Affine IETs with a singular conjugacy to an IET

Frank Trujillo; Corinna Ulcigrai

paper

Affine IETs with a singular conjugacy to an IET

arXiv:2210.14202

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.