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Real and complex earthquakes

Author: Dragomir Saric
Journal: Trans. Amer. Math. Soc. 358 (2006), 233-249
MSC (2000): Primary 30F60, 30F45, 32H02, 32G05; Secondary 30C62
Published electronically: February 4, 2005
MathSciNet review: 2171231
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Abstract: We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane $\mathbb{H} ^2$. We show that an earthquake restricted to the boundary $S^1$ of $\mathbb{H} ^2$ is a quasisymmetric map if and only if its earthquake measure is bounded. Multiplying an earthquake measure by a positive parameter we obtain an earthquake path. Consequently, an earthquake path with a bounded measure is a path in the universal Teichmüller space. We extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part. Such obtained complex earthquake (or bending) is holomorphic in the parameter. Moreover, the restrictions to $S^1$ of a bending with complex parameter of small imaginary part is a holomorphic motion of $S^1$in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.

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Additional Information

Dragomir Saric
Affiliation: Department of Mathematics, The Gradute School and University Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016
Address at time of publication: Institute for Mathematical Sciences, SUNY Stony Brook, Stony Brook, New York 11794-3660

Keywords: Earthquake, transverse measure, bending
Received by editor(s): March 1, 2003
Received by editor(s) in revised form: February 1, 2004
Published electronically: February 4, 2005
Article copyright: © Copyright 2005 American Mathematical Society