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

This paper investigates the (semi)group action of $\mathrm{SL}_3(\mathbb{R})$ on $\mathbb{P}(\mathbb{R}^3)$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We study the Hausdorff dimension of a dynamically defined limit set in $\mathbb{P}(\mathbb{R}^3)$ and generalize the classical Patterson-Sullivan formula using the approach of stationary measures. The two main examples are Anosov representations in $\mathrm{SL}_3(\mathbb{R})$ and the Rauzy gasket. 1. For Anosov representations in $\mathrm{SL}_3(\mathbb{R})$, we establish a sharp lower bound for the dimension of their limit sets in $\mathbb{P}(\mathbb{R}^3)$. Coupled with the upper bound in Pozzetti-Sambarino-Wienhard, it shows that their Hausdorff dimensions equal the affinity exponents. The merit of our approach is that it works uniformly for all the components of irreducible Anosov representations in $\mathrm{SL}_3(\mathbb{R})$. As an application, it reveals a surprising dimension jump phenomenon in the Barbot component, which is a local generalization of Bowen's dimension rigidity result. 2. For the Rauzy gasket, we confirm a folklore conjecture about the Hausdorff dimension of the gasket and improve the numerical lower bound to $3/2$. These results originate from a dimension formula of stationary measures on $\mathbb{P}(\mathbb{R}^3)$. Let $\nu$ be a probability measure on $\mathrm{SL}_3(\mathbb{R})$ whose support is finite and spans a Zariski dense subgroup. Let $\mu$ be the associated stationary measure for the action on $\mathbb{P}(\mathbb{R}^3)$. Under the exponential separation condition on $\nu$, we prove that the Hausdorff dimension of $\mu$ equals its Lyapunov dimension, which extends Hochman-Solomyak and B\'{a}r\'{a}ny-Hochman-Rapaport to non-conformal and projective settings respectively.