Small Cycles Property of Some Cremer Rational Maps and Polynomials

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.

Keywords

Irrationally indifferent fixed point / Linearization problem / Small cycles property