Non-reversibility and self-joinings of higher orders for ergodic flows

TL;DR

This paper investigates the non-reversibility of ergodic flows by examining self-joinings, providing criteria for non-isomorphism with inverses, and applies these to special flows over irrational rotations, including analytic and polynomial cases.

Contribution

It introduces a new criterion based on self-joinings for determining flow non-reversibility and applies it to a broad class of special flows over irrational rotations.

Findings

Identified classes of non-reversible flows, including von Neumann's flows.

Provided a topological analysis of self-similarity in special flows.

Constructed examples of flows without topological self-similarities.

Abstract

By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on the two torus, so called von Neumann's flows and some special flows with piecewise polynomial roof functions.. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.