Topological entropy bounds measure-theoretic entropy
L. Wayne Goodwyn · Proceedings of the American Mathematical Society · 1969
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(