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Irrational Fatou components in non-Archimedean dynamics

Juan Rivera-Letelier

TL;DR

The paper solves the long-standing question of whether a non-Archimedean polynomial can admit a wandering Fatou component that is an irrational disk. By working in Benedetto’s polynomial family $P_a(z)=a z^p+(1-a) z^{p+1}$ over a complete algebraically closed non-Archimedean field with residue characteristic $p>0$, it explicitly computes the diameter dynamics of wandering domains along prescribed itineraries. A Cantor-set analysis of possible diameters shows that, for suitable itineraries, the diameter values are not confined to any proper divisible subgroup of $\mathbb{R}_{>0}$, yielding an irrational disk wandering Fatou component; conversely, certain itineraries produce rational-disk wandering domains. The approach hinges on two parts: (i) a realization theorem for itineraries via a phase-parameter perturbation scheme and (ii) an explicit, delicate diameter computation through wild ramification, culminating in a complete description of the wandering component's geometry in this non-Archimedean setting. This advances understanding of wild ramification in non-Archimedean dynamics and demonstrates the richness of Fatou component geometry in the Berkovich projective line.

Abstract

This paper studies the geometry of Fatou components in non-Archimedean dynamics. By explicitly computing a wandering domain constructed by Benedetto, it provides the first example of a Fatou component that is an irrational disk.

Irrational Fatou components in non-Archimedean dynamics

TL;DR

The paper solves the long-standing question of whether a non-Archimedean polynomial can admit a wandering Fatou component that is an irrational disk. By working in Benedetto’s polynomial family over a complete algebraically closed non-Archimedean field with residue characteristic , it explicitly computes the diameter dynamics of wandering domains along prescribed itineraries. A Cantor-set analysis of possible diameters shows that, for suitable itineraries, the diameter values are not confined to any proper divisible subgroup of , yielding an irrational disk wandering Fatou component; conversely, certain itineraries produce rational-disk wandering domains. The approach hinges on two parts: (i) a realization theorem for itineraries via a phase-parameter perturbation scheme and (ii) an explicit, delicate diameter computation through wild ramification, culminating in a complete description of the wandering component's geometry in this non-Archimedean setting. This advances understanding of wild ramification in non-Archimedean dynamics and demonstrates the richness of Fatou component geometry in the Berkovich projective line.

Abstract

This paper studies the geometry of Fatou components in non-Archimedean dynamics. By explicitly computing a wandering domain constructed by Benedetto, it provides the first example of a Fatou component that is an irrational disk.
Paper Structure (11 sections, 12 theorems, 130 equations)

This paper contains 11 sections, 12 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Strategy and organization
  3. Acknowledgments
  4. Main Theorem
  5. Preliminaries
  6. Realization of itineraries
  7. Phase parameter relation
  8. Realization of itineraries
  9. Computing the diameter of a wandering domain
  10. Proof of \ref{['p:explicit-diameter']}
  11. Proof of the \ref{['t:main']}

Key Result

Theorem 1

Suppose that the residue characteristic of $K$ is strictly positive and $|K^{\times}|$ is a proper subgroup of $\mathbb{R}_{> 0}$. Moreover, let $\mathscr{D}$ be a proper divisible subgroup of $\mathbb{R}_{> 0}$ containing $|K^{\times}|$. Then, there is a polynomial with coefficients in $K$ with a F

Theorems & Definitions (23)

  • Theorem 1: Irrational Fatou Components
  • Theorem 2: Rational Wandering Domains
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['t:wandering-irrationally']} supposing the \ref{['t:main']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • ...and 13 more