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{{Short description|Fractal related to the mandelbrot set}}A '''Douady rabbit''' is a [[fractal]] derived from the [[filled Julia set|Julia set]] of the function <math display="inline">f_c(z) = z^2+c</math>, when [[Parameter (computer programming)|parameter]] <math>c</math> is near the center of one of the [[Mandelbrot set#Main cardioid and period bulbs|period three bulb]]s of the [[Mandelbrot set]] for a [[complex quadratic map]]. 
It is named after [[France|French]] [[mathematician]] [[Adrien Douady]].

[[File:Douady rabbit, plotted with Matplotlb.svg|thumb|upright=1.2|An example of a Douady rabbit. The colors show the number of iterations required to [[Plotting algorithms for the Mandelbrot set#Escape time algorithm|escape]].]]

==Background==
The Douady rabbit is generated by iterating the [[Mandelbrot set|Mandelbrot set map]] <math>z_{n+1}=z_n^2+c</math> on the [[complex plane]], where parameter <math>c</math> is fixed to lie in one of the two period three bulb off the main [[cardioid]] and <math>z</math> ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value <math>z_0</math> and calculating the amount of [[iteration|iterations]] required before the value of <math>z_n</math> escapes a bounded region, after which it will diverge toward [[infinity]].

It can also be described using the ''[[Complex quadratic polynomial#Forms|logistic map form]]'' of the [[complex quadratic map]], specifically

:<math>z_{n+1} = \mathcal M z_n := \gamma z_n \left(1 - z_n\right).</math>

which is equivalent to

<math>w_{n+1}=w_n^2+c</math>.

Irrespective of the specific iteration used, the [[filled Julia set]] associated with a given value of <math>\gamma</math> (or <math>\mu</math>) consists of all starting points <math>z_0</math> (or <math>w_0</math>) for which the iteration remains bounded. Then, the [[Mandelbrot set]] consists of those values of <math>\gamma</math> (or <math>\mu</math>) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either <math>\gamma</math> or <math>\mu</math>.

{{multiple image
| align             = center
| image1            = MandelbrotLambda.jpg
| width1            = 373
| image2            = MandelbrotMuDouadyRabbit.jpg
| width2            = 225
| caption1          = The Mandelbrot set in the <math>\gamma</math> plane
| caption2          = The Mandelbrot set in the <math>\mu</math> plane
}}

Noting that <math>\mu</math> is invariant under the substitution <math>\gamma \to 2 - \gamma</math>, the Mandelbrot set with respect to <math>\gamma</math> has additional horizontal symmetry. Since <math>z</math> and <math>w</math> are [[affine transformation]]s of one another, or more specifically a similarity transformation, consisting of only scaling, rotation and translation, the filled Julia sets look [[Similarity_(geometry)|similar]] for either form of the iteration given above.

==Detailed description==
[[File:DouadyRabbitInExponentialFamily.jpg|right|thumb|Douady rabbit in an exponential family]]
[[File:Rabbit-lamination.png|right|thumb|[[Lamination (topology)|Lamination]] of the rabbit Julia set]]
[[File:Douday rabbit rough dynamics.png|thumb|Representation of the dynamics inside the rabbit]]
You can also describe the Douady rabbit utilising the Mandelbrot set with respect to <math>\gamma</math> as shown in the graph above. 
In this figure, the Mandelbrot set superficially appears as two back-to-back [[unit disk]]s 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 <math>\gamma</math> 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 <math>\gamma</math>, it can be shown that <math>\mathcal M</math> has <math>z=0</math> and one other point as unstable (repelling) fixed points, and <math>z=\infty</math> as an attracting fixed point. 
Moreover, the map <math>{\mathcal{M}}^3</math> has three attracting fixed points. 
A Douady rabbit consists of the three attracting fixed points <math>z_1</math>, <math>z_2</math>, and <math>z_3</math> and their basins of attraction.

For example, Figure 4 shows the Douady rabbit in the <math>z</math> plane when <math>\gamma=\gamma_D=2.55268-0.959456i</math>, a point in the five-o'clock sprout of the right disk. For this value of <math>\gamma</math>, the map <math>\mathcal M</math> has the repelling fixed points <math>z=0</math> and <math>z=.656747-.129015i</math>. The three attracting fixed points of <math>{\mathcal M}^3</math> (also called period-three fixed points) have the locations

:<math>\begin{align}
z_1 &= 0.499997032420304 - (1.221880225696050\times10^{-6})i{\;}{\;}{\mathrm {(red)}},\\
z_2 &= 0.638169999974373 - (0.239864000011495)i{\;}{\;}{\mathrm {(green)}},\\
z_3 &= 0.799901291393262 - (0.107547238170383)i{\;}{\;}{\mathrm {(yellow)}}.
\end{align}</math>

The red, green, and yellow points lie in the basins <math>B(z_1)</math>, <math>B(z_2)</math>, and <math>B(z_3)</math> of <math>{\mathcal M}^3</math>, respectively. 
The white points lie in the basin <math>B(\infty)</math> of <math>\mathcal M</math>.

