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An optimal Yoccoz inequality for near-parabolic quadratic polynomials

Alex Kapiamba

TL;DR

This work addresses sharpening the Pommerenke-Levin-Yoccoz (PLY) bound for Mandelbrot limbs from the classical $O(1/q)$ to the conjectured $O(1/q^2)$ in a near-parabolic setting. It develops a unified framework based on parabolic implosion, Lavaurs maps, and near-parabolic renormalization, together with Log-multiplier families and parameter rays, to control the limiting geometry of external rays and their parameter analogues. The main contributions include the first explicit $O(1/q^2)$ bounds for certain limbs $\mathcal{L}_{p/q}$ (and their generalization to nested satellite copies via satellite towers), plus a robust toolkit (pre-petals, virtually parabolic Lavaurs maps, holomorphic motions, quasiconformal methods) that ties dynamical structures to parameter space. The results provide a sharp quantitative handle on Mandelbrot set geometry near parabolic points and lay groundwork for potential progress toward local connectivity statements for satellite combinatorics and beyond into broader unicritical families.

Abstract

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.

An optimal Yoccoz inequality for near-parabolic quadratic polynomials

TL;DR

This work addresses sharpening the Pommerenke-Levin-Yoccoz (PLY) bound for Mandelbrot limbs from the classical to the conjectured in a near-parabolic setting. It develops a unified framework based on parabolic implosion, Lavaurs maps, and near-parabolic renormalization, together with Log-multiplier families and parameter rays, to control the limiting geometry of external rays and their parameter analogues. The main contributions include the first explicit bounds for certain limbs (and their generalization to nested satellite copies via satellite towers), plus a robust toolkit (pre-petals, virtually parabolic Lavaurs maps, holomorphic motions, quasiconformal methods) that ties dynamical structures to parameter space. The results provide a sharp quantitative handle on Mandelbrot set geometry near parabolic points and lay groundwork for potential progress toward local connectivity statements for satellite combinatorics and beyond into broader unicritical families.

Abstract

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.

Paper Structure

This paper contains 16 sections, 59 theorems, 222 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Parabolic Implosion
  3. Modified continued fractions
  4. Petals for parabolic maps
  5. Petals for near-parabolic maps
  6. Lavaurs maps
  7. Escaping sets
  8. Pre-petals
  9. Virtually parabolic Lavaurs maps
  10. Parameter spaces
  11. Limits of External rays
  12. Parameter rays
  13. Induced Log-multiplier families
  14. Bounding limbs in $\mathcal{M}$
  15. Satellite towers
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $\alpha\in \mathcal{M}$ is not contained in an infinite nested sequence of small Mandelbrot copies, then $\mathcal{M}$ is locally connected at $\alpha$. If $\alpha$ is contained in an infinite nested sequence of small Mandelbrot copies as above and if $\emph{Diam}\, \mathcal{M}_{\alpha_n}\to 0$ when $n\to 0$, then $\mathcal{M}$ is locally connected at $\alpha$.

Figures (6)

  • Figure 1: The boundary of $\mathcal{M}$ with some limbs shaded.
  • Figure 2: Left: The filled Julia set for $Q_\alpha$ for $\alpha$ close to one with an external ray drawn in red. Right: The analogue of the filled Julia set for a Lavaurs map of $Q_1$ with an analogue of an external ray drawn in red.
  • Figure 3: A zoomed-in subset of the boundary of $\mathcal{M}$ near $1/n$ for some large $n$ (top) and the boundary of the analogue of $\mathcal{M}$ in the space of Lavaurs maps for $Q_1$ (bottom). In red, certain subsets of external rays of $\mathcal{M}$ which approximate analogues of external rays in the space of Lavaurs maps are drawn in. The bound on the size of these subsets of external rays of $\mathcal{M}$ induces the bound on the size of limbs in Theorem \ref{['main']}.
  • Figure 4: The sets $E_1$ (in light gray) and $E_2$ (in dark gray) for a $2$-nonescaping Lavaurs map of $Q_1$.
  • Figure 5: The parameter space of Lavaurs maps of $Q_1$. The $d$-escaping parameter set is shown in light gray for $d=1$, dark gray for $d=2$, and white for $d>2$.
  • ...and 1 more figures

Theorems & Definitions (102)

  • Theorem 1.1: Yocooz hubbard
  • Theorem 1.2: Pommerenke pommerenke, Levinlevin, Yoccoz hubbard
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • ...and 92 more