On the superadditive pressure for 1-typical, one-step, matrix-cocycle potentials

TL;DR

This paper extends the existence of Gibbs-type measures to superadditive potentials associated with 1-typical matrix cocycles for negative parameters near zero, and proves the analyticity of the topological pressure function.

Contribution

It generalizes the existence of Gibbs-type measures to superadditive potentials for 1-typical cocycles, beyond previously known cases, and analyzes the pressure function's analyticity.

Findings

Gibbs-type measures exist for $t<0$ close to 0 in this setting.

The topological pressure function is analytic near $t=0$.

The derivative of the pressure function equals the Lyapunov exponents.

Abstract

Let $(\Sigma_T,\sigma)$ be a subshift of finite type with primitive adjacency matrix $T$, $\psi:\Sigma_T \rightarrow \mathbb{R}$ a H\"older continuous potential, and $\mathcal{A}:\Sigma_T \rightarrow \mathrm{GL}_d(\mathbb{R})$ a 1-typical, one-step cocycle. For $t \in \mathbb{R}$ consider the sequences of potentials $\Phi_t=(\varphi_{t,n})_{n \in \mathbb{N}}$ defined by $$\varphi_{t,n}(x):=S_n \psi(x) + t\log \|\mathcal{A}^n(x)\|, \: \forall n \in \mathbb{N}.$$ Using the family of transfer operators defined in this setting by Park and Piraino, for all $t<0$ sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials $\Phi_t$. This extends the results of the well-understood subadditive case where $t \geq 0$. Prior to this, Gibbs-type measures were only known to exist for $t<0$ in the conformal, the reducible, the positive, or the dominated, planar settings, in which case they are Gibbs measures in the classical sense. We further prove that the topological pressure function $t \mapsto P_{\mathrm{top}}(\Phi_t,\sigma)$ is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.