The action of <math>\mathcal M</math> on these fixed points is given by the relations <math>{\mathcal M}z_1=z_2</math>, <math>{\mathcal M}z_2=z_3</math>, and <math>{\mathcal M}z_3=z_1</math>.

Corresponding to these relations there are the results

:<math>\begin{align}
{\mathcal M}B(z_1)&=B(z_2) {\;} {\mathrm {or}} {\;} {\mathcal M}{\;}{\mathrm {red}}\subseteq{\mathrm {green}},\\
{\mathcal M}B(z_2)&=B(z_3) {\;} {\mathrm {or}} {\;} {\mathcal M}{\;}{\mathrm {green}}\subseteq{\mathrm {yellow}},\\
{\mathcal M}B(z_3)&=B(z_1) {\;} {\mathrm {or}} {\;} {\mathcal M}{\;}{\mathrm {yellow}}\subseteq{\mathrm {red}}.
\end{align}</math>

[[File:ColorDouadyRabbit1.jpg|thumb|500px|center|Figure 4: Douady rabbit for <math>\gamma=2.55268-0.959456i</math> or <math>\mu=0.122565-0.744864i</math>]]

As a second example, Figure 5 shows a Douady rabbit when <math>\gamma=2-\gamma_D=-.55268+.959456i</math>, a point in the eleven-o'clock sprout on the left disk (<math>\mu</math> is invariant under this transformation). 
This rabbit is more symmetrical in the plane. 
The period-three fixed points then are located at

:<math>\begin{align}
z_1&= 0.500003730675024 + (6.968273875812428 \times 10^{-6})i {\;}{\;} ({\mathrm {red}}),\\
z_2&=-0.138169999969259 + (0.239864000061970)i {\;}{\;} ({\mathrm {green}}),\\
z_3&= -0.238618870661709 - (0.264884797354373)i {\;}{\;} ({\mathrm {yellow}}).
\end{align}</math>

The repelling fixed points of <math>\mathcal M</math> itself are located at <math>z=0</math> and <math>z=1.450795 + 0.7825835i</math>. 
The three major lobes on the left, which contain the period-three fixed points <math>z_1</math>,<math>z_2</math>, and <math>z_3</math>, meet at the fixed point <math>z=0</math>, and their counterparts on the right meet at the point <math>z=1</math>. 
It can be shown that the effect of <math>\mathcal M</math> on points near the origin consists of a counterclockwise rotation about the origin of <math>\arg(\gamma)</math>, or very nearly <math>120^\circ</math>, followed by scaling (dilation) by a factor of <math>|\gamma|=1.1072538</math>.

[[File:ColorDouadyRabbit2.jpg|thumb|500px|center|Figure 5: Douady rabbit for <math>\gamma=-0.55268+0.959456i</math> or <math>\mu=0.122565-0.744864i</math>]]

==Variants==
A ''twisted rabbit''<ref>{{Cite web |url=https://e.math.cornell.edu/people/belk/talkslides/TwistedRabbitTalkMichigan.pdf |title=A Geometric Solution to the Twisted Rabbit Problem by Jim Belk, University of St Andrews |access-date=2022-05-03 |archive-date=2022-11-01 |archive-url=https://web.archive.org/web/20221101023328/https://e.math.cornell.edu/people/belk/talkslides/TwistedRabbitTalkMichigan.pdf |url-status=live}}</ref> is the composition of a rabbit [[polynomial]] with <math>n</math> powers of [[Dehn twist]]s about its ears.<ref>{{Cite journal |author=Laurent Bartholdi |author2=Volodymyr Nekrashevych |arxiv=math/0510082 |title=Thurston equivalence of topological polynomials|journal=Acta Mathematica |date=2006 |volume=197 |pages=1–51 |doi=10.1007/s11511-006-0007-3}}</ref>

The ''corabbit'' is the symmetrical image of the rabbit. Here parameter <math> c \approx -0.1226 -0.7449i</math>. It is one of 2 other [[Polynomial|polynomials]] inducing the same [[permutation]] of their post-critical set are the rabbit.

===3D===
The Julia set has no direct analog in three dimensions.

===4D===
A [[quaternion]] Julia set with parameters <math>c = -0.123 + 0.745i</math> and a [[Cross section (geometry)|cross-section]] in the <math>xy</math> plane. The Douady rabbit is visible in the cross-section.

