Most Fatou and Julia components are small for polynomials

TL;DR

This paper demonstrates that for most polynomials, Julia and Fatou components tend to be small in size, regardless of local connectivity of the Julia set, with specific results for polynomials lacking irrationally neutral cycles.

Contribution

It establishes that Julia components are generally small and extends this to Fatou components in polynomials without irrationally neutral cycles.

Findings

Julia components are typically small in diameter.

Fatou components are also small when no irrationally neutral cycles are present.

Results hold even when the Julia set is not locally connected.

Abstract

We prove that Julia components of polynomials are generally small in diameter. For polynomials without irrationally neutral cycles, Fatou components are also typically small, even when the Julia set is not locally connected.