On the inverse limit stability of endomorphisms

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 $(T)$.