Ergodic Decomposition

TL;DR

This paper investigates how non-ergodic systems can be decomposed into ergodic components and explores property inheritance among these components, utilizing the Rokhlin Disintegration Theorem.

Contribution

It provides a measurable decomposition framework for non-ergodic systems into ergodic parts, extending the understanding of their structure and properties.

Findings

Non-ergodic systems can be decomposed into ergodic components.

Decomposition preserves certain properties across components.

The approach uses Rokhlin Disintegration Theorem.

Abstract

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones? Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.