Continuity properties of Lyapunov exponents for surface diffeomorphisms
Abstract
We study the entropy and Lyapunov exponents of invariant measures μ for smooth surface diffeomorphisms f, as functions of (f,μ). The main result is an inequality relating the discontinuities of these functions. One consequence is that for a C∞ 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.
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