Affine IETs with a singular conjugacy to an IET

TL;DR

This paper constructs affine interval exchange transformations (AIETs) that are topologically conjugate to standard IETs via a singular conjugacy, revealing new dynamical behaviors and measure-theoretic properties.

Contribution

It introduces AIETs with singular conjugacies to IETs, expanding understanding of conjugacy types and invariant measures in interval exchange dynamics.

Findings

Existence of AIETs topologically conjugate to IETs via singular conjugacies.

For almost every IET, certain AIETs are topologically conjugate with singular conjugacies.

The conjugacy's nature depends on the log-slopes vector's relation to the stable space.

Abstract

We produce affine interval exchange transformations (AIETs) which are topologically conjugated to (standard) interval exchange maps (IETs) via a singular conjugacy, i.e. a diffeomorphism $h$ of $[0,1]$ which is $C^0$ but not $C^1$ and such that the pull-back of the Lebesgue measure is a singular invariant measure for the AIET. In particular, we show that for almost every IET $T_0$ of at least two intervals and any vector $w$ belonging to the central-stable space $E_{cs}(T_0)$ for the Rauzy-Veech renormalization, any AIET T with log-slopes given by $w$ and semi-conjugated to $T_0$ is topologically conjugated to $T$. If in addition, if $w$ does not belong to $E_s(T_0)$, the conjugacy between $T$ and $T_0$ is singular.