Multiplier scales of a sequence of rational maps

TL;DR

This paper investigates the behavior of multipliers in degenerating sequences of rational maps, revealing bounds and growth patterns, and establishing a maximum number of significant multiplier scales based on the degree.

Contribution

It introduces a new analysis of multiplier scales in degenerating rational maps and proves an upper bound on the number of such scales using Ahbyankar's theorem.

Findings

Most periodic points have bounded multipliers or exploding multipliers at a common scale.

The set of scales induced by multiplier growth is finite, with at most 2d-2 such scales.

The paper establishes bounds on the number of non-trivial multiplier scales.

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.