Degeneration of Hitchin representations along internal sequences

TL;DR

This paper explores the parameterization of PSL(n,R) Hitchin representations of a surface, providing bounds on curve lengths and demonstrating sequences where topological entropy diminishes to zero.

Contribution

It introduces a lower bound on curve lengths in Hitchin representations and constructs sequences with entropy tending to zero, advancing understanding of Hitchin component degenerations.

Findings

Established a lower bound on curve lengths in Hitchin representations.

Constructed sequences in the Hitchin component with entropy approaching zero.

Linked parameterization to entropy behavior in degenerating sequences.

Abstract

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" of any closed curve on S in terms of the parameters describing this representation. We then show that this lower bound is good enough to produce large families of sequences in the Hitchin component along which the topological entropy converges to 0.