[[File:Quaternion Julia Douady rabbit.jpg|400 px]]

===Embedded===
[[File:Parabolic julia set c = -1.125 + 0.21650635094611*i.png|400 px]]

A small embedded [[homeomorphic]] copy of rabbit in the center of a Julia set<ref>{{Cite web |url=https://www.ibiblio.org/e-notes/MSet/quadrat.htm |title=Period-n Rabbit Renormalization. 'Rabbit's show' by Evgeny Demidov |access-date=2022-05-03 |archive-date=2022-05-03 |archive-url=https://web.archive.org/web/20220503170632/https://www.ibiblio.org/e-notes/MSet/quadrat.htm |url-status=live}}</ref>

===Fat===
The ''fat rabbit'' or ''chubby rabbit'' has c at the root of the 1/3-[[Mandelbrot set#Geometry|limb]] of the [[Mandelbrot set]]. It has a [[Periodic points of complex quadratic mappings|parabolic fixed point]] with 3 [[:b:Fractals/Iterations in the complex plane/parabolic|petals]].<ref>[http://www.math.nagoya-u.ac.jp/~kawahira/works/rims0402.pdf Note on dynamically stable perturbations of parabolics by Tomoki Kawahira] {{webarchive |url=https://web.archive.org/web/20061002174848/http://www.math.nagoya-u.ac.jp/~kawahira/works/rims0402.pdf |date=October 2, 2006}}</ref>
<gallery>
Parabolic Julia set for internal angle 1 over 3.png|Fat rabbit
Parabolic chessboard for internal angle 1 over 3.png|Parabolic chessboard
</gallery>

===n-th eared===
In general, the rabbit for the <math>period-(n+1)</math>th bulb of the main cardioid will have <math>n</math> ears<ref>{{Cite web |url=https://e.math.cornell.edu/people/belk/projects/AdamChodoff.pdf |title=Twisted Three-Eared Rabbits: Identifying Topological Quadratics Up To Thurston Equivalence by Adam Chodof |access-date=2022-05-03 |archive-date=2022-05-03 |archive-url=https://web.archive.org/web/20220503093346/https://e.math.cornell.edu/people/belk/projects/AdamChodoff.pdf |url-status=live}}</ref> For example, a period four bulb rabbit has three ears.

===Perturbed===
Perturbed rabbit<ref>{{Cite web |url=http://math.bu.edu/people/bob/papers.html |title=Recent Research Papers (Only since 1999) Robert L. Devaney: Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets |access-date=2020-04-07 |archive-date=2019-10-23 |archive-url=https://web.archive.org/web/20191023110636/http://math.bu.edu/people/bob/papers.html |url-status=live}}</ref>
<gallery caption="Perturbed rabbit">
Perturbated Rabbit Julia set.png|Perturbed rabbit
PerturbatedRabbitJuliaSetZoom.png|Perturbed rabbit zoom
</gallery>

==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|What is Thurston equivalence?|date=November 2023}} of functions of [[complex number]]s that usually are not given by a formula (these are called topological polynomials):<ref>{{Cite web |url=http://www.math.lsa.umich.edu/mathclub/winter2020/042320.pdf |title=Polynomials, dynamics, and trees by Becca Winarski |access-date=2022-05-08 |archive-date=2022-11-01 |archive-url=https://web.archive.org/web/20221101023328/http://www.math.lsa.umich.edu/mathclub/winter2020/042320.pdf |url-status=live}}</ref> 
* given a [[Topology|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<ref>{{Cite arXiv |author=Laurent Bartholdi |author2=Volodymyr Nekrashevych |eprint=math/0510082v3 |title=Thurston equivalence of topological polynomials|date=2005}}</ref> using iterated [[monodromic group]]s. The generalization of the problem to the case where the number of post-critical points is arbitrarily large has been solved as well.<ref>{{Cite arXiv |eprint=1906.07680v1 |title=Recognizing Topological Polynomials by Lifting Trees |author1=James Belk |author2= Justin Lanier |author3=Dan Margalit |author4=Rebecca R. Winarski|date=2019 |class=math.DS}}</ref>

==Gallery==
<gallery>
File:Lapin de Douady 01.png|Gray levels indicate the speed of convergence to infinity or to the attractive cycle
File:LCMJ rabbit.png|Boundaries of level sets
File:Douady Rabbit Julia set with modified binary decomposition.png|Binary decomposition
File:Douady Rabbit Julia set with modified binary decomposition LCM.png
File:Rabbit Julia set with spine.svg|With [[Filled Julia set#Spine|spine]]
File:Julia set with 3 external rays.svg|With [[external ray]]s
File:Fr253 rabbit4.jpg|Multibrot-4 Douady rabbit
File:Fr158.jpg|A Douady rabbit on a red background
File:Fr249.jpg|A chain of Douady rabbits
</gallery>

==See also==
* [[Dragon curve]]
* [[Herman ring]]
* [[Siegel disc]]

==References==
<references />

==External links==
* {{MathWorld |title=Douady Rabbit Fractal |id=DouadysRabbitFractal}}
* {{cite web |last=Dragt |first=A. |url=https://www.physics.umd.edu/dsat/dsatliemethods.html |title=Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics |date=}}
* [https://www.youtube.com/watch?v=JttLtB0Gkdk Adrien Douady: La dynamique du lapin (1996) - video on the YouTube]
{{PlanetMath attribution|id=9602|title=Douady Rabbit}}

{{Fractals|state=collapsed}}

[[Category:Fractals]]
[[Category:Limit sets]]