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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Real and complex earthquakes
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by Dragomir Šarić PDF
Trans. Amer. Math. Soc. 358 (2006), 233-249 Request permission


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 Šarić
  • 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:
  • Received by editor(s): March 1, 2003
  • Received by editor(s) in revised form: February 1, 2004
  • Published electronically: February 4, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 233-249
  • MSC (2000): Primary 30F60, 30F45, 32H02, 32G05; Secondary 30C62
  • DOI:
  • MathSciNet review: 2171231