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 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: for any finite cover a of A. Finally, we note that if a>fi, then N (a) ^ A (/3), and h (a, T) = ^ (j3, T). The topological entropy of T is defined as h (T) = sup h (a, T), where the supremum is taken over all finite open covers of A. It is easily seen that if 0 is a homomorphism from (A, P) onto (F, S) and if a is a finite cover of F, then h (<p-l (a), T) =h (a, S). It follows that h (S) £h (T).