Entropy and dyadic equivalence of random walks on a random scenery

For any 1-1 measure-preserving map T of a probability space. consider the [ T, T-1] endomorphism and the corresponding decreasing sequence of sigma -al gebras. Wt: demonstrate that if the decreasing sequence of sigma -algebras generated bq. [T, T-1] and [S, S-1] are isomorphic. then T and S must hale equal entropies. As a consequence. if the [T, T-1] endomorphism is isomorph ic to the [S, S-1] endomorphism. then the entropy of T is equal to the entr opy of S. Central to this is a relationship between Feldman's (f) over bar metric (1976. Israel J. Math. 24. 16-38) and Vershik's r metric (1970. Dokl . Akad. Nauk SSSR 193. 748 751). (C) 2000 Academic Press.