbolic systems, and are the simplest nonuniformly hyperbolic systems. Results obtained earlier for some particular systems, "almost Anosov" systems in the two-dimensional torus, show that such systems may-and sometimes do-exhibit totally different long term behavior. In this project we will study ergodic properties of more general "almost hyperbolic" systems. We are particular interested in orbit distrubitions, rate of mixing, and stochastic stability under small perturbation for such systems.