A set of necessary conditions for C¹ stability of noninvertible maps is presented. It is proved that the conditions are sufficient for C¹ 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 C¹ stable map (noninvertible and nonexpanding) in a manifold of dimension two, while a wide class of examples are already known in every other dimension.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-3,
author = {J. Iglesias and A. Portela and A. Rovella},
title = {C$^1$ stability of endomorphisms on two-dimensional manifolds},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {37-58},
zbl = {1263.37046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-3}
}
J. Iglesias; A. Portela; A. Rovella. C¹ stability of endomorphisms on two-dimensional manifolds. Fundamenta Mathematicae, Tome 219 (2012) pp. 37-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-1-3/