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，因为这通常允许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相当完整，也是该领域最成功的故事之一，但二维系统远没有得到很好的理解，这一领域的基本问题仍然悬而未决。该项目的目的是利用最近的事态发展和过去几年提供的精心设计的新工具，以解决表面动力学中的一些核心问题。主要集中在以下三个主题上。表面上零熵系统的分类2。表面动力学向混沌的过渡。更具体地说，项1旨在推广Franks和Handel最近对具有零熵的球面的保面积C-无穷微分同态的分类。项2解决了C.Tresser在1983年提出的一个猜想，即在表面动力学的混沌边界上出现倍周期级联。项目3包括对1991年提出的Franks-Misiurewicz猜想的其余情况的研究，该猜想涉及不存在某些类型的非周期动力学以及两个环面上的相关旋转集。

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.