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On Geodesics in Asymptotic Universal Teichmüller Space

Part of: Complex dynamical systems

Published online by Cambridge University Press:  08 January 2016

Zhou Zemin  and
Chen Jixiu
Zhou Zemin
Affiliation:
Department of Mathematics, Renmin University of China, Beijing, 100872, People's Republic of China ([email protected])
Chen Jixiu
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433, People's Republic of China ([email protected])
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Abstract

Let AT(Δ) be the asymptotic universal Teichmüller space, viewed as the space of all asymptotic Teichmüller equivalence classes [[μ]]. We show that if μ is asymptotically extremal in AT(Δ) and hp([[μ]]) < h([[μ]]) for some boundary point p of Δ, then there are infinitely many geodesics joining [[0]] and [[μ]] in AT(Δ). As a corollary, a necessary condition for a complex dilatation to be uniquely extremal in AT(Δ) is given.

Keywords

asymptotic Teichmüller spacequasi-conformal mappinggeodesic

MSC classification

Secondary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 4 , November 2016 , pp. 1065 - 1074
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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