Douady rabbit

Douady rabbit
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 a complex quadratic map. It is named after French mathematician Adrien Douady.
Formula
The rabbit is generated by iterating the Mandelbrot set map on the complex plane, with fixed to lie in the period three bulb off the main cardioid and ranging over the plane.
The pixels in the image are then colored to show whether a particular value of on iteration of the map remains bounded.
Variants
The Twisted rabbit is the composition of the rabbit polynomial with powers of the Dehn twists about its ears.
The Corabbit is the symmetrical image of the rabbit. Here parameter . 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
A Quaternion Julia set with parameters 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
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.
n-th eared
period 4 bulb rabbit = Three-Eared Rabbit
period 5 bulb rabbit = Four-Eared Rabbit
In general, the rabbit for the bulb off the main cardioid will have ears
Perturbed
Perturbed rabbit
Forms of the complex quadratic map
There are two common forms for the complex quadratic map . The first, also called the complex logistic map, is written as
where is a complex variable and is a complex parameter. The second common form is
Here is a complex variable and is a complex parameter. The variables and are related by the equation
and the parameters and are related by the equations
Note that is invariant under the substitution .
Mandelbrot and filled Julia sets
There are two planes associated with . One of these, the (or ) plane, will be called the mapping plane since sends this plane into itself. The other, the (or ) plane, will be called the control plane.
The nature of what happens in the mapping plane under repeated application of depends on where (or ) 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 . 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 is the control parameter, and Figure 2 shows the Mandelbrot set when is the control parameter. Since and are affine transformations of one another (a linear transformation plus a translation), the filled Julia sets look much the same in either the or planes.
The Douady 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 is within one of these four sprouts, the associated filled Julia set in the mapping plane is a Douady rabbit. For these values of , it can be shown that has and one other point as unstable (repelling) fixed points, and as an attracting fixed point. Moreover, the map has three attracting fixed points. Douady's rabbit consists of the three attracting fixed points and and their basins of attraction.
For example, Figure 3 shows Douady's rabbit in the plane when , a point in the five-o'clock sprout of the right disk.
For this value of , the map has the repelling fixed points and . The three attracting fixed points of (also called period-three fixed points) have the locations
The red, green, and yellow points lie in the basins and of , respectively. The white points lie in the basin of .
The action of on these fixed points is given by the relations
Corresponding to these relations there are the results
As a second example, Figure 4 shows a Douady rabbit when , a point in the eleven-o'clock sprout on the left disk (is invariant under this transformation). The rabbit now sits more symmetrically in the plane. The period-three fixed points then are located at
The repelling fixed points of itself are located at and . The three major lobes on the left, which contain the period-three fixed points and , meet at the fixed point , and their counterparts on the right meet at the point . It can be shown that the effect of on points near the origin consists of a counterclockwise rotation about the origin of , or very nearly , followed by scaling (dilation) by a factor of .
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):
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 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.
Gallery
See also
Dragon curve
Herman ring
Siegel disc
References
External links
Adrien Douady: La dynamique du lapin (1996) - video on the YouTube
Fractals
Limit sets