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).