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

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 A over topologically mixing subshifts of finite type admit a unique Gibbs equilibrium state μₜ associated with the non-additive family of potentials log \|Aⁿ\| ∈ N, for a range of parameters t ∈ (-t_*, +∞), where t_* > 0. Furthermore, these equilibrium states are $ψ$-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 C¹-open sets of $α$-fiber-bunched hyperbolic repellers.