Asymptotics of Transversality in Periodic Curves of Quadratic Rational Maps

TL;DR

The paper computes the Euler characteristic of the moduli spaces ${\pazocal{S}}_p$ of quadratic rational maps with a periodic marked critical point by relating zeros and poles of a specially crafted meromorphic one-form $d\tau$ on a compactification $\widehat{{\pazocal{S}}_p}$. The core technical device is a Main Lemma that connects parameter-space transversality to dynamical transversality at punctures, with the proof carried through a detailed Puiseux-series/dynamics analysis over a non-Archimedean field $\mathbb{L}$ and its completion $\mathbb{K}$. The method uses parabolic rescalings and rescaled polynomial models to control the local behavior near punctures, translating these into global topological data via degree theory and Bezout-type identities. The main result expresses $\chi({\pazocal{S}}_p)$ in terms of the explicit counts $\eta'(p)$ and $\eta'_{II}(p)$ as $\chi({\pazocal{S}}_p)=\dfrac{2\eta'(p)}{3}-\eta'_{II}(p)$ for $p\ge3$, with a smooth compactification adding $N_p$ ideal points so that $\chi(\widehat{\pazocal{S}}_p)=N_p+\dfrac{2\eta'(p)}{3}-\eta'_{II}(p)$. The results draw a strong analogy with cubic polynomials, reveal a rich interplay between dynamical and parameter-space transversality, and highlight Mandelbrot-torus-type structures in the moduli space via rescaling limits.

Abstract

We compute the Euler characteristic of the moduli space of quadratic rational maps with a periodic marked critical point of a given period.

Theorems & Definitions (53)

• Theorem 1
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3: KiwiRees
• Lemma 2.4
• Lemma 3.1: Main
• Proposition 3.2
• proof : Proof of Theorem \ref{['A']}, assuming Lemma \ref{['l-main']}
• Lemma 3.3