Anosov representations with Lipschitz limit set

TL;DR

This paper investigates a class of Anosov representations with Lipschitz regularity in their limit sets, introducing a new functional to analyze orbit growth and applying it to rigidity results in higher rank symmetric spaces.

Contribution

It introduces the unstable Jacobian functional for Anosov representations with Lipschitz limit sets and demonstrates its relevance to higher rank representations and rigidity phenomena.

Findings

Many higher rank representations, including Θ-positive ones, have Lipschitz limit sets.

The unstable Jacobian's orbit growth rate is integral for this class of representations.

Applications include new rigidity results related to orbit growth in symmetric spaces.

Abstract

We study Anosov representations whose limit set has intermediate regularity, namely is a Lipschitz submanifold of a flag manifold. We introduce an explicit linear functional, the unstable Jacobian, whose orbit growth rate is integral on this class of representations. We prove that many interesting higher rank representations, including $\Theta$-positive representations, belong to this class, and establish several applications to rigidity results on the orbit growth rate in the symmetric space.