Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions. Let(X, G) be a system, where X is a compact metric space and G is a finite family of continuous maps on X. Given a continuous function f on X, we define Pesin–Pitskel topological pressure PG(Z, f)for any subset Z ■ X and measure-theoretical pressure Pμ,G(X, f) for any μ∈ M(X), where M(X)denotes the set of all Borel probability measures on X. For any non-empty compact subset Z of X, we show that PG(Z, f) = sup{Pμ,G(X, f) : μ∈ M(X), μ(Z) = 1}.