Reciprocal Transformations and Their Discrete Maharam Extensions

TL;DR

This paper introduces reciprocal transformations and discrete Maharam extensions, new methods for constructing and analyzing measurable dynamical systems with specific ergodic properties, and explores their theoretical implications.

Contribution

It presents novel constructions for dynamical systems, linking reciprocal transformations with Maharam extensions, and investigates their ergodic properties and theoretical significance.

Findings

Reciprocal transformations distort measures in a controlled way.

Discrete Maharam extensions relate to infinite measure-preserving transformations.

Preliminary ergodic properties of these constructions are established.

Abstract

We introduce two abstract constructions for building new measurable dynamical systems from existing ones and study their ergodic properties. The first of these constructions, a "reciprocal transformation," produces a type of non-singular transformation where the measures of subsets are distorted in a simple way. We then introduce the "discrete Maharam extension" which associates an infinite measure-preserving transformation to each reciprocal transformations. We give some preliminary results about the ergodic theory of each of these constructions, mention ongoing work, as well as conjectures and questions for future research.