Variational principle for topological pressures on subsets

This paper studies the relations between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measures, which extends Feng and Huang's recent result on entropies [13] for pressures. More precisely, this paper defines the measure theoretic pressure Pμ(T,f) for any Borel probability measure, and shows that PB(T,f,K)=sup⁡{Pμ(T,f):μ∈M(X),μ(K)=1}, where M(X) is the space of all Borel probability measures, K⊆X is a non-empty compact subset and PB(T,f,K) is the Pesin-Pitskel topological pressure on K. Furthermore, if Z⊆X is an analytic subset, then PB(T,f,Z)=sup⁡{PB(T,f,K):K⊆Zis compact}. This paper also shows that Pesin-Pitskel topological pressure can be determined by the measure theoretic pressure.