Mathematical Sciences: Rotation and Entropy of Surface Homeomorphisms

本文主要研究曲面上的同胚理论，特别是旋转和拓扑熵。该项目的核心是通过规定旋转行为在精确意义上是足够复杂的，以映射的旋转熵来衡量，来验证环面或环面同胚中本质同宿轨道的存在。这项研究还将试图在更高亏格的曲面上建立一个正确的旋转向量的概念，作为建立特定旋转复杂性的周期轨道存在的新标准的一种手段。本文的研究是在曲面拓扑学和动力学系统的交界处进行的。它利用了低维系统中各种拓扑和动力学概念之间的有趣联系。

This research will focus on the theory of homeomorphisms on surfaces, specifically rotation and topological entropy. The core of the project deals with the problem of verifying the existence of essential homoclinic orbits in annulus or toral homeomorphisms by specifying that the rotational behavior be sufficiently complex, in a precise sense, as measured by the rotational entropy of the mapping. The research will also attempt to formulate a correct notion of a rotational vector on a surface of higher genus, as a means of establishing new criteria for the existence of periodic orbits of a specified rotational complexity. This research is at the interface of surface topology and dynamical systems. It makes use of interesting connections between various topological and dynamical notions in low dimensional systems.