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Renormalization of Analytic Multicritical Circle Maps with Bounded Type Rotation Numbers

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Abstract

In this paper, we study homeomorphisms of the circle with several critical points and bounded type rotation number. We prove complex a priori bounds for these maps. As an application, we get that bi-cubic circle maps with same bounded type rotation number are \(C^{1+\alpha }\) rigid.

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Notes

  1. The statement in Yampolsky (1999) is formulated for maps in the Epstein class, it is possible to extend (Yampolsky 1999, Lemma 4.2) to real-analytic circle maps using Lemma 3.5, see also Yampolsky (2019, Lemma 4.8)

  2. It is helpful to think here what happens in the limiting situation, that is, when \(\beta \) is one of the endpoints of \({{\mathcal {P}}}_m\). Then f renormalizes to a commuting pair with a critical point of criticality \(d\times d_1\) at 0 and the estimate in Theorem 3.3 is improved.

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Acknowledgements

We would like to thank the anonymous referee who provided generous and detailed comments on a previous version of the manuscript.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, Bloco H, Campus do Gragoatá, Niterói, Rio de Janeiro, RJ, 24.210-201, Brazil

    Gabriela Estevez

  2. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trab. São Carlense 400, São Carlos, SP, CEP 13566-590, Brazil

    Daniel Smania

  3. Department of Mathematics, University of Toronto 40, St George Street, Toronto, Ontario, Canada

    Michael Yampolsky

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  1. Gabriela Estevez
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  2. Daniel Smania
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  3. Michael Yampolsky
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Correspondence to Gabriela Estevez.

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G.E. was partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. D.S. was partially supported by CNPq 306622/2019-0, CNPq 430351/2018-6 and FAPESP Projeto Temático 2017/06463-3. M.Y. was partially supported by NSERC Discovery Grant.

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Estevez, G., Smania, D. & Yampolsky, M. Renormalization of Analytic Multicritical Circle Maps with Bounded Type Rotation Numbers. Bull Braz Math Soc, New Series 53, 1053–1071 (2022). https://doi.org/10.1007/s00574-022-00295-8

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  • DOI: https://doi.org/10.1007/s00574-022-00295-8

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