Self joinings of rigid rank one transformations arise as strong operator topology limits of convex combinations of powers

TL;DR

The paper demonstrates that for a generic set of measure-preserving transformations, self-joinings can be approximated as strong operator topology limits of convex combinations of their powers, extending previous results.

Contribution

It generalizes earlier work by showing that self-joinings of rigid rank one transformations are limits of convex combinations of powers in the strong operator topology.

Findings

Self-joinings are strong operator topology limits of convex combinations of powers.

This holds for a residual set of transformations in the weak topology.

The result extends the understanding of the structure of self-joinings in ergodic theory.

Abstract

This is a straightforward generalization Section 2 of arXiv:1805.11167. It shows that for a residual set of transformations in the space of measure preserving transformations, with the weak topology, any self-joining defines a Markov operator that is a strong operator topology limit of convex combinations of powers of the unitary operator given by the transformation.