Small Cycles Property of Some Cremer Rational Maps and Polynomials

Rong Fu , Ji Zhou

Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (1) : 123 -136.

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Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (1) : 123 -136. DOI: 10.1007/s11401-024-0006-8
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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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Irrationally indifferent fixed point / Linearization problem / Small cycles property

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Rong Fu, Ji Zhou. Small Cycles Property of Some Cremer Rational Maps and Polynomials. Chinese Annals of Mathematics, Series B, 2024, 45(1): 123-136 DOI:10.1007/s11401-024-0006-8

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