Affine IETs with a singular conjugacy to an IET

2022/10/25 by Frank Trujillo, Corinna Ulcigrai, Trujillo, Frank +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2210.14202

openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 at least two intervals and any vector w belonging to the central-stable space E_cs(T₀) for the Rauzy-Veech renormalization, any AIET T with log-slopes given by w and semi-conjugated to T₀ is topologically conjugated to T. If in addition, if w does not belong to E_s(T₀), the conjugacy between T and T₀ is singular.

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