Douady rabbit explained
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.
Background
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
zn+1=lMzn:=\gammazn\left(1-zn\right).
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
.
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
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
\gamma=\gammaD=2.55268-0.959456i
, 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
\begin{align}
z1&=0.499997032420304-(1.221880225696050 x 10-6)i{ }{ }{(red)
},\\z_2 &= 0.638169999974373 - (0.239864000011495)i,\\z_3 &= 0.799901291393262 - (0.107547238170383)i.\end
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
\begin{align}
{lM}B(z1)&=B(z2){ }{or
} \subseteq,\\B(z_2)&=B(z_3) \subseteq,\\B(z_3)&=B(z_1) \subseteq.\end
As a second example, Figure 5 shows a Douady rabbit when
\gamma=2-\gammaD=-.55268+.959456i
, 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
\begin{align}
z1&=0.500003730675024+(6.968273875812428 x 10-6)i{ }{ }({red
}),\\z_2&=-0.138169999969259 + (0.239864000061970)i,\\z_3&= -0.238618870661709 - (0.264884797354373)i .\end
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
.
Variants
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.
3D
The Julia set has no direct analog in three dimensions.
4D
A quaternion Julia set with parameters
and a
cross-section in the
plane. The Douady rabbit is visible in the cross-section.
Embedded
A small embedded homeomorphic copy of rabbit in the center of a Julia set[3]
Fat
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
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 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
References
- Web site: A Geometric Solution to the Twisted Rabbit Problem by Jim Belk, University of St Andrews . 2022-05-03 . 2022-11-01 . https://web.archive.org/web/20221101023328/https://e.math.cornell.edu/people/belk/talkslides/TwistedRabbitTalkMichigan.pdf . live.
- Laurent Bartholdi . Volodymyr Nekrashevych . math/0510082 . Thurston equivalence of topological polynomials. Acta Mathematica . 2006 . 197 . 1–51 . 10.1007/s11511-006-0007-3.
- Web site: Period-n Rabbit Renormalization. 'Rabbit's show' by Evgeny Demidov . 2022-05-03 . 2022-05-03 . https://web.archive.org/web/20220503170632/https://www.ibiblio.org/e-notes/MSet/quadrat.htm . live.
- http://www.math.nagoya-u.ac.jp/~kawahira/works/rims0402.pdf Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
- Web site: Twisted Three-Eared Rabbits: Identifying Topological Quadratics Up To Thurston Equivalence by Adam Chodof . 2022-05-03 . 2022-05-03 . https://web.archive.org/web/20220503093346/https://e.math.cornell.edu/people/belk/projects/AdamChodoff.pdf . live.
- Web site: Recent Research Papers (Only since 1999) Robert L. Devaney: Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets . 2020-04-07 . 2019-10-23 . https://web.archive.org/web/20191023110636/http://math.bu.edu/people/bob/papers.html . live.
- Web site: Polynomials, dynamics, and trees by Becca Winarski . 2022-05-08 . 2022-11-01 . https://web.archive.org/web/20221101023328/http://www.math.lsa.umich.edu/mathclub/winter2020/042320.pdf . live.
- Laurent Bartholdi . Volodymyr Nekrashevych . math/0510082v3 . Thurston equivalence of topological polynomials. 2005.
- 1906.07680v1 . Recognizing Topological Polynomials by Lifting Trees . James Belk . Justin Lanier . Dan Margalit . Rebecca R. Winarski. 2019 . math.DS.
External links
