Surface Attractors

TL;DR

This paper proves that for a continuous surface endomorphism with an attracting set where the restriction is a covering map, if the map is a local homeomorphism near the attractor, then it extends as a covering over the immediate basin.

Contribution

It establishes that local homeomorphism conditions near an attracting set imply a global covering property on its immediate basin.

Findings

If $f$ is a local homeomorphism in the basin, then $f$ is a $d:1$ covering there.

The result links local and global covering properties for surface endomorphisms.

Applicable to understanding the structure of attracting sets in surface dynamics.

Abstract

Let $f$ be a continuous endomorphism of a surface $M$, and $A$ an attracting set such that the restriction $f|_A: A \to A$ is a $d:1$ covering map. We show that if $f$ is a local homeomorphism in the immediate basin $B^0_A$ of $A$, then $f$ is also a $d:1$ covering of $B^0_A$.