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Transactions of the American Mathematical Society

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Real bounds and quasisymmetric rigidity of multicritical circle maps


Authors: Gabriela Estevez and Edson de Faria
Journal: Trans. Amer. Math. Soc. 370 (2018), 5583-5616
MSC (2010): Primary 37E10; Secondary 37E20, 37F10, 37A05, 37C15
DOI: https://doi.org/10.1090/tran/7177
Published electronically: February 19, 2018
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Abstract: Let $ f, g:S^1\to S^1$ be two $ C^3$ critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if $ h:S^1\to S^1$ is a topological conjugacy between $ f$ and $ g$ and $ h$ maps the critical points of $ f$ to the critical points of $ g$, then $ h$ is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods and is valid for non-integer critical exponents as well. We do not require $ h$ to preserve the power-law exponents at corresponding critical points.


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Additional Information

Gabriela Estevez
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP, Brasil
Email: gestevez@ime.usp.br

Edson de Faria
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP, Brasil
Email: edson@ime.usp.br

DOI: https://doi.org/10.1090/tran/7177
Keywords: Real bounds, multicritical circle maps, quasisymmetric rigidity, dynamical partitions.
Received by editor(s): December 23, 2015
Received by editor(s) in revised form: December 9, 2016
Published electronically: February 19, 2018
Additional Notes: This work was supported by “Projeto Temático Dinâmica em Baixas Dimensões” FAPESP Grant 2011/16265-2 and by CAPES (PROEX)
Article copyright: © Copyright 2018 American Mathematical Society

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