Douady rabbit

A Douady rabbit is a fractal derived from the Julia set of the function

is near the center of one of the period three bulbs of the Mandelbrot set for a complex quadratic map.

It is named after French mathematician Adrien Douady.

The Douady rabbit is generated by iterating the Mandelbrot set map

is fixed to lie in one of the two period three bulb off the main cardioid and

The resulting image can be colored by corresponding each pixel with a starting value

and calculating the amount of iterations required before the value of

escapes a bounded region, after which it will diverge toward infinity.

Irrespective of the specific iteration used, the filled Julia set associated with a given value of

are affine transformations of one another, or more specifically a similarity transformation, consisting of only scaling, rotation and translation, the filled Julia sets look similar for either form of the iteration given above.

You can also describe the Douady rabbit utilising the Mandelbrot set with respect to

In this figure, the Mandelbrot set superficially appears as two back-to-back unit disks with sprouts or buds, such as 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.

is within one of these four sprouts, the associated filled Julia set in the mapping plane is said to be a Douady rabbit.

A Douady rabbit consists of the three attracting fixed points

For example, Figure 4 shows the Douady rabbit in the

, a point in the five-o'clock sprout of the right disk.

(also called period-three fixed points) have the locations The red, green, and yellow points lie in the basins

The white points lie in the basin

Corresponding to these relations there are the results As a second example, Figure 5 shows a Douady rabbit when

, a point in the eleven-o'clock sprout on the left disk (

The three major lobes on the left, which contain the period-three fixed points

powers of Dehn twists about its ears.

[2] The corabbit is the symmetrical image of the rabbit.

It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit.

The Julia set has no direct analog in three dimensions.

It has a parabolic fixed point with 3 petals.

Perturbed rabbit[6] In the early 1980s, Hubbard posed the so-called twisted rabbit problem, a polynomial classification problem.

The goal is to determine Thurston equivalence types[definition needed] of functions of complex numbers that usually are not given by a formula (these are called topological polynomials):[7] The problem was originally solved by Laurent Bartholdi and Volodymyr Nekrashevych[8] using iterated monodromic groups.

The generalization of the problem to the case where the number of post-critical points is arbitrarily large has been solved as well.

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