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

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}