Mapping Class Actions on Moduli Spaces

It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic leaf of the Poisson moduli space of flat SU(2)-bundles over S, X(S, SU(2)). In our study of how individual mapping classes act on X, we show that ergodicity does not restrict to that of cyclic subgroups of MCG(S1,1), for S1,1 a punctured torus. The action of a mapping class on X(S1,1, SU(2)) induces a continuous deformation of discrete Hamiltonian dynamical systems on the 2-sphere S2. We discuss some of the dynamical phenomena associated to this action. We then present a method for extending this result to the actions of certain pseudo-Anosov classes on the moduli spaces of closed surfaces of genus g > 1. Int. J. Pure Appl. Math 9 (2003), 89-97.