Strictly toral dynamics

• 成果类型: Article
• 署名作者: Koropecki, Andres; Tal, Fabio Armando
• 署名单位: Universidade Federal Fluminense; Universidade de Sao Paulo
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0470-3
• 发表日期: 2014
• 页码: 339-381
• 关键词: torus homeomorphisms; rotation vectors; THEOREM; CURVES; sets

摘要

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points is shown to be a disjoint union of periodic topological disks (elliptic islands), while the set of essential points is an essential continuum, with typically rich dynamics (the chaotic region). This generalizes and improves a similar description by Jager. The key result is boundedness of these elliptic islands, which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in is as rich as in from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.