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

Abstract

By studying the weak closure of multidimensional off-diagonal self-joinings, we provide a sufficient condition 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 \(\mathbb{T}^2\), so-called von Neumann 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 non-trivial topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.