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Small Cycles Property of Some Cremer Rational Maps and Polynomials

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Abstract

This paper concerns the linearization problem on rational maps of degree d ≥ 2 and polynomials of degree d > 2 from the perspective of non-linearizability. The authors introduce a set \({{\cal C}_\infty }\) of irrational numbers and show that if \(\alpha \in {{\cal C}_\infty }\), then any rational map is not linearizable and has infinitely many cycles in every neighborhood of the fixed point with multiplier \(\lambda = {{\rm{e}}^{2\pi {\rm{i}}\alpha }}\). Adding more constraints to cubic polynomials, they discuss the above problems by polynomial-like maps. For the family of polynomials, with the help of Yoccoz’s method, they obtain its maximum dimension of the set in which the polynomials are non-linearizable.

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Authors and Affiliations

  1. Faculty of Science, Yibin University, Yibin, Sichuan, 644000, China

    Rong Fu

  2. Department of Mathematical Sciences, Sichuan Normal University, Chengdu, 610066, China

    Rong Fu & Ji Zhou

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  1. Rong Fu
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  2. Ji Zhou
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Correspondence to Rong Fu or Ji Zhou.

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Fu, R., Zhou, J. Small Cycles Property of Some Cremer Rational Maps and Polynomials. Chin. Ann. Math. Ser. B 45, 123–136 (2024). https://doi.org/10.1007/s11401-024-0006-8

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  • DOI: https://doi.org/10.1007/s11401-024-0006-8

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