Quasi-factors and relative entropy for infinite-measure-preserving transformations

Abstract

We extend the definition of quasi-factors for infinite-measure-preserving transformations. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. This extends (and is based upon) results of Glasner, Thouvenot and Weiss [6,7]. Following and extending Glasner and Weiss [8], we also prove that any conservative measure-preserving system with positive entropy in the sense of Danilenko and Rudolph [3] admits any probability-preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented.

Ergodic Poisson Splittings

“…(Here, N is the random variable defined by the identity on X, and in the case where µ is a probability measure, m is in fact concentrated on the subset of probability measures on X.) Meyerovitch in [13] extended this definition to the case where µ is infinite (but m is still a probability measure). Thus…”