A note on equivalences between various mixing scales

TL;DR

This paper identifies a defect in the existing geometric mixing scale, proposes an improved strong geometric mixing scale, and proves their equivalence with the functional mixing scale in terms of weak convergence.

Contribution

It introduces the strong geometric mixing scale and establishes its equivalence with other mixing scales, improving the understanding of mixing measures.

Findings

Identification of a defect in the geometric mixing scale

Introduction of the strong geometric mixing scale

Proof of equivalence among mixing scales

Abstract

In this note, we provide with a simple example to show a defect in the definition of the geometric mixing scale, and then introduce an improved scale, called as the strong geometric mixing scale. The main theorem in this note is the equivalence between geometric mixing scale, strong geometric mixing scale and functional mixing scale, in the sense of weak convergences.