## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 53, Number 4 (2001), 847-874.

### Non-linearizability of n-subhyperbolic polynomials at irrationally indifferent fixed points

#### Abstract

We study the non-linearlizability conjecture (NLC) for polynomials at non-Brjuno irrationally indifferent fixed points. A polynomial is $n$-subhyperbolic if it has exactly $n$ recurrent critical points corresponding to irrationally indifferent cycles, other ones in the Julia set are preperiodic and no critical orbit in the Fatou set accumulates to the Julia set. In this article, we show that NLC and, more generally, the cycle-version of NLC are true in a subclass of $n$-subhyperbolic polynomials. As a corollary, we prove the cycle-version of the Yoccoz Theorem for quadratic polynomials.

We also study several specific examples of $n$-subhyperbolic polynomials. Here we also show the scaling invariance of the Brjuno condition: if an irrational number $\alpha $ satisfies the Brjuno condition, then so do $m\alpha $ for every positive integer $m$.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 53, Number 4 (2001), 847-874.

**Dates**

First available in Project Euclid: 29 May 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1212067576

**Digital Object Identifier**

doi:10.2969/jmsj/05340847

**Mathematical Reviews number (MathSciNet)**

MR1852886

**Zentralblatt MATH identifier**

1030.37032

**Subjects**

Primary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets

Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 37F25: Renormalization 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

**Keywords**

$n$-subhyperbolic polynomiaj quadratic polynomiaj irrationally indifferent cycle Brjuno condition non-linearizability renormalization Teichm{\"u}ller space

#### Citation

OKUYAMA, Yûsuke. Non-linearizability of n-subhyperbolic polynomials at irrationally indifferent fixed points. J. Math. Soc. Japan 53 (2001), no. 4, 847--874. doi:10.2969/jmsj/05340847. https://projecteuclid.org/euclid.jmsj/1212067576