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Quasisymmetric distortion and rigidity of expanding endomorphisms of $S^{1}$

Author: Edson de Faria
Journal: Proc. Amer. Math. Soc. 124 (1996), 1949-1957
MSC (1991): Primary 58F03, 30C62; Secondary 26A16
MathSciNet review: 1307509
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Abstract: In this paper we examine a result of D. Sullivan according to which two $C^{1+\alpha }$ expanding endomorphisms of the circle are $C^{1+\alpha }$ 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.

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Additional Information

Edson de Faria
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05389-970 São Paulo SP- Brasil

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
Article copyright: © Copyright 1996 American Mathematical Society