The Douady rabbit is any of various particular filled Julia sets whose parameter is near the center of a period 3 bud of the Mandelbrot set for complex quadratic map. It is named after French mathematician Adrien Douady.

Formula

The rabbit is generated by iterating the Mandelbrot set map ${\displaystyle z=z^{2}+c}$ on the complex plane with $c$ fixed to lie in the period three bulb off the main cardiod and $z$ ranging over the plane. The pixels in the image are then colored to show whether for a particular value of $z$ the iteration converged or diverged.

Variants

Twisted rabbit^[1] or rabbits with twisted ears = is the composition of the “rabbit” polynomial with n-th powers of the Dehn twists about its ears.^[2]

Corabbit is symmetrical image of rabbit. Here parameter ${\displaystyle c\approx -0.1226-0.7449i}$ It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit.

3D

The Julia set has no direct analog in 3D.

4D

Quaternion Julia set with parameters c = −0,123 + 0.745i and a cross section in the XY plane. The "Douady Rabbit" Julia set is visible in the cross section.

Embedded

A small "embedded" homeomorphic copy of rabbit in the center of a Julia set^[3]

Fat

The fat rabbit or chubby rabbit has c at the root of 1/3-limb of the Mandelbrot set. It has a parabolic fixed point with 3 petals.^[4]

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  fat rabbit

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  parabolic chessboard

n-th eared

• period 4 bulb rabbit = Three-Eared Rabbit
• period 5 bulb rabbit = Four-Eared Rabbit

In general, the rabbit for the period-(n+1) bulb off the main cardiod will have n ears^[5]

Perturbed

Perturbed rabbit^[6]

• Perturbed Rabbit
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  Perturbed Rabbit

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  Perturbed rabbit zoom

Forms of the complex quadratic map

There are two common forms for the complex quadratic map ${\mathcal {M))$. The first, also called the complex logistic map, is written as

$z_{n+1}={\mathcal {M))z_{n}=\gamma z_{n}\left(1-z_{n}\right),$

where $z$ is a complex variable and $\gamma$ is a complex parameter. The second common form is

$w_{n+1}={\mathcal {M))w_{n}=w_{n}^{2}-\mu .$

Here $w$ is a complex variable and $\mu$ is a complex parameter. The variables $z$ and $w$ are related by the equation

$z=-{\frac {w}{\gamma ))+{\frac {1}{2)),$

and the parameters $\gamma$ and $\mu$ are related by the equations

$\mu =\left({\frac {\gamma -1}{2))\right)^{2}-{\frac {1}{4))\quad ,\quad \gamma =1\pm {\sqrt {1+4\mu )).$

Note that $\mu$ is invariant under the substitution $\gamma \to 2-\gamma$.

Mandelbrot and filled Julia sets

There are two planes associated with ${\mathcal {M))$. One of these, the $z$ (or $w$) plane, will be called the mapping plane, since ${\mathcal {M))$ sends this plane into itself. The other, the $\gamma$ (or $\mu$) plane, will be called the control plane.

The nature of what happens in the mapping plane under repeated application of ${\mathcal {M))$ depends on where $\gamma$ (or $\mu$) is in the control plane. The filled Julia set consists of all points in the mapping plane whose images remain bounded under indefinitely repeated applications of ${\mathcal {M))$. The Mandelbrot set consists of those points in the control plane such that the associated filled Julia set in the mapping plane is connected.

Figure 1 shows the Mandelbrot set when $\gamma$ is the control parameter, and Figure 2 shows the Mandelbrot set when $\mu$ is the control parameter. Since $z$ and $w$ are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the $z$ or $w$ planes.

Figure 1: The Mandelbrot set in the $\gamma$ plane.

Figure 2: The Mandelbrot set in the $\mu$ plane.

The Douady rabbit

^{[clarification needed]}

The Douady rabbit is most easily described in terms of the Mandelbrot set as shown in Figure 1 (above). In this figure, the Mandelbrot set, at least when viewed from a distance, appears as two back-to-back unit discs with sprouts. Consider the sprouts at the one- and five-o'clock positions on the right disk or the sprouts at the seven- and eleven-o'clock positions on the left disk. When $\gamma$ is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of $\gamma$, it can be shown that ${\mathcal {M))$ has $z=0$ and one other point as unstable (repelling) fixed points, and $z=\infty$ as an attracting fixed point. Moreover, the map ${\mathcal {M))^{3}$ has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points $z_{1}$, $z_{2}$, and $z_{3}$ and their basins of attraction.

