Variational principles for topological entropies of subsets

Let (X,T)(X,T) be a topological dynamical system. We define the measure-theoretical lower and upper entropies h̲μ(T), h¯μ(T) for any μ∈M(X)μ∈M(X), where M(X)M(X) denotes the collection of all Borel probability measures on X. For any non-empty compact subset K of X, we show thathtopB(T,K)=sup{h̲μ(T):μ∈M(X),μ(K)=1},htopP(T,K)=sup{h¯μ(T):μ∈M(X),μ(K)=1}, where htopB(T,K) denotes the Bowen topological entropy of K  , and htopP(T,K) the packing topological entropy of K  . Furthermore, when htop(T)<∞htop(T)<∞, the first equality remains valid when K is replaced by any analytic subset of X. The second equality always extends to any analytic subset of X.