  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.
