Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.