Unsolved problems on joinings, multiple mixing, spectrum, and rank

TL;DR

This paper reviews open problems in measure-preserving transformations related to mixing, spectrum, rank, and self-joinings, discussing known results and hypothetical examples that could resolve longstanding questions.

Contribution

It highlights the potential existence of a special automorphism with unique spectral and mixing properties that could answer several open problems in ergodic theory.

Findings

Discussion of open problems and known results in ergodic theory.

Hypothetical example of an automorphism with specific spectral properties.

No current evidence confirming the existence of such an automorphism.

Abstract

The note is devoted to multiple mixing, spectrum, rank and self-joinings of measure-preserving transformations. We recall famous open problems, discuss related questions and some known results. A hypothetical example of an automorphism of the class $MSJ(2)\setminus MSJ(3)$ has Lebesgue spectrum, infinite rank and does not have multiple mixing. If its spectrum is simple, then we get a solution to the problems of Banach, Rokhlin and del Junco-Rudolph. The existence of such an example, of course, seems unlikely, but any facts confirming the impossibility of this amazing situation have not yet been discovered. They not found even under the condition that its local rank is positive, which ensures the finite spectral multiplicity.