Quasi-factors for infinite-measure preserving transformations

TL;DR

This paper investigates quasi-factors in infinite-measure preserving systems, demonstrating entropy properties and extending existing results, with applications to Poisson-suspensions and measure-preserving systems.

Contribution

It extends Glasner's and Weiss's results on entropy and quasi-factors to infinite-measure systems and establishes new connections with Poisson-suspensions.

Findings

Existence of zero-entropy systems with positive-entropy quasi-factors.

Relative zero-entropy is preserved under quasi-factors.

Any positive-entropy system admits any positive-entropy probability system as a factor.

Abstract

This paper is a study of Glasner's definition of quasi-factors in the setting of infinite-measure preserving system. 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. Following and extending Glasner and Weiss, we also prove that any conservative measure preserving system with positive entropy in the sense of Danilenko and Rudolph admits any probability preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented.