# Hausdorff dimension of limit sets for projective Anosov representations

## TL;DR

This paper investigates the relationship between critical exponents and Hausdorff dimensions of limit sets in projective Anosov representations, establishing bounds for the Hausdorff dimension based on these exponents.

## Contribution

It provides new bounds for the Hausdorff dimension of symmetric limit sets in projective Anosov representations using critical exponents related to highest weights and simple roots.

## Key findings

- Hausdorff dimension is bounded between two critical exponents.
- Established bounds connect geometric and algebraic properties of representations.
- Results apply to limit sets in projective spaces.

## Abstract

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.