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

DOI:
https://doi.org/10.1090/S0002-9947-05-03651-2

Published electronically:
February 4, 2005

MathSciNet review:
2171231

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane . We show that an earthquake restricted to the boundary of 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 of a bending with complex parameter of small imaginary part is a holomorphic motion of 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

Email:
saric@math.sunysb.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03651-2

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