Entropy and Dyadic Equivalence of Random Walks on a Random Scenery

Abstract

For any 1-1 measure-preserving map T of a probability space, consider the [T, T-1] endomorphism and the corresponding decreasing sequence of σ-algebras. We demonstrate that if the decreasing sequence of σ-algebras generated by [T, T-1] and [S, S-1] are isomorphic, then T and S must have equal entropies. As a consequence, if the [T, T-1] endomorphism is isomorphic to the [S, S-1] endomorphism, then the entropy of T is equal to the entropy of S. Central to this is a relationship between Feldman's f metric (1976, Israel J. Math.24, 16-38) and Vershik's v metric (1970, Dokl. Akad. Nauk SSSR193, 748-751). © 2000 Academic Press.