Dynamics on surfaces

在动力系统理论中，对低维系统的理解是非常重要的，因为这通常可以阐明和描述在更广泛的背景下被证明是相关的基本机制和范例。虽然对一维系统的理解是相当完整的，并且是该领域的伟大成功故事之一，但二维系统远没有那么好理解，并且在这一领域的基本问题仍然是开放的。该项目的目的是利用最近的发展，并详细说明在过去几年中出现的新工具，以解决表面动力学中的一些核心问题。主要重点在于以下三个主题。1.曲面上零熵系统的分类.表面动力学中的混沌过渡.旋转理论在二维更具体地说，第1项旨在推广最近的分类面积保持C-无穷大同构的领域与零熵的弗兰克斯和亨德尔。第2项解决了C. Tresser从1983年开始对表面动力学中混沌边界上倍周期级联的发生进行了研究。第3项包括对1991年提出的Franks-Misiurewicz猜想的其余情况的研究，该猜想涉及两个环面上不存在某些类型的非周期动力学和相关的旋转集。

In dynamical systems theory the understanding of low-dimensional systems is of majorimportance, since this often allows to elucidate and describe basic mechanisms and paradigmatic examples that prove to be relevant in a much broader context. While the understanding of one-dimensional systems is fairly complete and one of the great success stories of the field, two-dimensional systems are far less well-understood and fundamental problems in this area are still wide open. The aim of the project is to make use of recent developments and elaborate new tools that have become available in the last years in order to address a number of central problems in surface dynamics. The main focus lies on the following three topics.1. Classification of zero entropy systems on surfaces.2. Transition to chaos in surface dynamics.3. Rotation theory in dimension twoMore specifically, item 1 aims at a generalisation of a recent classification of area-preserving C-infinity diffeomorphisms of the sphere with zero entropy by Franks and Handel. Item 2 addresses a conjecture by C. Tresser from 1983 on the occurrence of period doubling cascades on the boundary of chaos in surface dynamics. Item 3 includes the study of the remaining cases of the Franks-Misiurewicz conjecture, stated in 1991, concerning the non-existence of certain types of aperiodic dynamics and the related rotation sets on the two-torus.