$C^1$ stability of endomorphisms on two-dimensional manifolds
Volume 219 / 2012
Fundamenta Mathematicae 219 (2012), 37-58
MSC: Primary 37C75; Secondary 37C20.
DOI: 10.4064/fm219-1-3
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.
