Continuity properties of Lyapunov exponents for surface diffeomorphisms

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.

References in corpus (7)

• Genericity of zero Lyapunov exponents
• Continuity of Lyapunov Exponents for Cocycles with Invariant Holonomies
• Escape of mass and entropy for geodesic flows
• Effectivity of Uniqueness of the Maximal Entropy Measure on $p$-adic homogeneous spaces
• Entropies and volume growth of unstable manifolds
• Escape of entropy for countable Markov shifts
• C^r surface diffeomorphisms with no maximal entropy measure

Cited by in corpus (1)

• SRB measures for C $\infty$ surface diffeomorphisms