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Discontinuity-growth of interval-exchange maps. (English) Zbl 1183.37077
Summary: 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 Aut$$(\mathcal{E}) \cong \mathcal{E} \rtimes \mathbb{Z}/ 2 \mathbb{Z}$$.

##### MSC:
 37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 20F28 Automorphism groups of groups
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