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

Jialun Li, Wenyu Pan, Disheng Xu

Published 2023-11-17Version 1

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. This allows us to study the limit sets of Anosov representations and Rauzy gasket. We show that their Hausdorff dimensions are equal to the affinity exponents. In particular, there is a dimension gap in the Barbot component.