Random dynamics on character varieties

Character varieties are spaces which parametrise the deformations of representations of the fundamental group of a surface into a Lie group.

The symmetry group of the surface, the mapping class group, naturally act on these space, and this action has interesting dynamical properties.

In this talk I will consider a very particular case: the relative character varieties of the once-punctured torus with values in SL(2,C).

In this case, the spaces are affine algebraic surfaces given by an explicit equation, the famous family of Markov surfaces, and the action of the mapping class group is given by explicit polynomial transformations.

I will describe the stationary measures for this action: these are measures that describe the statistical distribution of the orbits of the group action.