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Julia sets and geometrically finite maps over finite extensions of the $p$-adic field

Shilei Fan, Lingmin Liao, Hongmin Nie, Yuefei Wang

Abstract

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $φ\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $φ$ is the natural restriction of $\mathbb{C}_p$-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that $φ$ is geometrically finite, we prove that the dynamics on the $K$-Julia set of $φ$ is a countable state Markov shift.

Julia sets and geometrically finite maps over finite extensions of the $p$-adic field

Abstract

Let be a finite extension of the field of -adic numbers, and be a rational map of degree at least . We prove that the -Julia set of is the natural restriction of -Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that is geometrically finite, we prove that the dynamics on the -Julia set of is a countable state Markov shift.

Paper Structure

This paper contains 30 sections, 45 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Main results
  4. Relation of the Julia sets
  5. Strategy of the proofs
  6. Preliminaries
  7. Notation
  8. $p$-adic disks
  9. Analytic functions
  10. Julia sets and Fatou sets
  11. Fatou components
  12. Brief background of symbolic dynamics
  13. Restriction of the Julia set
  14. Critical distances
  15. Points in $\partial J_{\mathbb{C}_p}^{K}(\phi)$
  16. ...and 15 more sections

Key Result

Theorem 1.3

Let $\phi\in K(z)$ be a semi-hyperbolic rational map of degree at least $2$. Then

Figures (1)

  • Figure 1: The portrait of $A_n$, $A'_n$, $B_n$ and $B'_n$ under $F$.

Theorems & Definitions (92)

  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • proof : Proof of Theorem \ref{['Thm:main1']} assuming Theorem \ref{['Thm:main']}
  • Lemma 2.1: Khrennikov04
  • ...and 82 more