On the Superadditive Pressure for 1-Typical, One-Step, Matrix-Cocycle Potentials

Let \((\Sigma _T,\sigma )\) $({\mathrm{\Sigma }}_T,\sigma )$ be a subshift of finite type with primitive adjacency matrix \(T\) $T$ , \(\psi :\Sigma _T \rightarrow \mathbb {R}\) $\psi :{\mathrm{\Sigma }}_T\to \mathbb{R}$ a Hölder continuous potential, and \(\mathcal {A}:\Sigma _T \rightarrow \textrm{GL}_d(\mathbb {R})\) $\mathcal{A}:{\mathrm{\Sigma }}_T\to {\text{GL}}_d(\mathbb{R})$ a 1-typical, one-step cocycle. For \(t \in \mathbb {R}\) $t\in \mathbb{R}$ consider the sequences of potentials \(\Phi _t=(\varphi _{t,n})_{n \in \mathbb {N}}\) ${\mathrm{\Phi }}_t={({\phi }_{t,n})}_{n\in \mathbb{N}}$ defined by \(\begin{aligned}\varphi _{t,n}(x):=S_n \psi (x) + t\log \Vert \mathcal {A}^n(x)\Vert , \, \forall n \in \mathbb {N}.\end{aligned}\) $$\begin{array}{c}\hfill {\phi }_{t,n}(x):=S_n\psi (x)+tlog\Vert {\mathcal{A}}^n(x)\Vert ,\forall n\in \mathbb{N}.\end{array}$$ Using the family of transfer operators defined in this setting by Park and Piraino, for all \(t<0\) $t<0$ sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials \(\Phi _t\) ${\mathrm{\Phi }}_t$ . This extends the results of the well-understood subadditive case where \(t \ge 0\) $t\ge 0$ . Prior to this, Gibbs-type measures were only known to exist for \(t<0\) $t<0$ 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 \(t \mapsto P_{\textrm{top}}(\Phi _t,\sigma )\) $t\mapsto P_{\text{top}}({\mathrm{\Phi }}_t,\sigma )$ is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.