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On conformal distortion and Sullivan's sector theorem
Author(s):
Edson
de Faria
Journal:
Proc. Amer. Math. Soc.
126
(1998),
67-74.
MSC (1991):
Primary 30C55, 30D05;
Secondary 58F23
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Abstract:
Using general bounds on the conformal distortion of univalent maps, we prove a strong version of Sullivan's sector theorem, which gives certain sufficient conditions for an arbitrarily long composition of univalent Herglotz functions to map the upper half-plane into a proper sub-sector.
References:
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- 2.
- C. Carathéodory, Conformal representation, Cambridge University Press, 1952. MR 13:734a
- 3.
- E. de Faria, Proof of universality for critical circle mappings, Ph.D. Thesis, CUNY, 1992.
- 4.
- H. Epstein, Fixed points of composition operators II, Nonlinearity 2 (1989), 305-310. MR 90j:58086
- 5.
- M. Flexor, Théoreme du secteur d'après D. Sullivan, Prepublication UPS Orsay (1992).
- 6.
- C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329-342. MR 88a:30057
- 7.
- W. de Melo and S. van Strien, One dimensional dynamics, Springer-Verlag, Berlin and New York, 1993. MR 95a:58035
- 8.
- D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, Mathematics into the Twenty-First Century, Amer. Math. Soc. Centennial Publication, vol.2, Amer. Math. Soc., Providence, RI, 1991, pp. 417-466. MR 93k:58194
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Additional Information:
Edson
de Faria
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Butantã CEP05508-900 São Paulo SP, Brasil
Email:
edson@ime.usp.br
DOI:
10.1090/S0002-9939-98-04487-6
PII:
S 0002-9939(98)04487-6
Keywords:
Non-linearity,
conformal distortion,
univalent mappings
Received by editor(s):
November 21, 1994
Additional Notes:
This work has been supported by FAPESP Grant 95/3187-4.
Communicated by:
Linda Keen
Copyright of article:
Copyright
1998,
American Mathematical Society
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