The pressure function for products of non-negative matrices

Abstract

Let (ΣA, σ) be a subshift of finite type and let M(x) be a continuous function on ΣA taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M1, ⋯, Mm. The pressure function reduces to P(q) := limn→∞ 1/n log ΣJ∈ΣA,n ∥MJ∥q. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.