ANOSOV MAPPING CLASS-ACTIONS ON THE SU(2)-REPRESENTATION VARIETY OF APUNCTURED TORUS

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 sho w that this property does not extend to that of cyclic subgroups of MC G(S), for S a punctured torus. The symplectic leaves of X(T-2 - pt., S U(2)) are topologically copies of the 2-sphere S-2, and we view mappin g class actions as a continuous family of discrete Hamiltonian dynamic al systems on S-2. These deformations limit to finite rotations on the degenerate leaf corresponding to - Id. boundary holonomy. Standard KA M techniques establish that the action is not ergodic on the leaves in a neighborhood of this degenerate leaf.