Most Interval Exchanges Have No Roots

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Let T be an m-interval exchange transformation. By the rank of T we mean the dimension of the 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+ m/2, 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 IETs whose discontinuities all belong to a single orbit.