Continuity properties of Lyapunov exponents for surface diffeomorphisms

TL;DR

This paper investigates how entropy and Lyapunov exponents vary with invariant measures for smooth surface diffeomorphisms, establishing inequalities and conditions linking their continuity properties and implications for SRB measures.

Contribution

It introduces a new inequality relating the discontinuities of entropy and Lyapunov exponents, and provides criteria for the existence of SRB measures with positive entropy.

Findings

Continuity of entropy implies continuity of Lyapunov exponents on certain measure sets.

Hausdorff dimension is upper semi-continuous for measures with entropy bounded away from zero.

A new criterion for the existence of SRB measures with positive entropy.

Abstract

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.