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

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