On the dominated splitting of Lyapunov stable aperiodic classes

TL;DR

This paper proves that for generic $C^1$ diffeomorphisms, Lyapunov stable aperiodic classes with a non-trivial dominated splitting are necessarily partially hyperbolic, with one bundle hyperbolic, advancing understanding of dynamical stability.

Contribution

It establishes that a non-trivial dominated splitting in Lyapunov stable aperiodic classes implies partial hyperbolicity with a hyperbolic bundle, under generic conditions.

Findings

Lyapunov stable aperiodic classes with dominated splitting are partially hyperbolic.

One of the bundles in the splitting must be hyperbolic.

The result applies to $C^1$-generic diffeomorphisms far from homoclinic bifurcations.

Abstract

Recent works related to Palis conjecture of J. Yang, S. Crovisier, M. Sambarino and D. Yang showed that any aperiodic class of a $C^1$-generic diffeomorphism far away from homoclinic bifurcations (or homoclinic tangencies) is partially hyperbolic. We show in this paper that, generically, a non-trivial dominated splitting implies partial hyperbolicity for an aperiodic class if it is Lyapunov stable. More precisely, for $C^1$-generic diffeomorphisms, if a Lyapunov stable aperiodic class has a non-trivial dominated splitting $E\oplus F$, then one of the two bundles is hyperbolic (either $E$ is contracted or $F$ is expanded).