The Hitchin component is a connected component of the representation variety of the fundamental group of a hyperbolic surface \(S\) into the Lie group \(PSL(d,\mathbb{R})\).
 When \(d=2\), the Hitchin component coincides with the space of hyperbolic structures on \(S\), an important object in many fields of mathematics. When \(d>2\), Hitchin representations retain significant algebraic, dynamical, and geometric properties. For example, Labourie connected Hitchin representations to the theory of Anosov flows to show that they are discrete and faithful.

In this talk, we will focus on marked length spectra of Hitchin representations. These are functions that associate to every closed curve on the surface a positive number which generalizes the notion of hyperbolic length of a closed geodesic. We will see how the Thermodynamic Formalism of (countable) Markov shifts can be used to understand the correlation number for these length spectra, and establish orbit counting and equidistribution results for the length spectra of cusped Hitchin representations.

This talk is based on joint work with Dai and with Bray, Canary and Kao.