$C^1$ stability of endomorphisms on two dimensional manifolds

TL;DR

This paper establishes necessary and sufficient conditions for $C^1$ stability of noninvertible maps on two-dimensional manifolds, providing the first known example of such a stable map in this setting.

Contribution

It proves the sufficiency of these conditions for $C^1$ stability and presents the first example of a noninvertible, nonexpanding $C^1$ stable map in two dimensions.

Findings

Necessary conditions for $C^1$ stability are identified.

These conditions are shown to be sufficient in two-dimensional manifolds.

An explicit example satisfying these conditions is provided.

Abstract

A set of necessary conditions for $C^1$ stability of noninvertible maps is presented. It is proved that the conditions are sufficient for $C^1$ stability in compact oriented manifolds of dimension two. An example given by F.Przytycki in 1977 is shown to satisfy these conditions. It is the first example known of a $C^1$ stable map (noninvertible and nonexpanding) in a manifold of dimension two, while a wide class of examples are already known in every other dimension. \end{abstract}