(Dis)continuity of Lyapunov exponents

Joint work with J. Buzzi and S. Crovisier.
Suppose $f$ is a topologically transitive $C^\infty$ surface diffeomorphism. The maps which associate to an ergodic invariant measure $\mu$ the entropy of $\mu$ and the top Lyapunov exponent of $\mu$ are not continuous in the weak star topology. We compare their discontinuities.