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December 2013 Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces, II
Chaohui ZHANG
Tokyo J. Math. 36(2): 289-307 (December 2013). DOI: 10.3836/tjm/1391177972

Abstract

Let $S$ be a Riemann surface containing at least two punctures $z$ and $z_0$. Let $\mathscr{F}(S)$ be the set of pseudo-Anosov maps of $S$ that are isotopic to the identity on $S\cup \{z\}$. We show that for any $f\in \mathscr{F}(S)$ and any twice punctured disk $\Delta$ enclosing $z$ and $z_0$, the pair $(\partial \Delta, f(\partial \Delta))$ fills $S$, where $\partial \Delta$ denotes the boundary of $\Delta$. Fix such a $\Delta$, and denote by $\mathscr{T}(\Delta)$ the set of twice punctured disks $\Delta'$ on $S$ enclosing $z$ and $z_0$ with the property that $(\partial \Delta, \partial \Delta')$ fills $S$. Let $\Delta_0\in \mathscr{T}(\Delta)$. We describe all possible pseudo-Anosov maps $f$ in $\mathscr{F}(S)$ sending $\Delta$ to $\Delta_0$, and classify elements of $\mathscr{F}(S)$ in terms of $\mathscr{T}(\Delta)$. We also show that there are infinitely many elements $f_k\in \mathscr{F}(S)$ with $f_k(\Delta)=\Delta_0$ such that their dilatations $\lambda(f_k)\rightarrow +\infty$ as $k\rightarrow +\infty$.

Citation

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Chaohui ZHANG. "Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces, II." Tokyo J. Math. 36 (2) 289 - 307, December 2013. https://doi.org/10.3836/tjm/1391177972

Information

Published: December 2013
First available in Project Euclid: 31 January 2014

zbMATH: 1288.32021
MathSciNet: MR3161559
Digital Object Identifier: 10.3836/tjm/1391177972

Subjects:
Primary: 53C35
Secondary: 53C40

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

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Vol.36 • No. 2 • December 2013
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