Topological entropy bounds measure-theoretic entropy

Let T be a homeomorphism from a compact space A onto itself and let p be a P-invariant probability measure on the Borel sets of A. It was conjectured in [l] that the measure-theoretic entropy of P with respect to u is less than or equal to the topological entropy of P. The purpose of this paper is to show, under the assumption that A is metric, that the inequality holds when T is assumed only to be a continuous map from A into itself. We shall first prove the inequality under the assumption that A is a closed subset of the Hilbert cube which is invaraint under a certain type of shift operator, and P is the restriction of the operator to A. The generalization will be obtained by considering representations of T as such shifts. By a flow we mean a pair (A, P), where A is a compact metric space and T is a continuous map from A into itself. Throughout the paper, (A, P) and (F, S) will denote arbitrary flows. A continuous map <p: A—> F will be called a homomorphism from (A, T) into (F, S) iid)oT = Sod>. Ii a is any finite cover of A, we let N(a) be the number of members in a subcover of a of minimal cardinality. As in [l], we write aVj8= { UCW: P£a, F£|3} and we write a>fi to mean that a is a refinement of fi, though this is contrary to the notation of many authors. As in [l], it follows from the fact that N(a\/fi) ^N(a)-N(fi), that the limit exists in the following definition: