Homeomorphic Bernoulli Trial Measures and Ergodic Theory

We survey the some of the main results, ideas and conjectures concerning two problems and their connections. The first problem concerns determining when two Bernoulli trial measures are homeomorphic to each other, i.e. when one is the image measure of the other via a homeomorphism of the Cantor space. The second problem concerns the following. Given a positive integer k characterize those Bernoulli trial measures m for which there is a homeomorphism preserving m and which has exactly k ergodic measures with m being one of them. We will also discuss some of the history leading to these problems. A measure on a Cantor space is taken to mean a probability measure on the Borel subsets of a space homeomorphic to Cantor space which gives non-empty open sets positive measure and gives points measure zero. That is, all measures are assumed to be full, non-atomic probability measures. In this paper, C is a Cantor space means it is a topological space homeomorphic to {0, 1}N provided with the product topology where {0, 1} has the discrete topology. Some particular representations of the Cantor space will hold our attention. We are interested in two problems. One is to determine when two such measures or two measures of a given type, μ on a Cantor space X, and ν on a Cantor space Y are “homeomorphic,” i.e., when is there a homeomorphism h of X onto Y such that μ = ν ◦ h−1? The second problem concerns the ergodic properties of such measures. When is there a homeomorphism h preserving μ for which μ is the unique ergodic measure or when is μ one of exactly k ergodic measures for h? Besides presenting some new results and recounting some previously obtained, we will indicate some of the origins of these problems particularly as they concern Bernoulli trial measures on {0, 1}N or {0, 1}Z. 2000 Mathematics Subject Classification. Primary 37B05; Secondary 28D05, 28C15.