Scenery entropy as an invariant of RWRS processes

Abstract: Probabilistic models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems. A point in the state spaces consists of a walk-trajectory and a scenery, and its `motion' corresponds to shifting the time-origin. These models were proposed as natural examples of non-Bernoulli K-automorphisms by Adler, Ornstein and Weiss. This was proved in a famous analysis by Kalikow using Ornstein's Very Weak Bernoulli characterization of Bernoulli processes. Since then, various authors have generalized this construction to give other examples, including some smooth examples due to Katok and Rudolph. However, the methods used to prove non-Bernoullicity do not obviously show that these examples are distinct from one another. This paper introduces a new isomorphism-invariant of probability-preserving systems, and shows that in a large class of the above examples it essentially captures the Kolmogorov-Sinai entropy of the scenery process alone. As a result, constructions that use different scenery-entropies give continuum-many non-isomorphic examples. Conditionally on an invariance principle for certain local times, these include a continuum of distinct smooth non-Bernoulli K-automorphisms on a fixed compact manifold.