Multiple mixing, 75 years of Rokhlin's problem

TL;DR

This paper reviews the history and unresolved questions surrounding Rokhlin's problem of whether mixing implies k-fold mixing in measure-preserving transformations, highlighting 75 years of ongoing research.

Contribution

It provides a historical overview and discusses open problems related to Rokhlin's invariants and the implications of mixing properties in ergodic theory.

Findings

Summary of key results over 75 years

Discussion of open problems in mixing theory

Clarification of the relationship between mixing and k-fold mixing

Abstract

In 1949, V.A. Rokhlin introduced new invariants for measure-preserving transformations, called k-fold mixing. Does mixing imply k-fold mixing? -- this problem remains open. We recall shortly some results and discuss related problems.