Lyapunov spectrum of homoclinic classes

TL;DR

This paper investigates the Lyapunov spectrum of ergodic measures in isolated homoclinic classes of $C^1$-generic diffeomorphisms, revealing that the spectrum has nonempty interior and can be realized by measures supported on the class.

Contribution

It demonstrates that the Lyapunov spectrum of these classes has nonempty interior and characterizes the spectra realizable by ergodic measures.

Findings

Lyapunov spectrum has nonempty interior.

Any interior vector is realizable by an ergodic measure.

Discusses the averaged Lyapunov spectrum extension.

Abstract

We study the Lyapunov spectrum of the ergodic measures of isolated homoclinic classes of $C^1$-generic diffeomorphisms. We show that this spectrum has nonempty interior and that any vector in its interior is the spectrum of some ergodic measure fully supported on the homoclinic class. We also discuss the averaged Lyapunov spectrum of homoclinic classes (an extension of the Lyapunov graph).