Anosov mapping class actions on the SU(2)-representation variety of a punctured torus, Ergod

Abstract. Recently, Goldman [2] proved that the mapping class group of a compact surface S, MCG(S), acts ergodically on each symplectic stratum of the Poisson moduli space of flat SU(2)-bundles over S, X(S, SU(2)). We show that this property does not extend to that of cyclic subgroups of MCG(S), for S a punctured torus. The symplectic leaves of X(T 2 − pt., SU(2)) are topologically copies of the 2-sphere S2, and we view mapping class actions as a continuous family of discrete Hamiltonian dynamical systems on S2. These deformations limit to finite rotations on the degenerate leaf corresponding to −Id. boundary holonomy. Standard KAM techniques establish that the action is not ergodic on the leaves in a neighborhood of this degenerate leaf