Structural stability of the inverse limit of endomorphisms

TL;DR

This paper proves that certain endomorphisms satisfying specific hyperbolicity and transversality conditions are structurally stable in the inverse limit sense, providing a characterization and applications to homoclinic tangencies.

Contribution

It establishes the $C^1$-inverse limit structural stability of endomorphisms under Axiom A and transversality, confirming conjectured necessary and sufficient conditions.

Findings

Proves $C^1$-inverse limit structural stability under Axiom A and transversality.

Characterizes $C^1$-inverse limit structurally stable covering maps.

Applies results to unfolding of homoclinic tangencies.

Abstract

We prove that every endomorphism which satisfies Axiom A and the strong transversality conditions is $C^1$-inverse limit structurally stable. These conditions were conjectured to be necessary and sufficient. This result is applied to the study of unfolding of some homoclinic tangencies. This also achieves a characterization of $C^1$-inverse limit structurally stable covering maps.