Variational Principles for Topological Entropies of Subsets

Let (X,T ) be a topological dynamical system. We define the measuretheoretical lower and upper entropies hμ(T ), hμ(T ) for any μ ∈ M(X), where M(X) denotes the collection of all Borel probability measures on X. For any non-empty compact subset K of X, we show that htop(T,K) = sup{hμ(T ) : μ ∈M(X), μ(K) = 1}, htop(T,K) = sup{hμ(T ) : μ ∈M(X), μ(K) = 1}, where htop(T,K) denotes Bowen’s topological entropy of K, and h P top(T,K) the packing topological entropy of K. Furthermore, when 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.