We first describe the geometric theory of Hitchin representations of geometrically finite Fuchsian groups, which generalizes the work of Labourie and Fock-Goncharov on Hitchin representations of closed surface groups. Geodesic flows of geometrically finite Fuchsian groups are modelled by countable Markov shifts and we develop counting and equidistribution results for well-behaved potentials in the spirit of Lalley's results for finite Markov shifts which apply in the setting of cusped Hitchin representations. This work is part of a program to develop a theory of the augmented Hitchin component which parallels the classical theory of augmented Teichmuller space. (Joint work with Harry Bray, Nyima Kao, Giuseppe Martone, Tengren Zhang and Andy Zimmer) Speaker(s): Richard Canary (U Michigan)