Universal Weighted Averaging for Ergodic Flows

TL;DR

This paper introduces a new framework for analyzing ergodic flows using weighted averages, extending existing results, and proposing the concept of almost mixing, with examples and proofs regarding mixing properties.

Contribution

It develops homothetic and weighted averaging methods for flows, strengthening prior results and introducing the concept of almost mixing in ergodic theory.

Findings

Strengthens Kozlov-Treshchev's results on nonuniform averages

Proposes the concept of almost mixing for flows

Provides examples of non-mixing almost mixing flows

Abstract

This paper studies homothetic and more general weighted averages for flows. Absolutely continuous convolutions of singular weights are considered, thereby strengthening Kozlov-Treshchev's result on nonuniform averages for ergodic flows. The concept of almost mixing, formulated in terms of homothetic weighted average convergences, is proposed. An example of a non-mixing almost mixing flow is given. It is proven that rigid flows are not almost mixing.