Connectedness of the set of central Lyapunov exponents

TL;DR

This paper proves that for a generic set of diffeomorphisms with certain hyperbolic properties, the set of central Lyapunov exponents associated with ergodic measures with full support or positive entropy forms an interval.

Contribution

It establishes the connectedness of the set of central Lyapunov exponents for generic partially hyperbolic homoclinic classes with one-dimensional center.

Findings

The set of central Lyapunov exponents is an interval for generic diffeomorphisms.

This applies to ergodic measures with full support or positive entropy.

The result holds for partially hyperbolic homoclinic classes with one-dimensional center.

Abstract

We show that there is a residual subset $\mathcal{R}$ of $Diff^1(M)$ such that for any $f\in\mathcal{R}$ and any partially hyperbolic homoclinic class $H(p,f)$ with one dimensional center direction, the set of central Lyapunov exponents associated with the ergodic with either full support or positive entropy is an interval.