Journal of Modern Dynamics (JMD)

Discontinuity-growth of interval-exchange maps

Pages: 379 - 405, Issue 3, July 2009      doi:10.3934/jmd.2009.3.379

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Christopher F. Novak - Department of Mathematics and Statistics, University of Michigan - Dearborn, 4901 Evergreen Rd., Dearborn,MI 48128, United States (email)

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}$.

Keywords:  Interval-exchange transformation, group actions, discontinuity-growth, distortion elements, centralizers, automorphism group.
Mathematics Subject Classification:  Primary: 37E05; Secondary: 37C85, 20F28.

Received: November 2008;      Revised: May 2009;      Available Online: August 2009.