Discontinuity-growth of interval-exchange maps

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July 2009, 3(3): 379-405. doi: 10.3934/jmd.2009.3.379

Discontinuity-growth of interval-exchange maps

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

Received November 2008 Revised May 2009 Published August 2009

Abstract
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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}$.

Keywords: distortion elements, automorphism group., centralizers, discontinuity-growth, group actions, Interval-exchange transformation.

Mathematics Subject Classification: Primary: 37E05; Secondary: 37C85, 20F2.

Citation: Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379

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