On the inverse limit stability of endomorphisms

TL;DR

This paper investigates the stability of endomorphisms under inverse limits in the $C^1$ topology, establishing conditions under which such stability occurs and providing examples and conjectures related to these dynamics.

Contribution

It introduces a new perspective on inverse limit stability for endomorphisms, linking it to axiom A and a strong transversality condition, and offers examples and conjectures in this framework.

Findings

Existence of a $C^1$-inverse stable endomorphism that is robustly transitive with persistent critical set.

All $C^1$-inverse limit stable endomorphisms satisfying axiom A also satisfy a strong transversality condition $(T)$.

Endomorphisms satisfying axiom A and $(T)$ are $C^1$-inverse limit stable, with applications to Hénon maps and rational functions.

Abstract

We present several results suggesting that the concept of $C^1$-inverse limit stability is free of singularity theory. We describe an example of a $C^1$-inverse stable endomorphism which is robustly transitive with persistent critical set. We show that every (weak) axiom A, $C^1$-inverse limit stable endomorphism satisfies a certain strong transversality condition $(T)$. We prove that every attractor-repellor endomorphism satisfying axiom A and Condition $(T)$ is $C^1$-inverse limit stable. The latter is applied to H\'enon maps, rational functions of the sphere and others. This leads us to conjecture that $C^1$-inverse stable endomorphisms are those which satisfy axiom A and the strong transversality condition $(T)$.