Universal Weighted Averaging for Ergodic Flows

TL;DR

This work extends Kozlov-Treshchev by studying weighted ergodic averages for ergodic flows with both absolutely continuous and singular weights and by introducing universality notions for sequences of measures on the real line. It shows that universality is preserved under convolution roots via Theorem 1.2 and Theorem 2.1, linking the behavior of ν and its convolution powers ν^{*n}. The paper also develops the concept of almost mixing, proving that mixing implies almost mixing while demonstrating that non-mixing flows can still be almost mixing (Theorems 3.1 and 3.2), and finally establishes that flows with rigid factors are not almost mixing (Theorem 4.1). Overall, the results broaden the class of weights for which ergodic averages converge and clarify the relationships among universality, almost mixing, and rigidity, with implications for flows such as toral windings.

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.