Entropy andσ-algebra equivalence of certain random walks on random sceneries

Abstract

LetX=X 0,X 1,…be a stationary sequence of random variables defining a sequence space Σ with shift mapσ and let (T t, Ω) be an ergodic flow. Then the endomorphismT X(x, ω)=(σ(x),T x 0(ω)) is known as a random walk on a random scenery. In [4], Heicklen, Hoffman and Rudolph proved that within the class of random walks on random sceneries whereX is an i.i.d. sequence of Bernoulli-(1/2, 1/2) random variables, the entropy ofT t is an isomorphism invariant. This paper extends this result to a more general class of random walks, which proves the existence of an uncountable family of smooth maps on a single manifold, no two of which are measurably isomorphic.