Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

TL;DR

This paper establishes conditions under which entropy gaps at infinity lead to counting and equidistribution results, applying these to cusped Hitchin and Anosov representations of geometrically finite Fuchsian groups.

Contribution

It introduces a framework linking entropy gaps at infinity with counting and equidistribution, extending these results to specific classes of geometric group representations.

Findings

Entropy gap at infinity implies counting and equidistribution results.

Application to cusped Hitchin and Anosov representations.

Use of renewal theorem for these applications.

Abstract

We show that if an eventually positive, non-arithmetic, locally H\"older continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseb\"ohmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.