Douady rabbit

The rabbit is generated by iterating the Mandelbrot set map ${\displaystyle z=z^{2}+c}$ on the complex plane with ${\displaystyle c}$ fixed to lie in the period three bulb off the main cardioid and ${\displaystyle z}$ ranging over the plane.

The pixels in the image are then colored to show whether a particular value of ${\displaystyle z}$ the iteration converged or diverged.

The Twisted rabbit[1] is the composition of the rabbit polynomial with ${\displaystyle n}$the powers of the Dehn twists about its ears.[2]

The Corabbit is the symmetrical image of the 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.

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

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

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

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

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

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

and the parameters ${\displaystyle \gamma }$ and ${\displaystyle \mu }$ are related by the equations

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

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

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

The nature of what happens in the mapping plane under repeated application of ${\displaystyle {\mathcal {M}}}$ depends on where ${\displaystyle \gamma }$ (or ${\displaystyle \mu }$) 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 ${\displaystyle {\mathcal {M}}}$. 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 ${\displaystyle \gamma }$ is the control parameter, and Figure 2 shows the Mandelbrot set when ${\displaystyle \mu }$ is the control parameter. Since ${\displaystyle z}$ and ${\displaystyle w}$ are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the ${\displaystyle z}$ or ${\displaystyle w}$ planes.

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

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

Douady rabbit in an exponential family

Lamination of rabbit Julia set

Representation of the dynamics inside the rabbit.

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 ${\displaystyle \gamma }$ is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of ${\displaystyle \gamma }$, it can be shown that ${\displaystyle {\mathcal {M}}}$ has ${\displaystyle z=0}$ and one other point as unstable (repelling) fixed points, and ${\displaystyle z=\infty }$ as an attracting fixed point. Moreover, the map ${\displaystyle {\mathcal {M}}^{3}}$ has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, and ${\displaystyle z_{3}}$ and their basins of attraction.

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

The action of ${\displaystyle {\mathcal {M}}}$ 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

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

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

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

Figure 3: Douady's rabbit for ${\displaystyle \gamma =2.55268-0.959456i}$ or ${\displaystyle \mu =0.122565-0.744864i}$.

As a second example, Figure 4 shows a Douady rabbit when ${\displaystyle \gamma =2-\gamma _{D}=-.55268+.959456i}$, a point in the eleven-o'clock sprout on the left disk. (As noted earlier, ${\displaystyle \mu }$ 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 ${\displaystyle {\mathcal {M}}}$ itself are located at ${\displaystyle z=0}$ and ${\displaystyle z=1.450795+0.7825835i}$. The three major lobes on the left, which contain the period-three fixed points ${\displaystyle z_{1}}$,${\displaystyle z_{2}}$, and ${\displaystyle z_{3}}$, meet at the fixed point ${\displaystyle z=0}$, and their counterparts on the right meet at the point ${\displaystyle z=1}$. It can be shown that the effect of ${\displaystyle {\mathcal {M}}}$ on points near the origin consists of a counterclockwise rotation about the origin of ${\displaystyle \arg(\gamma )}$, or very nearly ${\displaystyle 120^{\circ }}$, followed by scaling (dilation) by a factor of ${\displaystyle |\gamma |=1.1072538}$.

Figure 4: Douady's rabbit for ${\displaystyle \gamma =-0.55268+0.959456i}$ or ${\displaystyle \mu =0.122565-0.744864i}$.

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 has been solved as well.[9]

• 
  Gray levels indicate the speed of convergence to infinity or to the attractive cycle
• 
  boundaries of level sets
• 
  binary decomposition
• 
• 
  with spine
• 
  with external rays

• Dragon curve
• Herman ring
• Siegel disc

• Weisstein, Eric W. "Douady Rabbit Fractal". MathWorld.
• Dragt, A. "Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics".
• Adrien Douady: La dynamique du lapin (1996) - video on the YouTube

This article incorporates material from Douady Rabbit on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.