Linearizability of Saturated Polynomials

TL;DR

This paper proves Douady's conjecture for a class of polynomials, showing that non-M"obius rational functions cannot have Siegel disks with non-Brjuno rotation numbers, extending known results about linearizability.

Contribution

It establishes Douady's conjecture for polynomials with a specific critical orbit structure, including all polynomials of the form z^d + c.

Findings

Douady's conjecture holds for the specified polynomial class.

The conjecture is confirmed for all polynomials of the form z^d + c.

The result links critical orbit structure to linearizability and Siegel disk properties.

Abstract

Brjuno and R\"ussmann proved that every irrationally indifferent fixed point of an analytic function with a Brjuno rotation number is linearizable, and Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured that this is sharp for all rational functions of degree at least 2, i.e., that non-M\"obius rational functions cannot have Siegel disks with non-Brjuno rotation numbers. We prove that Douady's conjecture holds for the class of polynomials for which the number of infinite tails of critical orbits in the Julia set equals the number of irrationally indifferent cycles. As a corollary, Douady's conjecture holds for the polynomials $P(z) = z^d + c$ for all $d > 1$ and all complex $c$.