  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.
