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.
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