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 $m$.