Markov intertwining operators, joinings, and asymptotic properties of dynamical systems

TL;DR

This paper reviews the development of the theory of Markov intertwining operators and joinings in measure-preserving dynamical systems, highlighting open problems and their implications for asymptotic properties and multiple mixing.

Contribution

It provides a comprehensive overview of the author's work from 1991 to 2001 on joinings, Markov operators, and their applications to dynamical systems, emphasizing unresolved questions.

Findings

Many results on joinings have remained unchanged over time.

Open questions about minimal self-joinings and pairwise independence remain.

The work highlights the complexity and open problems in the theory of dynamical systems.

Abstract

This text is written based on the author's publications during the period from 1991 to 2001. The work is devoted to the theory of Markov intertwining operators and joinings of measure-preserving group actions, as well as to their applications to study asymptotic properties of dynamical systems. Special attention is paid to Rokhlin's problems on multiple mixing and multiple spectrum. The development of these topics over the past twenty years has not been discussed. In fact many results on joinings have frozen in time, many questions have remained open without losing their relevance, but probably have ceased to excite interest due to difficulties. For example, it is not known whether the minimal self-joinings of order 2 imply all orders? Is there a non-trivial pairwise independent joining for a weakly mixing system of zero entropy? What can be said about such joinings for transformations with small local rank? These questions are ripe for a long time, and the author reminds the reader about them, combining his story with numerous partial and related results.