Dynamical Systems on Non-compact Spaces

非紧空间动力系统

• 批准号: 0652966
• 负责人: Omri Sarig
• 金额: --
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652966&HistoricalAwards=false
• 关键词: Dynamical; Systems; Non; compact; Spaces

This project deals with ergodic theory and dynamical systems. The research will concentrate on three fundamental problems associated with various aspects of noncompactness in dynamical systems. The first problem considers measure rigidity phenomena for horocycle flows on infinite volume hyperbolic surfaces. The existing theory treats finite invariant measures, whereas the project involves a program to study the infinite, but locally finite, case. The second and third problems are concerned with the thermodynamic formalism for maps with infinite Markov partitions and for nonuniformly hyperbolic surface diffeomorphisms. Building on previous work of the principal investigator and others, the project will investigate an analogy between certain phenomena associated with such systems and critical phenomena in statistical physics (namely, phase transitions). The principal investigator will pursue a detailed program for utilizing this analogy to explore in a systematic way the ergodic theoretic effects of noncompactness or nonuniform hyperbolicity. A dynamical system is a model that describes a system (think of a physical system) that can be in one of many possible states, together with a law that prescribes how the state of the system evolves in time. Such models are used frequently in mathematics, physics, biology, and engineering. Most of the mathematical research in dynamical systems hitherto has focused on systems whose collections of possible states are "small" in an appropriate sense (the precise mathematical term is "compact"). In contrast, this project studies dynamical systems with noncompact collections of states. It focuses on various dynamical phenomena that can appear only in the noncompact setting, most notably a new type of rigidity on the level of infinite invariant measures, as well as a certain collection of phenomena similar to critical phenomena that one encounters in physics. There are numerous potential applications of these ideas to other areas of mathematics, including geometry and mathematical physics. In particular, it is hoped that the results of the research will shed light on the theory of phase transitions in statistical physics.

这个项目涉及遍历理论和动力系统。研究将集中在与动力系统中的非紧性的各个方面相关的三个基本问题。第一个问题考虑无限体积双曲曲面上的均圈流动的测度刚性现象。现有的理论对待有限不变的措施，而该项目涉及一个程序来研究无限的，但局部有限的情况下。第二和第三个问题是关于具有无穷马尔可夫划分的映射和非一致双曲曲面自同构的热力学形式。在主要研究者和其他人以前工作的基础上，该项目将研究与此类系统相关的某些现象与统计物理学中的临界现象（即相变）之间的类比。主要研究者将追求一个详细的计划，利用这个类比，以系统的方式探索非紧性或非均匀双曲性的遍历理论效应。动态系统是一个模型，它描述了一个系统（想想一个物理系统），它可以处于许多可能的状态之一，以及规定系统状态如何随时间演变的定律。这些模型经常在数学、物理、生物学和工程学中使用。迄今为止，大多数动力系统的数学研究都集中在可能状态的集合在适当意义上是“小”的系统（精确的数学术语是“紧凑”）。相反，这个项目研究具有非紧状态集合的动力系统。它专注于只能在非紧环境中出现的各种动力学现象，最值得注意的是无限不变测度水平上的一种新型刚性，以及类似于物理学中遇到的临界现象的某些现象。这些思想在数学的其他领域有许多潜在的应用，包括几何和数学物理。特别是，希望研究结果将阐明统计物理学中的相变理论。