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Non-smooth geodesic flows and the earthquake flow on Teichmüller space*

Published online by Cambridge University Press:  19 September 2008

Howard Weiss
Howard Weiss
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125, U.S.A.
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Abstract

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Thurston generalized the notion of a twist deformation about a simple closed geodesic on a hyperbolic Riemann surface to a twisting or shearing along a much more complicated object called a measure geodesic lamination. This new deformation is called an earthquake and it generates a flow on the tangent bundle of Teichmüller space.

In this paper we study the earthquake flow. We show that the flow is not smooth and that it is not the geodesic flow for an affine connection. We also derive the explicit form of the system of differential equations which earthquake trajectories satisfy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 571 - 586
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[AH]Ahlfors, L.V.. Lectures on Quasiconformal Mappings. Van Nostrand: New York, NY, 1966.Google Scholar
[BE]Beardon, A.. The Geometry Discrete Groups. Springer Verlag: New York N.Y., 1983.CrossRefGoogle Scholar
[CL]Coddington, E. & Levinson, N.. Theory of Ordinary Differential Equations. McGraw Hill: New York, NY, 1955.Google Scholar
[FN]Fenchel, W. & Nielsen, J.. Discontinuous groups of non-Euclidean motions. Unpublished manuscript.Google Scholar
[GKM]Gromoll, D., Klingenberg, W. & Meyer, W.. Riemannsche Geometrie in Grossen, Lecture Notes in Mathematics 55. Springer-Verlag: NY, 1967.Google Scholar
[KE]Keen, L.. A Rough Fundamental Domain for Teichmuller Spaces. Bull. Amer. Math. Soc. 83 (6) (1977), 1190–1226.CrossRefGoogle Scholar
[K1]Kerckhoff, S.. The Nielsen Realization Problem. Ann. of Math. 117 (1983), 235–265.CrossRefGoogle Scholar
[K2]Kerckhoff, S.. Earthquakes are Analytic. Comment. Math. Helv. 60 (1985).CrossRefGoogle Scholar
[K3]Kerckhoff, S.. Lines of Minima in Teichmuller Space, To appear.Google Scholar
[ML]Milnor, J.. Morse Theory. Ann Math. Studies 51. Princeton Univ. Press: Princeton, NJ, 1983.Google Scholar
[MOR]Morgan, J.. Columbia University Lecture Notes.Google Scholar
[SP]Spivak, M.. A Comprehensive Introduction to Differential Geometry, Vol. 1–5, Publish or Perish. Berkeley: CA 1970, 1975.Google Scholar
[T]Thurston, W.P.. The Geometry and Topology of 3-Manifolds. Princeton University Lecture Notes.Google Scholar
[W1]Wolpert, S.. University of Maryland Lecture Notes.Google Scholar
[W2]Wolpert, S.. On the symplectic geometry of deformation of a hyperbolic surface. Ann. of Math. 117 (1983), 207–234.CrossRefGoogle Scholar
[WW]Waterman, P. & Wolpert, S.. Earthquake Tessellations of Teichmüller Space. Trans. Amer. Math. Soc. 275 (1983), 157–167.Google Scholar