Infinitesimal bending and twisting in onedimensional dynamics
Author:
Frederick P. Gardiner
Journal:
Trans. Amer. Math. Soc. 347 (1995), 915937
MSC:
Primary 30C65; Secondary 30F30, 30F60, 32G15, 47B99
MathSciNet review:
1290717
Fulltext PDF Free Access
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Abstract: An infinitesimal theory for bending and earthquaking in onedimensional dynamics is developed. It is shown that any tangent vector to Teichmüller space is the initial data for a bending and for an earthquaking ordinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear operators from a Banach space of measures on the Möbius strip to tangent vectors to Teichmüller space, and the construction of a nonlinear right inverse for these operators. The inverse is constructed by establishing an infinitesimal version of Thurston's earthquake theorem.
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 [2]
 L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385404. MR 0115006 (22:5813)
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 L. Bers, A nonstandard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113134. MR 0192046 (33:273)
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 [7]
 , Lacunary series as quadratic differentials in conformal dynamics, Contemporary Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 307330. MR 1292907 (95g:58189)
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 Oliver A. Goodman, Metrized laminations and quasisymmetric maps, Ph.D. thesis, Warwick University, 1989.
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 , Quasiconformal homeomorphisms and dynamics. I, Solution of the FatouJulia problem on wandering domains, Ann. of Math. (2) 122 (1985), 401418. MR 819553 (87i:58103)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512907174
PII:
S 00029947(1995)12907174
Article copyright:
© Copyright 1995 American Mathematical Society
