ExoticHard

McMullen 2–3 Map

A rational map where squaring drives points outward and an inverse-cubic term sends them through the pole at the origin.

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$A pale blue McMullen fractal forming a many-sided ring of rounded cells around a large central opening$

FractalPark’s McMullen 2–3 rendering: nested bead-like boundaries circling a broad central opening.

Overview

The McMullen 2–3 map belongs to the family usually written as $f_\lambda(z)=z^n+\lambda/z^d$. FractalPark calls the parameter $c$ and chooses $n=2$, $d=3$. It first resembles a quadratic map with an extra term. Then $z^3$ appears in the denominator and changes the rulebook: the origin is a pole, not an ordinary point.

One formula now has two territories. Far out, $z^2$ dominates and pushes large values farther away. Near zero, $c/z^3$ takes over and sends points toward infinity through the pole. Between them lies a moving frontier where neither term wins outright.

That frontier may close into rings, break into rounded cells, or join into a connected web. It depends on $c$ and on the critical-point orbits. There is no single McMullen outline.

The Mathematics

A quadratic term and a cubic pole

z_{n+1}=z_n^2+\frac{c}{z_n^3}

At each step, square the current complex value, then add $c/z^3$. When $|z|$ is large, the first term grows roughly like $|z|^2$, while the reciprocal term fades like $1/|z|^3$. Near zero the balance flips: the reciprocal term becomes enormous.

The same expression can be written as

$$ f_c(z)=\frac{z^5+c}{z^3}. $$

The map has a pole of order three at $z=0$, and—when $c\neq0$—is a rational map of degree five. The “2–3” names the exponents in the original expression, not the rational-map degree.

In the exact definition, the pole sends zero to infinity on the Riemann sphere. A numerical renderer cannot divide by zero, so FractalPark replaces values extremely close to the origin with a tiny nonzero value and guards the denominator. That is a computational safety rail, not a different definition.

History

The family takes its name from Curtis T. McMullen. In Section 7 of his 1988 paper *Automorphisms of Rational Maps*, McMullen constructed rational maps whose Julia sets break into a Cantor set of Jordan curves—infinitely many disjoint loops arranged with Cantor-set structure. Later authors adopted “McMullen maps” for the family $z^n+\lambda/z^d$ associated with this construction. McMullen is now Cabot Professor of Mathematics at Harvard, where his personal academic homepage and publication list remain available.

The 2–3 choice is the smallest pair of exponents satisfying the characteristic inequality

$$ \frac{1}{n}+\frac{1}{d}<1, $$

because $1/2+1/3=5/6$. It also gives the lowest possible degree, $n+d=5$, in this part of the theory. The inequality alone does not guarantee that every parameter produces circles: the relevant critical orbits must also enter the appropriate escape region.

Work after McMullen mapped out that dependence in much greater detail. Robert Devaney, Daniel Look, and David Uminsky proved an “escape trichotomy” in 2005: when the free critical orbit escapes, the Julia set can be a Cantor set, a Cantor set of circles, or a Sierpiński curve, depending on how it reaches infinity.

Visual Characteristics

One large pale opening anchors the image. An uneven ring of rounded cells circles it; farther out, smaller cells make a fine lace. What looks like a smooth blue band at first resolves into a crowded rim.

The outer edge is gently faceted rather than round, and no two cells quite match. That belongs to this parameter, frame, and colouring—not every McMullen map. The pale palette favours relief and nested boundaries over an escape-speed gradient.

Remix and Examples

Start from the canonical Document

Open the same approved formula state used by this guide, then change the view, coloring, transforms, or animation in the interactive Explorer.

Remix in Explorer Open Formula Editor

Azure Whorl

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Frequently Asked Questions

What do 2 and 3 mean in the name?

They name the two powers in $z^2+c/z^3$: the polynomial side squares $z$, while the reciprocal side divides by its cube. In the general McMullen family they are $n$ and $d$.

Do not read them as “degree two to degree three.” Combine the terms and the map is $(z^5+c)/z^3$, so for nonzero $c$ its rational-map degree is five.

What happens at z = 0?

The denominator $z^3$ vanishes, so zero is a pole, not an ordinary starting point. On the extended complex plane it maps to infinity. Nearby points are flung away with strength growing like $1/|z|^3$.

That pole is not a rendering accident; it is the feature that distinguishes this family from a polynomial such as $z^2+c$. FractalPark only adds a tiny numerical guard near zero so the shader can represent the same limiting behavior without performing an undefined division.