Douady rabbit

The Douady rabbit is generated by iterating the Mandelbrot set map $z\_\{n+1\}=z\_n^2+c$ on the complex plane, where parameter $c$ is fixed to lie in one of the two period three bulb off the main cardioid and $z$ ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value $z\_0$ and calculating the amount of iterations required before the value of $z\_n$ 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

:$z\_\{n+1\}\; =\; \backslash mathcal\; M\; z\_n\; :=\; \backslash gamma\; z\_n\; \backslash left(1\; -\; z\_n\backslash right).$

which is equivalent to

$w\_\{n+1\}=w\_n^2+c$.

Irrespective of the specific iteration used, the filled Julia set associated with a given value of $\backslash gamma$ (or $\backslash mu$) consists of all starting points $z\_0$ (or $w\_0$) for which the iteration remains bounded. Then, the Mandelbrot set consists of those values of $\backslash gamma$ (or $\backslash mu$) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either $\backslash gamma$ or $\backslash mu$.

{{multiple image

| align = center

| image1 = MandelbrotLambda.jpg

| width1 = 373

| image2 = MandelbrotMuDouadyRabbit.jpg

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| caption1 = The Mandelbrot set in the $\backslash gamma$ plane

| caption2 = The Mandelbrot set in the $\backslash mu$ plane

}}

Noting that $\backslash mu$ is invariant under the substitution $\backslash gamma\; \backslash to\; 2\; -\; \backslash gamma$, the Mandelbrot set with respect to $\backslash gamma$ has additional horizontal symmetry. Since $z$ and $w$ 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.

File:DouadyRabbitInExponentialFamily.jpg

File:Rabbit-lamination.png of the rabbit Julia set]]

File:Douday rabbit rough dynamics.png

You can also describe the Douady rabbit utilising the Mandelbrot set with respect to $\backslash gamma$ 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 $\backslash gamma$ 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 $\backslash gamma$, it can be shown that $\backslash mathcal\; M$ has $z=0$ and one other point as unstable (repelling) fixed points, and $z=\backslash infty$ as an attracting fixed point.

Moreover, the map $\{\backslash mathcal\{M\}\}^3$ has three attracting fixed points.

A Douady rabbit consists of the three attracting fixed points $z\_1$, $z\_2$, and $z\_3$ and their basins of attraction.

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

:$\backslash begin\{align\}$

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}

The red, green, and yellow points lie in the basins $B(z\_1)$, $B(z\_2)$, and $B(z\_3)$ of $\{\backslash mathcal\; M\}^3$, respectively.

The white points lie in the basin $B(\backslash infty)$ of $\backslash mathcal\; M$.

The action of $\backslash mathcal\; M$ on these fixed points is given by the relations $\{\backslash mathcal\; M\}z\_1=z\_2$, $\{\backslash mathcal\; M\}z\_2=z\_3$, and $\{\backslash mathcal\; M\}z\_3=z\_1$.

Corresponding to these relations there are the results

:$\backslash begin\{align\}$

{\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}

File:ColorDouadyRabbit1.jpg

As a second example, Figure 5 shows a Douady rabbit when $\backslash gamma=2-\backslash gamma\_D=-.55268+.959456i$, a point in the eleven-o'clock sprout on the left disk ($\backslash mu$ is invariant under this transformation).

This rabbit is more symmetrical in the plane.

The period-three fixed points then are located at

:$\backslash begin\{align\}$

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}

The repelling fixed points of $\backslash mathcal\; M$ itself are located at $z=0$ and $z=1.450795\; +\; 0.7825835i$.

The three major lobes on the left, which contain the period-three fixed points $z\_1$,$z\_2$, and $z\_3$, meet at the fixed point $z=0$, and their counterparts on the right meet at the point $z=1$.

It can be shown that the effect of $\backslash mathcal\; M$ on points near the origin consists of a counterclockwise rotation about the origin of $\backslash arg(\backslash gamma)$, or very nearly $120^\backslash circ$, followed by scaling (dilation) by a factor of $|\backslash gamma|=1.1072538$.

File:ColorDouadyRabbit2.jpg

A twisted rabbit{{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}} is the composition of a rabbit polynomial with $n$ powers of Dehn twists about its ears.{{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}}

The corabbit is the symmetrical image of the rabbit. Here parameter $c\; \backslash approx\; -0.1226\; -0.7449i$. 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 $c\; =\; -0.123\; +\; 0.745i$ and a cross-section in the $xy$ plane. The Douady rabbit is visible in the cross-section.

400 px

400 px

A small embedded homeomorphic copy of rabbit in the center of a Julia set{{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}}

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.[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}}

Parabolic Julia set for internal angle 1 over 3.png|Fat rabbit

Parabolic chessboard for internal angle 1 over 3.png|Parabolic chessboard

In general, the rabbit for the $period-(n+1)$th bulb of the main cardioid will have $n$ ears{{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}} For example, a period four bulb rabbit has three ears.

Perturbed rabbit{{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}}

Perturbated Rabbit Julia set.png|Perturbed rabbit

PerturbatedRabbitJuliaSetZoom.png|Perturbed rabbit zoom

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 numbers that usually are not given by a formula (these are called topological polynomials):{{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}}

• 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{{Cite arXiv |author=Laurent Bartholdi |author2=Volodymyr Nekrashevych |eprint=math/0510082v3 |title=Thurston equivalence of topological polynomials|date=2005}} 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.{{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}}

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 spine

File:Julia set with 3 external rays.svg|With external rays

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

• Dragon curve
• Herman ring
• Siegel disc

• {{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