Every transformation is disjoint from almost every IET

TL;DR

This paper proves that almost every transformation is disjoint from almost every interval exchange transformation, leading to new insights into ergodic properties and rigidity sequences in dynamical systems.

Contribution

It establishes that every transformation is disjoint from almost every IET, answering a longstanding question and strengthening previous results on rigidity sequences.

Findings

Almost every pair of IETs is disjoint.

The product of almost every pair of IETs is uniquely ergodic.

Any density 1 sequence contains a rigidity sequence for almost every IET.

Abstract

We show that every transformation is disjoint from almost every interval exchange transformation (IET), answering a question of Bufetov. In particular, we prove that almost every pair of IETs is disjoint. It follows that the product of almost every pair is uniquely ergodic. A key step in the proof is showing that any sequence of density 1 contains a rigidity sequence for almost every IET, strengthening a result of Veech.