Dynamics on character varieties, random orbits

Consider the 2-dimensional real torus with one puncture; its fundamental group is the free group F on 2 generators. The character variety of F describing representations of F in SL(2,\C) modulo conjugacy is an affine space of dimension 3. The mapping class group GL(2,\Z) acts on it by automorphisms (the action is by pre-composition of representations). My goal will be to describe some dynamical properties of this action. (based on a joint work with Dupont and Martin-Baillon)