On separated families of Anosov representations

TL;DR

This paper introduces new notions of separation for families of Anosov representations, linking their asymptotic behavior to spectral data and deriving bounds on Thurston's metric, with applications to convex projective structures.

Contribution

It provides novel separation concepts for Anosov representations and connects their divergence to spectral invariants, extending previous examples in convex projective geometry.

Findings

Critical exponent asymptotic to a combinatorial spectral invariant

Derived bounds on Thurston's asymmetric metric

Generalized McMullen's example of convex projective structures

Abstract

We introduce different notions of separation for families of Anosov representations. We show that, along a diverging sequence of such families, the critical exponent is asymptotic to a combinatorial invariant computable from the spectral data of a finite graph. Our method allows us to derive bounds on the Thurston asymmetric metric. As an application, we study specific degenerations of convex projective structures on a pair of pants, generalizing an example of McMullen.