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Multiplier scales of a sequence of rational maps

Chen Gong

TL;DR

This work extends the study of multiplier behavior in degenerating families of rational maps from the complex plane to a non-Archimedean framework, using ultrafilters and Berkovich geometry. It proves a dichotomy: either most periodic points acquire multipliers growing at a uniform rate or they explode at a common scale, with a refined measurement via multiplier scales. The core technical contributions include constructing a non-Archimedean limit map $f_\omega^{\epsilon}$, establishing a link between the Lyapunov exponent of the limit and the asymptotics of multipliers, and proving that there are at most $2d-2$ non-trivial multiplier scales through Abhyankar's inequality and transcendence-degree bounds. The results unify and extend prior dichotomies for meromorphic families and provide sharp bounds demonstrated by explicit examples, highlighting the role of higher-rank valuations and complex Robinson fields in degeneration phenomena.

Abstract

We analyze the behavior of multipliers of a degenerating sequence of complex rational maps. We show either most periodic points have uniformly bounded multipliers, or most of them have exploding multipliers at a common scale. We further explore the set of scales induced by the growth of multipliers. Using Ahbyankar's theorem, we prove that there can be at most 2d-2 such non-trivial multiplier scales.

Multiplier scales of a sequence of rational maps

TL;DR

This work extends the study of multiplier behavior in degenerating families of rational maps from the complex plane to a non-Archimedean framework, using ultrafilters and Berkovich geometry. It proves a dichotomy: either most periodic points acquire multipliers growing at a uniform rate or they explode at a common scale, with a refined measurement via multiplier scales. The core technical contributions include constructing a non-Archimedean limit map , establishing a link between the Lyapunov exponent of the limit and the asymptotics of multipliers, and proving that there are at most non-trivial multiplier scales through Abhyankar's inequality and transcendence-degree bounds. The results unify and extend prior dichotomies for meromorphic families and provide sharp bounds demonstrated by explicit examples, highlighting the role of higher-rank valuations and complex Robinson fields in degeneration phenomena.

Abstract

We analyze the behavior of multipliers of a degenerating sequence of complex rational maps. We show either most periodic points have uniformly bounded multipliers, or most of them have exploding multipliers at a common scale. We further explore the set of scales induced by the growth of multipliers. Using Ahbyankar's theorem, we prove that there can be at most 2d-2 such non-trivial multiplier scales.

Paper Structure

This paper contains 17 sections, 20 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Dynamics of rational maps over a complete valued field
  3. Berkovich projective line
  4. Julia sets, Lyapunov exponents and periodic cycles
  5. Tangent map and reduction map
  6. Bernoulli map
  7. Non-standard analysis and higher rank valuations
  8. Ultrafilters
  9. Scales
  10. Complex Robinson fields
  11. Scales and higher rank valuations
  12. Finiteness of scales on fields of finite transcendence degree
  13. Proof of the main results
  14. Construction of the non-Archimedean limit map
  15. Dichotomy in the asymptotic behavior of multipliers
  16. ...and 2 more sections

Key Result

Theorem A

Fix a non-principal ultrafilter $\omega$. Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of rational maps of degree $d$ degenerating in moduli space. Denote by $\chi_{f_n}$ be the Lyapunov exponent of $f_n$, and set Then exactly one of the following holds:

Theorems & Definitions (54)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Theorem B
  • Example 1.6
  • Theorem 2.1: Fav25FRL25
  • Proposition 2.2
  • ...and 44 more