Most interval exchanges have no roots
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.
Keywords: Interval exchange transformation, root.
Received: April 2016; Revised: January 2017; Available Online: March 2017. |