Blow-up of multipliers in meromorphic families of rational maps

TL;DR

The work links degenerations of one-parameter families of degree $d\ge2$ rational maps to non-Archimedean dynamics on the Berkovich line, revealing a sharp dichotomy: either all cycle multipliers stay uniformly bounded or almost all multipliers blow up at the degeneration point with rate governed by a non-Archimedean Lyapunov exponent $\\lambda(f_{\\mathrm{na}})$. By transferring complex dynamical questions to the non-Archimedean setting, the authors prove that if $\\lambda(f_{\\mathrm{na}})=0$ then multipliers are controlled, while if $\\lambda(f_{\\mathrm{na}})>0$ then a positive proportion of cycles exhibit exponential growth in $|t|^{-1}$, implying widespread repulsion. They establish existence and bounds for repelling cycles in polynomials and cubic maps, leveraging Puiseux expansions and non-Archimedean stability arguments, and deduce sharp consequences for multiplier maps, including a proper birational embedding result for cubic rational maps. The work thereby bridges complex degeneration phenomena with non-Archimedean dynamics to obtain precise information about how multipliers behave in degenerating families and provides structural insights into the moduli spaces via multiplier data.

Abstract

We study the blow-up of the multipliers of periodic cycles in one-parameter holomorphic degenerating families of rational maps of the Riemann sphere.

Figures (1)

• Figure 1: Proof of Theorem \ref{['thm:period3']}

Theorems & Definitions (24)

• Theorem A
• Theorem B
• Corollary C
• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• proof
• Lemma 1.3
• Remark 1.4
• Theorem 1.5