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 anyμ∈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}.

Sub-additive Unstable Topological Pressure of Diffeomorphisms

In this paper,we study unstable topological pressure for C^(1)-smooth partially hyperbolic diffeomorphisms with sub-additive potentials.Moreover,without any additional assumption,we have established the expected variational principle which connects this unstable topological pressure and the unstable measure theoretic entropy,as well as the corresponding Lyapunov exponent.