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Rigidity, universality, and hyperbolicity of renormalization for critical circle maps with non-integer exponents

Part of: Complex dynamical systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  25 September 2018

IGORS GORBOVICKIS  and
MICHAEL YAMPOLSKY
IGORS GORBOVICKIS
Affiliation:
Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany email i.gorbovickis@jacobs-university.de
MICHAEL YAMPOLSKY
Affiliation:
University of Toronto, Mathematics Department, 40 St George Street, Toronto, Ontario, Canada, M5S2E4 email yampol@math.toronto.edu
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Abstract

We construct a renormalization operator which acts on analytic circle maps whose critical exponent $\unicode[STIX]{x1D6FC}$ is not necessarily an odd integer $2n+1$, $n\in \mathbb{N}$. When $\unicode[STIX]{x1D6FC}=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps by Yampolsky [Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci.96 (2002), 1–41]. In the case when $\unicode[STIX]{x1D6FC}$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+\unicode[STIX]{x1D6FC}}$-rigidity for such maps.

Keywords

low-dimensional dynamicsrenormalizationuniversalityrigiditycritical circle map

MSC classification

Primary: 37E20: Universality, renormalization 37F25: Renormalization 37E10: Maps of the circle
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1282 - 1334
Copyright
© Cambridge University Press, 2018

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