Mathematical Sciences: Investigations in Probability and Erodic Theory

本文将从概率论和遍历论的几个方面进行研究：1)可数字母序列空间上吉布斯态的构造和基本性质；2)在具有可数符号动力学的动力系统中的应用，如非紧流形上的测地线流和带抛物点的共形动力系统；3)流动轨道统计规律的相关问题；4)分支扩散过程和分支随机游走的时空增长。这项研究是统计理论、动力系统和遍历理论的融合。有一类很广泛的动力系统，它们都有完善的符号动力学机制。这个类的特点是，相关的符号动力学只需要一个有限的字母（所以必须处理有限状态马尔可夫过程）。本研究将这类扩展到那些需要无限可数字母的系统，并开发了相应的机制。这将为黎曼流形上轨道在极限集上的聚类和封闭测地线统计的新结果提供工具。这项工作的结果将是统一许多已知的结果，这些结果是通过非常不同的方法获得的，因此统一，从而简化了具有完全不同结果的领域。

This research will be conducted in several areas of probability and ergodic theory: 1) Construction and fundamental properties of Gibbs states on sequence spaces with countable alphabets; 2) applications to dynamical systems with countable symbolic dynamics, e.g., geodesic flows on noncompact manifolds and conformal dynamical systems with parabolic points; 3) related problems concerning statistical regularities of orbits in flows; and 4) temporal and spatial growth in branching diffusion processes and branching random walks. This research is a blend of statistical theory, dynamical systems, and ergodic theory. There is a broad class of dynamical systems for which the well-developed machinery of symbolic dynamics is available. This class is characterized by the fact that the associated symbolic dynamics needs only a finite alphabet (so one has to deal with finite state Markov processes). This research extends this class to those systems for which an infinitely countable alphabet becomes necessary and develops the corresponding machinery. This will provide the tools for obtaining new results about the clustering of orbits at limit sets and about the statistics of closed geodesics on Riemannian manifolds. The outcome of this work will be to unite many known results which were won by very different methods, so unifying, thus simplifying, areas with quite separate results.