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

Abstract

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