Persistent bundles over a two dimensional compact set

TL;DR

This paper extends the concept of Axiom A and strong transversality to invariant compact sets in surface diffeomorphisms, demonstrating the persistence of certain bundle structures and providing new examples of persistent laminations beyond normal hyperbolicity.

Contribution

It generalizes the AS condition to invariant compact sets and proves the structural stability and persistence of bundles over these sets in surface diffeomorphisms.

Findings

Structural stability of AS invariant compact sets on surfaces.

Persistence of bundles over these sets under certain conditions.

Existence of non-trivially persistent laminations not normally hyperbolic.

Abstract

The $C^1$-structurally stable diffeomorphims of a compact manifold are those that satisfy Axiom A and the strong transversality condition (AS). We generalize the concept of AS from diffeomorphisms to invariant compact subsets. Among other properties, we show the structural stability of the AS invariant compact sets $K$ of surface diffeomorphisms $f$. Moreover if $\hat f$ is the dynamics of a compact manifold, which fibers over $f$ and such that the bundle is normally hyperbolic over the non-wandering set of $f_{|K}$, then the bundle over $K$ is persistent. This provides non trivial examples of persistent laminations that are not normally hyperbolic.