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Quasisymmetric orbit-flexibility of multicritical circle maps

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  30 September 2021

EDSON DE FARIA Open the ORCID record for EDSON DE FARIA [Opens in a new window]  and
PABLO GUARINO Open the ORCID record for PABLO GUARINO [Opens in a new window]
EDSON DE FARIA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil (e-mail: edson@ime.usp.br)
PABLO GUARINO*
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Brazil
*
e-mail: pablo_guarino@id.uff.br
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Abstract

Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

Keywords

critical circle mapsquasisymmetric orbit-flexibilityskew product

MSC classification

Primary: 37E10: Maps of the circle
Secondary: 37E20: Universality, renormalization 37C40: Smooth ergodic theory, invariant measures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 11 , November 2022 , pp. 3271 - 3310
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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