Discontinuity-growth of interval-exchange maps

Christopher F. Novak

doi:10.3934/jmd.2009.3.379

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2009, Volume 3, Issue 3: 379-405. Doi: 10.3934/jmd.2009.3.379

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Discontinuity-growth of interval-exchange maps

Christopher F. Novak^1,

1.
Department of Mathematics and Statistics, University of Michigan - Dearborn, 4901 Evergreen Rd., Dearborn,MI 48128

Received: October 31, 2008

Revised: April 30, 2009

Published: July 2009

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Abstract

For an interval-exchange map $f$, the number of discontinuities $d(f^n)$ either exhibits linear growth or is bounded independently of $n$. This dichotomy is used to prove that the group $\mathcal{E}$ of interval-exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval-exchanges. As a further application of this dichotomy, a classification of centralizers in $\mathcal{E}$ is given. This classification is used to show that $\text{Aut}(\mathcal{E}) \cong \mathcal{E}$ ⋊ $\mathbb{Z}$/$ 2 \mathbb{Z}$.

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Mathematics Subject Classification: Primary: 37E05; Secondary: 37C85, 20F28.

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