Image sets in measurable dynamics

TL;DR

This paper develops a consistent framework for working with image sets in non-invertible measurable dynamics, enabling rigorous analysis of ergodic properties and correcting misconceptions in classical results.

Contribution

It introduces a new approach to define and analyze image sets in non-invertible measure-preserving maps, improving theoretical understanding.

Findings

Provides a consistent definition of image sets in non-invertible dynamics

Enables rigorous proofs of recurrence, ergodicity, and exactness

Eliminates unnecessary assumptions on measurability in classical theorems

Abstract

While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of measure-preserving) maps. This concept is illustrated in the context of basic ergodic properties like recurrence, ergodicity, exactness and existence of generators. It allows us to turn various suasive but logically false statements about set-theoretic images into actual theorems, and to eliminate extra assumptions on the measurability of images from some classical results.