Variational principles for topological entropies of subsets

Let (X, T) be a topological dynamical system. We define the measure-theoretical lower and upper entropies (h) under bar (mu)(T), (h) over bar (mu)(T) for any mu is an element of M(X), where M(X) denotes the collection of all Bore probability measures on X. For any non-empty compact subset K of X, we show that h(top)(B)(T, K) = sup{(h) under bar (mu)(T); mu is an element of M(X), mu(K) = 1}, h(top)(P)(T, K) = sup{(h) over bar (mu)(T); mu is an element of M(X), mu(K) = 1}, where h(top)(B)(T, K) denotes the Bowen topological entropy of K, and h(top)(P)(T, K) the packing topological entropy of K. Furthermore, when h(top)(T) < infinity, 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. (C) 2012 Elsevier Inc. All rights reserved.