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Journal of Modern Dynamics (JMD)
 

Most interval exchanges have no roots

Pages: 249 - 262, Volume 11, 2017      doi:10.3934/jmd.2017011

 
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Daniel Bernazzani - Department of Mathematics, Rice University, MS 136, P.O. Box 1892, Houston, TX 77251-1892, United States (email)

Abstract: Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged intervals. We prove that if $T$ is minimal and the rank of $T$ is greater than $1+\lfloor m/2 \rfloor$, then $T$ cannot be written as a power of another interval exchange. We also demonstrate that this estimate on the rank cannot be improved.
    In the case that $T$ is a minimal 3-interval exchange transformation, we prove a stronger result: $T$ cannot be written as a power of another interval exchange if and only if $T$ satisfies Keane's infinite distinct orbit condition. In the course of proving this result, we give a classification (up to conjugacy) of those minimal interval exchange transformations whose discontinuities all belong to a single orbit.

Keywords:  Interval exchange transformation, root.
Mathematics Subject Classification:  Primary: 37E05; Secondary: 37A05.

Received: April 2016;      Revised: January 2017;      Available Online: March 2017.

 References