ERGODIC-THEORY ON MODULI SPACES

Let M be a compact surface with chi(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to S U(2) (for example SU(2) itself). Then the mapping class group Gamma(M) 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 Gamm a(M) preserves a symplectic structure on X(M) which has finite total v olume on M. More generally, the subspase of X(nl) corresponding to fla t bundles with fixed behavior over partial derivative M carries a Gamm a(M)-invariant symplectic structure. The main result is that Gamma(M) acts ergodically on X(M) with respect to the measure induced by the sy mplectic structure.