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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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