Simple Lyapunov spectrum of partially hyperbolic diffeomorphisms

TL;DR

This paper investigates the simplicity of Lyapunov spectra in partially hyperbolic diffeomorphisms, showing generic simplicity for certain classes and providing criteria based on periodic points.

Contribution

It establishes conditions under which the Lyapunov spectrum is simple and demonstrates generic simplicity for measures of maximal entropy in partially hyperbolic systems.

Findings

Certain volume-preserving partially hyperbolic diffeomorphisms are approximated by open sets with simple spectrum.

A criterion for simplicity based on periodic points homoclinically related to the measure.

Generic simplicity of Lyapunov spectrum for measures of maximal entropy.

Abstract

We study the simplicity of the Lyapunov spectrum of partially hyperbolic diffeomorphisms. We prove that a class of volume-preserving partially hyperbolic diffeomorphisms is $C^r$-accumulated by $C^2$-open sets with simple spectrum. Also we prove that a class of partially hyperbolic maps has simple spectrum generically for the measures of maximal entropy. In order to prove these results, we give a criterion for simplicity of the Lyapunov spectrum in terms of periodic points homoclinically related to the invariant measure.