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Pi in the Mandelbrot set everywhere

Thies Brockmoeller, Oscar Scherz, Nedim Srkalovic

TL;DR

This work addresses why the constant $\pi$ emerges in the escape-time statistics near bifurcation points of the Mandelbrot set. It leverages Oudkerk's parabolic perturbation theory for well-behaved maps, recasting the local dynamics via Fatou coordinates, gate structures, and lifted phases to derive precise asymptotics for the critical orbit, particularly along parameter rays. The main contributions are (i) the first proof of the $\pi$-occurrence at $c=-3/4$ and $c=-5/4$, and (ii) a generalization showing the same phenomenon at all primitive and satellite bifurcations, with explicit formulas $N(\alpha) = \pi\cdot\frac{2qn}{|\mu'_{qn}(c_0)\alpha|} + O(1)$ and a conceptual explanation of the mechanism. The results illuminate a universal mechanism linking parabolic perturbations to a $\pi$-scale escape dynamic and are reinforced by numerical experiments, including robustness statements via Koebe-type distortion bounds.

Abstract

The numerical phenomenon of $π$ appearing at parameters $c = 1/4$, $c=-3/4$ and $c=-5/4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter $c=1/4$. Very recently in 2023, an even sharper result for $c=1/4$ was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters $c=-3/4$ and $c=-5/4$, which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.

Pi in the Mandelbrot set everywhere

TL;DR

This work addresses why the constant emerges in the escape-time statistics near bifurcation points of the Mandelbrot set. It leverages Oudkerk's parabolic perturbation theory for well-behaved maps, recasting the local dynamics via Fatou coordinates, gate structures, and lifted phases to derive precise asymptotics for the critical orbit, particularly along parameter rays. The main contributions are (i) the first proof of the -occurrence at and , and (ii) a generalization showing the same phenomenon at all primitive and satellite bifurcations, with explicit formulas and a conceptual explanation of the mechanism. The results illuminate a universal mechanism linking parabolic perturbations to a -scale escape dynamic and are reinforced by numerical experiments, including robustness statements via Koebe-type distortion bounds.

Abstract

The numerical phenomenon of appearing at parameters , and in the Mandelbrot set has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter . Very recently in 2023, an even sharper result for was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters and , which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.
Paper Structure (5 sections, 17 theorems, 33 equations, 6 figures, 1 table)

This paper contains 5 sections, 17 theorems, 33 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of Oudkerk's theory
  4. Proof of the main theorem, satellite case
  5. Numerical experiments

Key Result

Theorem 1

Let $c_0$ be a parabolic parameter in $\mathcal{M}$ with parabolic periodic point $z_0$ and let $n$ be its period. Then $(p^{\circ n})'(z_0)$ is a $q$th root of unity. Fix an escape radius $R>2$. Let $c_k\in\mathbb{C}\setminus\mathcal{M}$ be a sequence such that $c_k\to c_0$ along one of the two par Here $\tau(c_0) = \frac{|\mu_{qn}'(c_0)|}{2qn}$ where $\mu_{qn}'(c_0)$ is the derivative of the nat

Figures (6)

  • Figure 1: The parameters $c=-3/4$, $c'=1/4$ and $c"=-5/4$ in the Mandelbrot set, together with the parameter rays $\mathcal{R}_{1/3}$, $\mathcal{R}_{0/1}$ and $\mathcal{R}_{6/15}$ and sequences $c_t$, $c'_t$ and $c"_t$ approaching the bifurcations. The images are taken with mandel.
  • Figure 2: Example of a gate structure for $q=2$. $\sigma_0$ is a double fixed point while $\sigma_1$ is a simple fixed point. $\mathbf{gate}_{}(f) = (1,\star)$, i.e. $\overline{\ell_{1,+,f}\cup\ell_{1,-,f}}$ is homeomorphic to a circle and $\overline{\ell_{2,+,f}}$ is homeomorphic to a circle.
  • Figure 3: A neighborhood $U$ of $c_0$ is mapped under $\mu_{qn}$.
  • Figure 4: Gate structures of $f_{\alpha_+}$ and $f_{\alpha_-}$ for $q=3$.
  • Figure 5: Taking $f$ sufficiently close to $f_0$ we can ensure that $|N_f - n_f|\leq 1$. Here, $S_{j,-,f}$ is sufficiently close to $S_{j,-,f_0}$ such that it is contained in an $\varepsilon$-neighborhood $U$ of $S_{j,-,f_0}$ (in red), which in turn is contained in $\textbf{S}_-\cup D_\delta(0)$. $S_{j,-,f}$ ends at 0 and some fixed point $\sigma(f)$ distinct from 0, while both ends of $S_{j,-,f_0}$ land at 0.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2: Near the parameter ray
  • Definition 1.1: Orbit, periodic points, multiplier
  • Definition 1.2: Parabolic periodic point
  • Definition 1.3: Hyperbolic components, multiplier map
  • Definition 1.4: Parabolic parameter
  • Proposition 1.5: Bifurcations
  • Definition 1.6: Bifurcation
  • Proposition 1.7: Roots of hyperbolic components
  • Definition 1.8: Dynamic rays
  • ...and 31 more