A Douady rabbit is a fractal derived from the Julia set of the function , when parameter 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.

An example of a Douady rabbit. The colors show the number of iterations required to escape.



The Douady rabbit is generated by iterating the Mandelbrot set map   on the complex plane, where parameter   is fixed to lie in one of the two period three bulb off the main cardioid and   ranging over the plane. 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.

It can also be described using the logistic map form of the complex quadratic map, specifically


which is equivalent to


Irrespective of the specific iteration used, the filled Julia set associated with a given value of   (or  ) consists of all starting points   (or  ) for which the iteration remains bounded. Then, the Mandelbrot set consists of those values of   (or  ) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either   or  .

The Mandelbrot set in the   plane
The Mandelbrot set in the   plane

Noting that   is invariant under the substitution  , the Mandelbrot set with respect to   has additional horizontal symmetry. Since   and   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.

Detailed description

Douady rabbit in an exponential family
Lamination of the rabbit Julia set
Representation of the dynamics inside the rabbit

You can also describe the Douady rabbit utilising the Mandelbrot set with respect to   as shown in the graph above. 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. When   is within one of these four sprouts, the associated filled Julia set in the mapping plane is said to be 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. A Douady rabbit consists of the three attracting fixed points  ,  , and   and their basins of attraction.

For example, Figure 4 shows the Douady 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  ,  , and  .

Corresponding to these relations there are the results

Figure 4: Douady rabbit for   or  

As a second example, Figure 5 shows a Douady rabbit when  , a point in the eleven-o'clock sprout on the left disk (  is invariant under this transformation). This rabbit is more symmetrical 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  .

Figure 5: Douady rabbit for   or  



A twisted rabbit[1] is the composition of a rabbit polynomial with   powers of Dehn twists about its ears.[2]

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.

The Julia set has no direct analog in three dimensions.

A quaternion Julia set with parameters   and a cross-section in the   plane. The Douady rabbit is visible in the cross-section.





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

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

n-th eared


In general, the rabbit for the  th bulb of the main cardioid will have   ears[5] For example, a period four bulb rabbit has three ears.



Perturbed rabbit[6]

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[definition needed] 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, determining which quadratic polynomial it is Thurston equivalent to
  • determining the equivalence class of twisted rabbits, i.e. composite of the rabbit polynomial with nth powers of Dehn twists about its ears.

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]


See also



  1. ^ "A Geometric Solution to the Twisted Rabbit Problem by Jim Belk, University of St Andrews" (PDF). Archived (PDF) from the original on 2022-11-01. Retrieved 2022-05-03.
  2. ^ Laurent Bartholdi; Volodymyr Nekrashevych (2006). "Thurston equivalence of topological polynomials". Acta Mathematica. 197: 1–51. arXiv:math/0510082. doi:10.1007/s11511-006-0007-3.
  3. ^ "Period-n Rabbit Renormalization. 'Rabbit's show' by Evgeny Demidov". Archived from the original on 2022-05-03. Retrieved 2022-05-03.
  4. ^ Note on dynamically stable perturbations of parabolics by Tomoki Kawahira Archived October 2, 2006, at the Wayback Machine
  5. ^ "Twisted Three-Eared Rabbits: Identifying Topological Quadratics Up To Thurston Equivalence by Adam Chodof" (PDF). Archived (PDF) from the original on 2022-05-03. Retrieved 2022-05-03.
  6. ^ "Recent Research Papers (Only since 1999) Robert L. Devaney: Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets". Archived from the original on 2019-10-23. Retrieved 2020-04-07.
  7. ^ "Polynomials, dynamics, and trees by Becca Winarski" (PDF). Archived (PDF) from the original on 2022-11-01. Retrieved 2022-05-08.
  8. ^ Laurent Bartholdi; Volodymyr Nekrashevych (2005). "Thurston equivalence of topological polynomials". arXiv:math/0510082v3.
  9. ^ James Belk; Justin Lanier; Dan Margalit; Rebecca R. Winarski (2019). "Recognizing Topological Polynomials by Lifting Trees". arXiv:1906.07680v1 [math.DS].

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