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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.16694 |
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Table of Contents:
- Let $(Σ_T,σ)$ be a subshift of finite type with primitive adjacency matrix $T$, $ψ:Σ_T \rightarrow \mathbb{R}$ a Hölder continuous potential, and $\mathcal{A}:Σ_T \rightarrow \mathrm{GL}_d(\mathbb{R})$ a 1-typical, one-step cocycle. For $t \in \mathbb{R}$ consider the sequences of potentials $Φ_t=(φ_{t,n})_{n \in \mathbb{N}}$ defined by $$φ_{t,n}(x):=S_n ψ(x) + t\log \|\mathcal{A}^n(x)\|, \: \forall n \in \mathbb{N}.$$ 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 $Φ_t$. 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 $t \mapsto P_{\mathrm{top}}(Φ_t,σ)$ is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.