A powerful tool in the study of geodesic flow of a closed hyperbolic surface is the connection to Markov shifts on a finite alphabet. When the surface has cusps, a countable rather than finite alphabet is needed for the dynamics to have enough regularity to recover, for instance, counting results. However these results are independently known for geodesic flow in the setting of Riemannian negative curvature. In this talk, we present counting and equidistribution results for a family of geometric structures on surfaces called Hitchin representations. These Hitchin representations generalize hyperbolic geometry to a non-Riemannian setting. This results arises as an application of an equidistribution theorem for the class of topologically mixing countable Markov shifts with Sarig’s BIP property.

This is based on joint work with Dick Canary, Nyima Kao, and Giuseppe Martone.