Continuity properties of Lyapunov exponents for surface diffeomorphisms

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 $C^{\mathrm{\infty }}$ 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.