For example, Figure 3 shows Douady's rabbit in the $z$ plane when $\gamma =\gamma _{D}=2.55268-0.959456i$, a point in the five-o'clock sprout of the right disk. For this value of $\gamma$, the map ${\mathcal {M))$ has the repelling fixed points $z=0$ and $z=.656747-.129015i$. The three attracting fixed points of ${\mathcal {M))^{3}$ (also called period-three fixed points) have the locations

${\displaystyle z_{1}=0.499997032420304-(1.221880225696050\times 10^{-6})i{\;}{\;}{\mathrm {(red)} },}$

${\displaystyle z_{2}=0.638169999974373-(0.239864000011495)i{\;}{\;}{\mathrm {(green)} },}$

${\displaystyle z_{3}=0.799901291393262-(0.107547238170383)i{\;}{\;}{\mathrm {(yellow)} }.}$

The red, green, and yellow points lie in the basins ${\displaystyle B(z_{1})}$, ${\displaystyle B(z_{2})}$, and ${\displaystyle B(z_{3})}$ of ${\mathcal {M))^{3}$, respectively. The white points lie in the basin $B(\infty )$ of ${\mathcal {M))$.

The action of ${\mathcal {M))$ on these fixed points is given by the relations

${\displaystyle {\mathcal {M))z_{1}=z_{2},}$

${\displaystyle {\mathcal {M))z_{2}=z_{3},}$

${\displaystyle {\mathcal {M))z_{3}=z_{1}.}$

Corresponding to these relations there are the results

${\displaystyle {\mathcal {M))B(z_{1})=B(z_{2}){\;}{\mathrm {or} }{\;}{\mathcal {M)){\;}{\mathrm {red} }\subseteq {\mathrm {green} },}$

${\displaystyle {\mathcal {M))B(z_{2})=B(z_{3}){\;}{\mathrm {or} }{\;}{\mathcal {M)){\;}{\mathrm {green} }\subseteq {\mathrm {yellow} },}$

${\displaystyle {\mathcal {M))B(z_{3})=B(z_{1}){\;}{\mathrm {or} }{\;}{\mathcal {M)){\;}{\mathrm {yellow} }\subseteq {\mathrm {red} }.}$

As a second example, Figure 4 shows a Douady rabbit when $\gamma =2-\gamma _{D}=-.55268+.959456i$, a point in the eleven-o'clock sprout on the left disk. (As noted earlier, $\mu$ is invariant under this transformation.) The rabbit now sits more symmetrically in the plane. The period-three fixed points then are located at

${\displaystyle z_{1}=0.500003730675024+(6.968273875812428\times 10^{-6})i{\;}{\;}({\mathrm {red} }),}$

${\displaystyle z_{2}=-0.138169999969259+(0.239864000061970)i{\;}{\;}({\mathrm {green} }),}$

${\displaystyle z_{3}=-0.238618870661709-(0.264884797354373)i{\;}{\;}({\mathrm {yellow} }),}$

The repelling fixed points of ${\mathcal {M))$ itself are located at $z=0$ and $z=1.450795+0.7825835i$. The three major lobes on the left, which contain the period-three fixed points $z_{1}$,$z_{2}$, and $z_{3}$, meet at the fixed point $z=0$, and their counterparts on the right meet at the point $z=1$. It can be shown that the effect of ${\mathcal {M))$ on points near the origin consists of a counterclockwise rotation about the origin of $\arg(\gamma )$, or very nearly $120^{\circ }$, followed by scaling (dilation) by a factor of $|\gamma |=1.1072538$.

Twisted rabbit problem

In the early 1980s, Hubbard posed the so-called twisted rabbit problem, a polynomial classification problem. The goal is to determine Thurston equivalence types of functions of complex numbers that usually are not given by a formula (these are called topological polynomials)."^[7]

• given a topological quadratic whose branch point is periodic with period three, to which quadratic polynomial is it Thurston equivalent ?
• determining the equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears.

It was originally solved by Laurent Bartholdi and Volodymyr Nekrashevych^[8] using iterated monodromy groups.

The generalization of the twisted rabbit problem to the case where the number of post-critical points is arbitrarily large is also solved^[9]

Gallery

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  Gray levels indicate the speed of convergence to infinity or to the attractive cycle

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  boundaries of level sets

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  binary decomposition

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  with spine

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  with external rays