Erodic Properties for 'Almost Hyperbolic' Systems

“几乎双曲”系统的侵蚀特性

• 批准号: 9970646
• 负责人: Huyi Hu
• 金额: $ 6.88万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970646&HistoricalAwards=false
• 关键词: Erodic; Properties; Almost; Hyperbolic; Systems

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.

本文研究了“几乎双曲”动力系统的遍历性，这里的“几乎双曲”系统是指除了有限点集之外处处是双曲的光滑动力系统。 这些系统包括“几乎Anosov”同构、“几乎双曲”不变集、单位区间上具有中立不动点的分段扩张映射以及具有中立不动点的扩张映射的不变集。 在光滑动力系统空间中，“几乎双曲”系统位于一致双曲系统集合的边界上。 由曲面上的“几乎Anosov”同态得到的结果表明，这类系统的遍历性质可能与一致双曲系统的遍历性质有很大的不同，尽管它们的拓扑性质是相似的. 例如，这些系统可能接受也可能不接受SRB测度，即使它们接受SRB测度，相关性的衰减也可能从指数变化到多项式.本项目着重研究SRB测度的存在性和性质，相关性的衰减率，以及一般“几乎双曲”系统在小扰动下的随机稳定性.一致双曲系统是60年代后期以来光滑动力系统的主要研究课题. 这些系统表现出多种复杂的动力学行为，其行为通常被认为是混沌的。 遍历理论关注理解系统的长期行为。 目前，一致双曲型方程组的遍历性已经得到了很好的理解，而非一致双曲型方程组的遍历性也引起了人们的兴趣。 “几乎双曲”意味着双曲条件在有限的点集处失效。 这些系统位于一致双曲系统集合的边界上，是最简单的非一致双曲系统。 对于某些特殊的系统，如二维环面中的“几乎Anosov”系统，先前得到的结果表明，这样的系统可能--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。 我们对此类系统的轨道分布、混合率和小扰动下的随机稳定性特别感兴趣。