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Locating critical points attracted to p-adic attracting cycles

Juan Rivera-Letelier

TL;DR

This work analyzes non-Archimedean rational dynamics and establishes a sharp threshold for when an attracting cycle must attract a critical point. The authors introduce a d-dependant multiplier bound via ${\lambda(d)=\min\{|m|: m\in\{1,\dots,d\}\}}$ and show that a cycle with ${|\lambda|<\lambda(d)^{\#\mathcal{O}}}$ guarantees a nearby critical point, with Lyapunov exponents providing a finiteness criterion for small-exponent cycles. They develop a robust toolkit around distorted log-size and inseparability, proving refined bounds for locating critical values (Section 3) and then locating critical points within immediate basins (Section 4), including the Cantor-type and non-Cantor-type basins. The paper concludes with sharpness examples (Section 5) demonstrating that the bounds are tight in various regimes, thereby clarifying the landscape of attracting dynamics in the non-Archimedean setting and extending the analogies with Fatou–Julia theory in this context.

Abstract

In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map, this paper establishes a sharp condition on the multiplier of an attracting cycle ensuring it attracts a critical point.

Locating critical points attracted to p-adic attracting cycles

TL;DR

This work analyzes non-Archimedean rational dynamics and establishes a sharp threshold for when an attracting cycle must attract a critical point. The authors introduce a d-dependant multiplier bound via and show that a cycle with guarantees a nearby critical point, with Lyapunov exponents providing a finiteness criterion for small-exponent cycles. They develop a robust toolkit around distorted log-size and inseparability, proving refined bounds for locating critical values (Section 3) and then locating critical points within immediate basins (Section 4), including the Cantor-type and non-Cantor-type basins. The paper concludes with sharpness examples (Section 5) demonstrating that the bounds are tight in various regimes, thereby clarifying the landscape of attracting dynamics in the non-Archimedean setting and extending the analogies with Fatou–Julia theory in this context.

Abstract

In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map, this paper establishes a sharp condition on the multiplier of an attracting cycle ensuring it attracts a critical point.
Paper Structure (17 sections, 19 theorems, 152 equations)

This paper contains 17 sections, 19 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Notes and references
  3. Strategy and organization
  4. Preliminaries
  5. Distorted log-size
  6. Critical points in the absence of poles
  7. Locating critical values of rational functions
  8. Proof of \ref{["t:critically-mapped'"]}
  9. Proof of \ref{["c:critically-mapped'"]}
  10. Proof of \ref{["t:strictly-critically-mapped'"]}
  11. Proof of \ref{['t:strictly-critically-mapped"']}
  12. Locating critical points in immediate basins
  13. Proof of \ref{["t:critically-attracted'"]}
  14. Proof of \ref{["t:infinitely-attracted'"]}
  15. Examples
  16. ...and 2 more sections

Key Result

Theorem 1

Let $R$ be a rational map of degree $d$ at least two with coefficients in $K$, and $\mathcal{O}$ an attracting cycle of $R$. If the multiplier $\lambda$ of $\mathcal{O}$ satisfies ${|\lambda| < \lambda(d)^{\#\mathcal{O}}}$, then $\mathcal{O}$ attracts a critical point of $R$.

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2
  • Corollary 1.3
  • Theorem 3
  • Corollary 1.4
  • Theorem 4
  • Conjecture 1.5: Riv03c
  • Lemma 2.1
  • ...and 18 more