Buff forms and invariant curves of near-parabolic maps

TL;DR

This paper develops a framework using Buff's meromorphic 1-form to analyze local dynamics of near-parabolic maps, proving the continuity of invariant curves and external rays as multipliers approach roots of unity.

Contribution

It introduces a general method to study invariant curves in near-parabolic maps and proves their stability and continuity properties under perturbations.

Findings

Invariant curves land at repelling periodic points near zero.

External rays of polynomial maps are Hausdorff continuous as multipliers approach roots of unity.

The framework applies to both polynomial and more general near-parabolic maps.

Abstract

We introduce a general framework to study the local dynamics of near-parabolic maps using the meromorphic $1$-form introduced by X.~Buff. As a sample application of this setup, we prove the following tameness result on invariant curves of near-parabolic maps: Let $g(z)=\lambda z+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $\lambda$ a primitive $q$th root of unity, and let $\gamma: \, ]-\infty,0] \to {\mathbb D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(\gamma(t))=\gamma(t+1)$ and $\lim_{t \to -\infty} \gamma(t)=0$. Take a sequence $g_n(z)=\lambda_n z+O(z^2)$ with $|\lambda_n|\neq 1$ such that $g_n \to g$ uniformly on ${\mathbb D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve $\gamma_n: \, ]-\infty,0] \to {\mathbb C}$ such that $\gamma_n \to \gamma$ uniformly on the fundamental segment $[-1,0]$. If $\lambda_n^q \to 1$ non-tangentially, then $\gamma_n$ lands at a repelling periodic point near $0$, and $\gamma_n \to \gamma$ uniformly on $]-\infty,0]$. In the special case of polynomial maps, this proves Hausdorff continuity of external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially.