Dynamical Systems on Non-compact Spaces

这个项目涉及遍历理论和动力系统。本研究将集中于与动力系统的非紧性有关的三个基本问题。第一个问题考虑无限体积双曲表面上环流的测量刚度现象。现有的理论处理有限不变测度，而该项目涉及一个程序来研究无限但局部有限的情况。第二个和第三个问题是关于无限马尔可夫分区映射和非一致双曲曲面微分同态映射的热力学形式。在首席研究员和其他人先前工作的基础上，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关键现象（即相变）之间的类比。主要研究者将追求一个详细的程序，利用这个类比，以系统的方式探索非紧性或非均匀双曲的遍历理论效应。动力系统是一个模型，它描述了一个系统（想想一个物理系统）可以处于许多可能的状态之一，以及一个规定系统状态如何随时间演变的定律。这些模型经常用于数学、物理、生物和工程。迄今为止，动力系统的大多数数学研究都集中在可能状态集合在适当意义上是“小”的系统上（精确的数学术语是“紧”）。相反，本项目研究具有非紧态集合的动力系统。它着重于各种只能在非紧化环境中出现的动力学现象，最引人注目的是无限不变测度水平上的新型刚性，以及类似于物理学中遇到的临界现象的某些现象集合。这些思想在其他数学领域有许多潜在的应用，包括几何和数学物理。特别希望研究结果能对统计物理中的相变理论有所启发。

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.