Ergodic averaging with and without invariant measures

TL;DR

This paper explores ergodic averaging in dynamical systems, focusing on the size of typical trajectory sets and the role of natural measures, especially when invariant measures are absent.

Contribution

It investigates ergodic averages without invariant measures and examines the connection to natural measures, extending classical ergodic theory.

Findings

Characterization of typical trajectories without invariant measures

Analysis of natural measures as limits of system images

Insights into ergodic averages in non-invariant contexts

Abstract

The classical Birkhoff ergodic theorem in its most popular version says that the time average along a single typical trajectory of a dynamical system is equal to the space average with respect to the ergodic invariant distribution. This result is one of the cornerstones of the entire ergodic theory and its numerous applications. Two questions related to this subject will be addressed: how large is the set of typical trajectories, in particular in the case when there are no invariant distributions, and how the answer is connected to properties of the so called natural measures (limits of images of "good" measures under the action of the system).