Real and complex earthquakes
Author:
Dragomir Šarić
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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
- R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
- C. J. Earle, I. Kra, and S. L. Krushkal′, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927–948. MR 1214783, DOI https://doi.org/10.1090/S0002-9947-1994-1214783-6
- D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
- D. B. A. Epstein, A. Marden, and V. Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. (2) 159 (2004), no. 1, 305–336. MR 2052356, DOI https://doi.org/10.4007/annals.2004.159.305
- Frederick P. Gardiner, Infinitesimal bending and twisting in one-dimensional dynamics, Trans. Amer. Math. Soc. 347 (1995), no. 3, 915–937. MR 1290717, DOI https://doi.org/10.1090/S0002-9947-1995-1290717-4
- Frederick Gardiner and Linda Keen, Holomorphic motions and quasi-Fuchsian manifolds, Complex geometry of groups (Olmué, 1998) Contemp. Math., vol. 240, Amer. Math. Soc., Providence, RI, 1999, pp. 159–174. MR 1703557, DOI https://doi.org/10.1090/conm/240/03578
- Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906
- F. P. Gardiner, J. Hu, and N. Lakic, Earthquake curves, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 141–195. MR 1940169, DOI https://doi.org/10.1090/conm/311/05452
- J. Hu, Earthquake Measure and Cross-ratio Distortion, Contemp. Math. vol. 355, 285–308, A.M.S., 2004.
- Linda Keen and Caroline Series, How to bend pairs of punctured tori, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359–387. MR 1476997, DOI https://doi.org/10.1090/conm/211/02830
- Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI https://doi.org/10.2307/2007076
- Steven P. Kerckhoff, Earthquakes are analytic, Comment. Math. Helv. 60 (1985), no. 1, 17–30. MR 787659, DOI https://doi.org/10.1007/BF02567397
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
- Curtis T. McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), no. 2, 283–320. MR 1478844, DOI https://doi.org/10.1090/S0894-0347-98-00259-8
- Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347–355. MR 1037218, DOI https://doi.org/10.1090/S0002-9939-1991-1037218-8
- William P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91–112. MR 903860
- William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30F60, 30F45, 32H02, 32G05, 30C62
Retrieve articles in all journals with MSC (2000): 30F60, 30F45, 32H02, 32G05, 30C62
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:
saric@math.sunysb.edu
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