# American Institute of Mathematical Sciences

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

## Most interval exchanges have no roots

 Department of Mathematics, Rice University, MS 136, P.O. Box 1892, Houston, TX 77251-1892, USA

Received  April 30, 2016 Revised  January 24, 2017 Published  March 2017

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.

Citation: Daniel Bernazzani. Most interval exchanges have no roots. Journal of Modern Dynamics, 2017, 11: 249-262. doi: 10.3934/jmd.2017011
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##### References:
A tower of type $(m, n)$ over a 2-IET
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