Simple models of random walks in random sceneries give rise to examples of probability-preserving dynamical systems with interesting ergodic theoretic properties. In particular, they provide examples (the only known `natural’ examples) of systems with the K property that are not measure theoretically isomorphic to Bernoulli shifts, as conjectured by Ornstein, Adler and Weiss and then proved in a famous analysis by Kalikow. However, while the construction gives a large family of these examples, it remained unknown whether they are all really distinct up to measure-theoretic isomorphism.
In this talk I will sketch a new invariant for probability-preserving systems which can be used to recover the entropy rate of the scenery as an isomorphism-invariant of these models. This implies that they form continuum-many distinct examples. In general, this new invariant of systems is defined by viewing the sequence of finite-dimensional marginals of a stationary stochastic process as a sequence of probability measures on the appropriate Hamming metric spaces, and then considering asymptotic features of the metric geometry of those spaces.