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Renormalization of circle maps and smoothness of Arnold tongues

Part of: Low-dimensional dynamical systems Complex dynamical systems

Published online by Cambridge University Press:  24 February 2025

NATALIYA GONCHARUK  and
MICHAEL YAMPOLSKY
NATALIYA GONCHARUK*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX, USA
MICHAEL YAMPOLSKY
Affiliation:
Mathematics Department, University of Toronto, Toronto, Canada (e-mail: yampol@math.toronto.edu)
*
e-mail: natasha_goncharuk@tamu.edu
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Abstract

We study the global behavior of the renormalization operator on a specially constructed Banach manifold that has cubic critical circle maps on its boundary and circle diffeomorphisms in its interior. As an application, we prove results on smoothness of irrational Arnold tongues.

Keywords

low-dimensional dynamicssmooth dynamics

MSC classification

Primary: 37E20: Universality, renormalization 37E10: Maps of the circle 37E45: Rotation numbers and vectors
Secondary: 37F25: Renormalization
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 45
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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