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.

We give here the first examples of C¹ structurally stable maps on manifolds of dimension greater than two that are neither diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C¹ stable. A necessary condition for the existence of such examples is also given.

A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.

Sufficient conditions for a map having nonwandering critical points to be Ω-stable are introduced. It is not known if these conditions are necessary, but they are easily verified for all known examples of Ω-stable maps. Their necessity is shown in dimension two. Examples are given of Axiom A maps that have no cycles but are not Ω-stable.

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