On the uniqueness of affine IETs semi-conjugated to IETs

TL;DR

This paper proves the uniqueness of affine interval exchange transformations semi-conjugated to a given IET for almost every irreducible case, addressing a question in the dynamics of these transformations.

Contribution

It establishes the existence and uniqueness of AIETs with prescribed log-slope vectors semi-conjugated to typical IETs, advancing understanding of their structure.

Findings

Unique AIETs exist for almost every irreducible IET and given log-slope vector.

Provides partial answer to a question by Marmi, Moussa, and Yoccoz.

Connects the dynamics of AIETs with the Kontsevich-Zorich cocycle.

Abstract

We prove that for almost every irreducible interval exchange transformation $T$ and for any vector $\omega$ in its associated central-stable space (with respect to the Kontsevich-Zorich cocycle) there exists a unique AIET, up to normalization of its domain, semi-conjugated to $T$ and whose log-slope vector equals $\omega$. This provides a partial answer to a question raised by S. Marmi, P. Moussa, and J.-C. Yoccoz.