Ergodic Theory on Moduli Spaces

Abstract

Let M be a compact surface with x(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2) itself). Then the mapping class group rM of M acts on the moduli space X(M) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then FM preserves a symplectic structure on X(M) which has finite total volume on M. More generally, the subspace of X(M) corresponding to flat bundles with fixed behavior over AM carries a rM-invariant symplectic structure. The main result is that rM acts ergodically on X(M) with respect to the measure induced by the symplectic structure.