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

Abstract: 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. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.