Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications

Abstract

We contribute to the thermodynamic formalism of Hölder continuous fiber-bunched matrix cocycles, Anosov diffeomorphisms, and hyperbolic repellers. Specifically, we prove that $1$-typical fiber-bunched cocycles $\mathcal{A}$ over topologically mixing subshifts of finite type admit a unique Gibbs equilibrium state ${\mu }_t$ associated with the non-additive family of potentials $\{t\log \Vert {\mathcal{A}}^n\Vert {\}}_{n\in \mathbb{N}}$, for a range of parameters $t\in (-t_{\ast },+\infty )$, where $t_{\ast }>0$. Furthermore, these equilibrium states are $\psi$-mixing, therefore weak Bernoulli. In addition, these results allow us to derive consequences for the thermodynamic formalism of open sets of hyperbolic repellers and Anosov diffeomorphisms. In particular, it provides a positive answer to a conjecture posed by Gatzouras and Peres for $C^1$-open sets of $\alpha$-fiber-bunched hyperbolic repellers.