|
Quasisymmetric distortion and rigidity of expanding endomorphisms of 
Author(s):
Edson
de Faria
Journal:
Proc. Amer. Math. Soc.
124
(1996),
1949-1957.
MSC (1991):
Primary 58F03, 30C62;
Secondary 26A16
MathSciNet review:
1307509
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper we examine a result of D. Sullivan according to which two  expanding endomorphisms of the circle are  conjugate as soon as they are symmetrically conjugate. We develop general a priori estimates on the local distortion of quasisymmetric mappings and combine them with the classical naive distortion lemma to present a complete proof of Sullivan's result. A new proof is offered at the end that renders unnecessary the use of Markov partitions or the control of eigenvalues at periodic points.
References:
- [Ah]
- L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton, NJ, 1966. MR 34:336
- [Ca]
- L. Carleson, On mappings conformal at the boundary, J. Analyse Math. 19 (1967), 1--13. MR 35:6821
- [Cu]
- G. Cui, Linear models of circle expanding maps, Academia Sinica, preprint (1994).
- [GS1]
- F. Gardiner and D. Sullivan, Symmetric structures on a closed curve, Amer. J. of Math. 114 (1992), 683--736. CMP 92:16
- [GS2]
- ------, Lacunary series as quadratic differentials in conformal dynamics, Contemporary Mathematics 169 (1994). CMP 94:17
- [PS]
- A. Pinto and D. Sullivan, The circle and the solenoid (to appear).
- [Sa]
- R. Sacksteder, The measures invariant under an expanding map, Lecture Notes in Math., vol. 392, Springer-Verlag, Berlin, 1972, pp. 179--194. MR 53:11024
- [SS]
- M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Erg. Th. & Dynam. Sys. 5 (1985), 285--289. MR 87g:58104
- [Su1]
- D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, Proc. Sympos. Pure Math. 48 (1988), 15--23. MR 90k:58141
- [Su2]
- ------, 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. MR 93k:58194
- [Sh]
- M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175--199. MR 39:2169
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (1991):
58F03, 30C62,
26A16
Retrieve articles in all Journals with
MSC (1991):
58F03, 30C62,
26A16
Additional Information:
Edson
de Faria
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05389-970 São Paulo SP- Brasil
Email:
edson@ime.usp.br
DOI:
10.1090/S0002-9939-96-03218-2
PII:
S 0002-9939(96)03218-2
Keywords:
Expanding maps,
symmetric conjugacies
Received by editor(s):
November 21, 1994
Additional Notes:
This work is part of Projeto Temático de Equipe ``Transição de Fase Dinâmica em Sistemas Evolutivos'', supported by FAPESP Grant 90/3918-5.
Communicated by:
Linda Keen
Copyright of article:
Copyright
1996,
American Mathematical Society
|
