Affine interval exchange maps with a singular conjugacy to an IET

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 C0 but not C1 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 T0 of d ≥ 2 intervals and any vector ω belonging to the central-stable space Ecs(T0), for the Rauzy-Veech renormalization, any AIET T with log-slopes given by ω and semi-conjugated to T0 is topologically conjugated to T. In addition, if ω ∉ Es (T0), the conjugacy between T and T0 is singular.