Variational principles for topological entropies of subsets

Let (X, T ) be a topological dynamical system. We define the measure-theoretical 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 hB top(T ,K) = sup hμ(T ): μ ∈ M(X), μ(K) = 1 , hP top(T ,K) = sup hμ(T ): μ ∈ M(X), μ(K) = 1 , where hB top(T ,K) denotes the Bowen topological entropy of K, and hP 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. © 2012 Elsevier Inc. All rights reserved