Homotopically unbounded disks for generic surface diffeomorphisms

TL;DR

This paper investigates homotopically unbounded periodic disks in surface diffeomorphisms isotopic to the identity, revealing their prevalence and dynamical significance under generic conditions, especially related to rotation sets and attractors.

Contribution

It establishes the generic existence and properties of unbounded periodic disks in surface diffeomorphisms with nonempty interior rotation sets, advancing understanding of their dynamical roles.

Findings

Unbounded periodic disks are common in generic surface diffeomorphisms.

Presence of nonempty interior rotation set implies existence of such disks.

Unbounded disks with rational prime ends rotation number are basins of attractors or repellers.

Abstract

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.