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

Let (Sigma(T), sigma)be a subshift of finite type with primitive adjacency matrix T, psi: Sigma(T )-> R a Holder continuous potential, and A : Sigma(T )-> GL(d)(R)a 1-typical, one-step cocycle. For t is an element of R consider the sequences of potentials Phi(t) = (phi(t,n))(n is an element of N )defined by phi(t,n)(x) := S-n psi(x) + t log & Vert;A(n)(x)& Vert;, for all n is an element of N. Using the family of transfer operators defined in this setting by Park and Piraino, for t < 0 all 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 >= 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 (sic) P-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.