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Discontinuity-growth of interval-exchange maps

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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}$.
    Mathematics Subject Classification: Primary: 37E05; Secondary: 37C85, 20F28.


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