论文标题
同型无界磁盘，用于通用表面差异性
Homotopically unbounded disks for generic surface diffeomorphisms
论文作者
论文摘要
在本文中，我们考虑了正属属和$ c^r $ -diffeomorphisms $ f：s \ rightarrow s $同位素的封闭式表面$ s $ s $ - diffeomorphisms $ f：s \ rightarrow s $同位素（$ r \ geq 1）$。主要目的是研究同型无限制的周期性开放拓扑磁盘（即，在通用覆盖物中升至无限的连接集）。我们表明这些磁盘并不少见，并且与重要的动力学现象有关。我们还在某些通用条件下研究了这些磁盘上的动力学。我们的第一个主要结果意味着，如果地图的旋转集具有非空的内部且不局部恒定，则对于圆环（或对任意表面，具有额外条件），则该地图为$ c^r $ $ $ $ $ $ $ $ $ $ $ $，表现出表现出周期性均匀无结合磁盘的差异性。我们的第二个结果表明，$ c^r $ - 基因上，如果旋转集具有非空内部装置（如果$ s $的属属大于$ 1 $，则是一个额外的假设），最大的周期性磁盘是无绑定的最大周期性磁盘，并且具有合理的质量旋转数，必须是某些紧凑的吸引者或驱虫器中包含的底座或disk中包含的库。作为副产品，我们获得了描述$ c^r $ generic设置中周期性轨道稳定或不稳定流形的补充的某些周期性成分的结果。
In this paper we consider closed orientable surfaces $S$ of positive genus and $C^r$-diffeomorphisms $f:S\rightarrow S$ isotopic to the identity ($r\geq 1)$. The main objective is to study periodic open topological disks which are homotopically unbounded (i.e. which lift to unbounded connected sets in the universal covering). We show that these disks are not uncommon, and are related to important dynamical phenomena. We also study the dynamics on these disks under certain generic conditions. Our first main result implies that for the torus (or for arbitrary surfaces, with an additional condition) if the rotation set of a map has nonempty interior and is not locally constant, then the map is $C^r$-accumulated by diffeomorphisms exhibiting periodic homotopically unbounded disks. Our second result shows that $C^r$-generically, if the rotation set has nonempty interior (plus an additional hypothesis if the genus of $S$ is greater than $1$) a maximal periodic disk which is unbounded and has a rational prime ends rotation number must be the basin of some compact attractor or repeller contained in the disk. As a byproduct we obtain results describing certain periodic components of the complement of the closure of stable or unstable manifolds of a periodic orbit in the $C^r$-generic setting.
