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

TL;DR

This paper establishes the existence and uniqueness of Gibbs equilibrium states for fiber-bunched matrix cocycles over hyperbolic systems, demonstrating their mixing properties and confirming a conjecture for hyperbolic repellers.

Contribution

It proves the uniqueness and mixing properties of equilibrium states for fiber-bunched cocycles, extending thermodynamic formalism to new classes of hyperbolic systems.

Findings

Unique Gibbs equilibrium states exist for certain fiber-bunched cocycles.

These states are $ ext{psi}$-mixing and weak Bernoulli.

Results confirm a conjecture for $C^1$-open sets of hyperbolic repellers.

Abstract

We contribute to the thermodynamic formalism of H\"older 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 $\{t \log \|\mathcal{A}^n\|\}_{n \in \mathbb{N}}$, for a range of parameters $t \in (-t_*, +\infty)$, where $t_* > 0$. 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.