On the dimension of limit sets on $\mathbb{P}(\mathbb{R}^3)$ via stationary measures: variational principles and applications

Yuxiang Jiao, Jialun Li, Wenyu Pan, Disheng Xu

Published 2023-11-17Version 1

In this article, we establish the variational principle of the affinity exponent of Borel Anosov representations. We also establish such a principle of the Rauzy gasket. In Li-Pan-Xu, they obtain a dimension formula of the stationary measures on $\mathbb{P}(\mathbb{R}^3)$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\mathrm{SL}_3(\mathbb{R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings.