# Fractals/Iterations in the complex plane/siegel

"A Siegel disk is a (maximal) domain on which a holomorphic map is conjugated to a rotation, whose angle divided by one turn is called the rotation number." A Cheritat
For Siegel disc parameter c :
* should be on boundary of hyperbolic component of Mandelbrot set ( internal radius = 1)
* internal angle should be irrational number between 0 and 1 .
Because set of irrationals is uncountable so the number of Julia sets with Siegel disc is infinite.

# Definitions

* Critical orbit tending to weakly attracting fixed point.When point c goes from center to boundary along internal ray for irrational internal angle then critical orbit changes from one point thru spiral to circle surranding fixed point alpha.
* Critical Orbit, Inner and outer circle for Golden Mean Quadratic Julia set

## Center

Center of Siegel disc is an irrationally indifferent periodic point.
Mane's theorem :
"... appart from its center, a Siegel disk cannot contain any periodic point, critical point, nor any iterated preimage of a critical or periodic point. On the other hand it can contain an iterated image of a critical point."

## Center component of Julia set

In case of c on boundary of main cardioid center component of Julia set is a component containing Siegel disc ( and its center). Critical orbit is a boundary of Siegel disc and center component. All other components are preimages of this component ( see animated image using inverse iterations ).

## Radius
### Conformal radius

Conformal radius
In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use in complex analysis, in particular in conformal maps and conformal geometry.
Definition Given a simply connected domain D ⊂ C, and a point z ∈ D, by the Riemann mapping theorem there exists a unique conformal map f : D → D onto the unit disk (usually referred to as the uniformizing map) with f(z) = 0 ∈ D and f′(z) ∈ R+. The conformal radius of D from z is then defined as
    $\mathrm {rad} (z,D):={\frac {1}{f'(z)}}\,.$
The simplest example is that the conformal radius of the disk of radius r viewed from its center is also r, shown by the uniformizing map x ↦ x/r. See below for more examples.
One reason for the usefulness of this notion is that it behaves well under conformal maps: if φ : D → D′ is a conformal bijection and z in D, then $\mathrm {rad} (\varphi (z),D')=|\varphi '(z)|\,\mathrm {rad} (z,D)$.
The conformal radius can also be expressed as $\exp(\xi _{x}(x))$ where $\xi _{x}(y)$ is the harmonic extension of $\log(|x-y|)$ from $\partial D$ to $D$.

### Inner radius

Inner radius of Siegel Disc =
* radius of inner circle, where inner circle with center at fixed point is the biggest circle inside Siegel Disc.
* minimal distance between center of Siel Disc and critical orbit
Code for computing internal radius :

#### Maxima CAS code

Here orbit is a list of complex points z.

#### C code
### Outer radius

Outer radius of Siegel Disc = radius of outer circle.
Outer circle with center at fixed point is minimal circle containing Siegel disc.
C code :

## Folding

Folding in scholaredia

## Siegel disk implosion

" ... an arbitrary small change of the multiplier of the Siegel point may lead to an implosion of the Siegel disk - its inner radius collapses to zero "
Examples :
* from Golden Mean Siegel Disc with rotation number = [0;1,1,1,1,1, ...]= 0.618033988957902 to parabolic Julia set with rotation number (internal angle) = 5/8 = [0;1,1,1,1,1]
* SEMI-CONTINUITY OF SIEGEL DISKS UNDER PARABOLIC IMPLOSION : P(n) = [0, 2, 2, n + r] with r = (√5−1)/ 2 , the first three images show the Siegel disk of P(n) for n = 10, 500, 10000, and the last is the virtual Siegel disk of p/q = 2/5 they tend to. Here P(n) > p/q.

# Types

for c :
* on boundary of main cardioid are siegel discs around fixed point alfa
* on boundary of period n component are periodic Siegel discs around n-periodic points
These Siegel Discs are :
* bounded (because the Julia set is bounded)
* not smooth (differentiable) in general

## Around fixed point

* Julia set consist of infinitely many curves bounding open regions
* $f_{c}\,$ maps each region into "larger" one, until the region containing the fixed point is reached
* inside component containing Siegel disc $f_{c}\,$ rotates points on invariant loops around the fixed point
so in other words :
"$K_{c}$ has infinitely many components. One of these contains the alpha fixed point. It is mapped to itself by $f_{c}$. This is the Siegel disk."
"The Siegel disk is one component and it has infinitely many preimages. If you zoom in to z = 0 with large nmax, you shall see that there are two components touching at z=0". ( Wolf Jung )

# Visualisation

Visualisation of dynamical plane :
* critical orbit by forward iteration of critical point
* Julia set by
** boundary of center component by drawing of critical orbit
** whole Julia set by inverse iteration of critical orbit
* interior
** whole interior by average velocity by Chris King
** Siegel disc orbits by forward iteration of point inside or on boundary of Siegel Disc

## Average velocity by Chris King

Discrete Velocity of non-attracting Basins and Petals: Compute, for the points that don't escape, the average discrete velocity on the orbit
* in c
* in octave

## Optimisation

See also general methods

### Trick 1

If point ( z0 or its forward image zn) is inside inner circle then it is interior point.

# Examples
## Methods

* Continued fractions
** t = [3,2,1000,1,.] = 0.28573467254058817026888613062003 . It gives c = -0.096294753554390530825955047963 + 0.648802699422348309293468536773i
* known aproximations of irrational numbers
** 1/pi
** eulers number
** golden ratio
** roots of numbers, like
*** square roots: sqrt(2), sqrt(3), sqrt(5), sqrt(7), sqrt(8), sqrt(15), sqrt(21), sqrt(35), sqrt(49), SQRT(65), sqrt(99), SQRT(100), SQRT(101), SQRT(121), sqrt(200), sqrt(278),
*** cubic roots, cbrt(3),
** logarithms, like $log_{2}3$
* descendants of above numbers:
** like 1/pi
** sum of irrational and rational number : if t is an irrational number and r is a rational the t+r is irrational
* fast converging series

### Sequences

Sequence of Continued fraction:
* [1, 1, 1, . . .]
* [1, 1, 1, 20, 1, . . .]
* [1, 1, 1, 20, 1, 1, 1, 30, 1, . . .]
sequence from Siegel disk to Lea-Fatou flower:
* plain
* digitated
* virtual
* Leau-Fatou flower
| cf(t) | t | c(t) | internal adress | center z | period | inner R | outer R |
| [0; 1, 1, 1,...] | 0.618033988957902 | -0.390540870218399-0.586787907346969i | | -0.368684439039160-0.337745147130762i | 1 | 0.25 | 0.4999 |
| [0;3,2,1000,1...] | .2857346725405882 | 0.113891513213121+0.595978335936124i | | -0.111322907374331+0.487449700270424i | 1 | .1414016645283217 | .5285729063154562 |

## values

* t = 0.3119660900888915 = [0, 3, 4, 1, 6, 1, 1, 65, 7, 1, 2, 63] ; c = -0.01183223669 + 0.63816572702*%i
* t = 0.3572354109849235 = [0, 2, 1, 3, 1, 54, 2, 1, 17, 1, 1, 1, 12, 1, 1, 1, 2, 2] ; c = -0.256625459 + 0.6345309*%i
* t = 0.3573615246360573 = [0, 2, 1, 3, 1, 22, 1, 1, 4, 1, 5, 1, 4, 3, 2, 21] ; c = -0.257341 + 0.634456*%i

## Boundary of main cardioid

* examples by Jay Hill
* Dynamical plane for Golden Mean with Critical orbit and some orbits inside Siegel Disc
* Golden Mean Quadratic Siegel Disc with interior coloured by average discrete velocity on the orbit
* high resolution gray version
* Julia set for Golden Mean made with IIM/J
* animated version - up to level 100
* Julia set of quadratic polynomial with Siegel disk for rotation number [3,2,1000,1...]

### Parametrization

The boundary of main cardioid is described by 2 simultaneous equations :
* describing fixed point ( period 1 ) under f
* describing multiplier ( stability ) of fixed point ( it should be indifferent)
${\begin{cases}z_{1}^{2}+c=z_{1}\\abs(\lambda (z_{1}))=1\end{cases}}$
where :
$z_{1}=\left\{z:f_{c}(z)=z\right\}\,$
$\lambda (z_{1})=2*z_{1}\,$
First equation :
$z_{1}^{2}+c=z_{1}$
$c=-z_{1}^{2}+z_{1}$
Second equation :
$\lambda (z_{1})=1\,$
$Abs(2z_{1})=1\,$
$2z_{1}=e^{2*\Pi *t*i}\,$
$z_{1}=e^{\Pi *t*i}=cos(t)+i*sin(t)\,$
As a result one gets function describing relation between parameter c and internal angle t :
$c=c(t)={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)={\frac {e^{\Pi *t*i}}{2}}-{\frac {e^{2*\Pi *t*i}}{4}}$
It is used for computing :
* c point of boundary of main cardioid
* when t is irrational number then filled Julia set has Siegel disc
One can compute boundary point c
$c=c_{x}+c_{y}*i$
of period 1 hyperbolic component ( main cardioid) for given t using
New parameter can be computed after increasing t using arbitrary step:

### near Liouville numbers

    $t=\sum _{k=1}^{\infty }2^{-k!}=2^{-1}+2^{-2}+2^{-6}+2^{-24}+\ldots ={\frac {12845057}{16777216}}\approx 0.7656250596046448$
t = 0.7656250596046448 gives :
* c = 0.294205040086005 -0.448819580822501*i on the main cardioid
* c = -0.975495621741693 -0.248796172491005*i on the period 2 component
* c = -0.219419079596579 +0.741123657495634*i on the period 3 component along internal ray 1/3
* c = -1.749557112879525 -0.008859942836393*i on the small period 3 cardioid on the main antenna
    $t=\sum _{k=1}^{\infty }4^{-k!}=4^{-1}+4^{-2}+4^{-6}+4^{-24}+\ldots \approx 0.3127441406250035$
    $t=\sum _{k=1}^{\infty }8^{-k!}=8^{-1}+8^{-2}+8^{-6}+8^{-24}+\ldots \approx 0.1406288146972656$
    $t=\sum _{k=1}^{\infty }16^{-k!}=16^{-1}+16^{-2}+16^{-6}+16^{-24}+\ldots \approx 0.06640630960464478$
where ! denotes factorial.
Image :
* DarkHeart by Chris King
Maxima cas code:

### radians

* 1/(2*pi) = 0.1591549430918953
* 2/(2*pi) = 0.3183098861837907
* 3/(2*pi) = 0.477464829275686
* 4/(2*pi) = 0.6366197723675814
* 5/(2*pi) = 0.7957747154594768
* 6/(2*pi) = 0.954929658551372
Images:
* examples by Kerry Mitchell examples by Kerry Mitchell

### the Golden Mean

rotation number t is the Golden Mean ( exactly Golden ratio conjugate, compare Julia set for Golden Mean which is disconnected Julia set
$t={\frac {{\sqrt {5}}-1}{2}}\approx 0.618033988749895$
Continued_fraction expansion :
$t=[0;1,1,1,\dots ]=[1,1,1,\dots ]={\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}$
In Maxima CAS :
so
$c\approx -0.390540870218399-0.586787907346969*i$
Compare with :
* Quadratic Julia sets depicted by combined methods
* Golden mean Siegel disk by Curtis T McMullen
* Siegel Disk Fractal by Jim Muth
* Siegel disk by Davoud Cheraghi
* a better Siegel disk program in Mathematica From Roger Bagula
* Xander's image
* Images by Arnaud Chéritat
* irrational by Faber

#### Rays landing on the critical point

There are 2 rays landing on the critical point z=0 with angles:
Here
where the powers of 2 form the Fibonacci sequence :
These rays are preimages of ray $\omega$ which land on the critical value.

### digitated

algorithm goes like this :
* start with rational number p/q ( q is a number of digits
* find continous fraction of p/q using for example : wolfram alpha
* lets choose 2/7 = [0; 3, 2]
* for n from 0 to 10 compute rotation number = t(n) = [0;3, 2 , 10^n, golden_ratio]
* for each t compute multiplier m = e^(2*pi*i*t) and c = (m*(1-m/2))/2,
* for each c draw critical orbit

#### 1/2

* Siegel disk for c = -0.749998153581339 +0.001569040474910 i
* Infolding Siegel disk
* Animation with external rays. Last frame show parabolic Julia set
using quad double precision :

#### 1/3

The result :
quad precision (t and c ):

#### 9/13

Example from scholarpedia
result :
These Siegel disks have 13 digits. SO it will be near t=n/13, probably :
this number is irreducible and it's decimal expansion has period 6
Numerical computations goes near :
It is irreducible fraction . It's decimal expansion has period 2
Important numbers :

#### QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIES

Example by XAVIER BUFF AND ARNAUD CHERITAT

#### 2/7

* finite continued fractions aproximation to [0;1,1,1,....
* Critical Orbit, Inner and outer circle of [0;3,2,1000,1...] Julia set
* Julia set [0;3,2,1000,1... ]
* Critical orbit
rotation number t is :
$t=[3,2,1000,1,...]=[0;3,2,1000,1\dots ]={\cfrac {1}{3+{\cfrac {1}{2+{\cfrac {1}{1000+{\cfrac {1}{1+\ddots }}}}}}}}$
In Maxima CAS one can compute it :
So
How one can find limit of [3,2,1000,1,...] ?
Here is explanation of Bill Wood
"I don't know if Maxima knows much about the algebra of continued fractions, but it can be of some help hacking out the manipulation details of a derivation. A most useful fact is that
provided the continued fraction converges. If we apply that three times by hand to [3, 2, 1000, 1, 1, ...] we obtain
Now it is known that [1, 1, ...] converges to the Golden Ratio = (1+sqrt(5))/2. So now we can use Maxima as follows:
You set a to [0, 3, 2, 1000, 1, 1, ...], which by our useful fact must be the reciprocal of [3, 2, 1000, 1, 1, ...], and indeed the reciprocal of 3.499750279196346 is 0.2857346725405882, which is what your float(t) evaluates to, so we seem to get consistent results.
If all of the continued fractions for the rotation numbers exhibited on the link you provided do end up repeating 1 forever then the method I used above can be used to determine their limits as ratios of linear expressions in sqrt(5)." Bill Wood
Check fraction:
and result :
or using quad double precision ( qd library) :

#### 2/9
# Questions

* is it possible to find function which maps circle to Siegel disc ?
* what happens between virtual Siegel disk and parabolic critical orbit ?

## which external rays ( parameter rays) land on the Siegel points c ? Rays for irrational angle ?

What are the angles of external rays that land on the Cremer and Siegel parameter points c ?
Those that do not belong to any closed wake. They are irrational and I guess there is not much more known. You can approximate them with angles of suitable wakes ...
This something analogous to the construction of the middle-1/3 standard Cantor set by removing open intervals successively.
* Start with [0,1].
* Remove (1/3, 2/3).
* Remove (1/7, 2/7) and (5/7, 6/7)
There is a Cantor set of angles remaining, which are the angles of all rays landing at the main cardioid. The rational angles belong to roots and the irrational angles to Siegel and Cremer parameters. Moreover, each rational angle is a boundary point of an interval removed after finitely many steps. So in the following construction of removing closed intervals, you do not get a Cantor set, and only the irrational angles remain:
* Start with [0,1].
* Remove [1/3, 2/3].
* Remove [1/7, 2/7] and [5/7, 6/7]

# Further reading

* cylinder reormalization
* Arnaud Chéritat : presentation Rome, Avril 2009 : Siegel disks
* Boundaries of Siegel disks - numerical studies of their dynamics and regularity by Rafael de la Llave, Nikola P. Petrov
* Fixed points and circle maps by Ricardo Pérez-Marco
* fractint- Julia Sets, Orbits,Siegel Disks
* youtube: Collapse of a Siegel Disk Gary by Welz
* small divisors
** video : john-milnor small-denominators
** Chronology of the KAM Theorem and the Outer Billiards Map

# References

* ↑ About Zhang’s premodels for Siegel disks of quadratic rational maps. by Arnaud Cheritat CNRS, Univ. Toulouse Feb. 2011
* ↑ wikipedia : siegel disc
* ↑ encyclopedia of math : Siegel disc ( see discrete case )
* ↑ More Fun With Irrational Internal Angles by Faber McMullen
* ↑ wikipedia : Irrational number
* ↑ Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009
* ↑ Davoud Cheraghi home page
* ↑ wikipedia : Conformal radius
* ↑ scholarpedia : Quadratic Siegel disks
* ↑ scholarpedia : Quadratic_Siegel_disks
* ↑ Image InfoldingSiegelDisk.gif from Scholarpedia
* ↑ Computability of Julia sets by Mark Braverman, Michael Yampolsky
* ↑ SEMI-CONTINUITY OF SIEGEL DISKS UNDER PARABOLIC IMPLOSION by ARNAUD CHERITAT
* ↑ Fractal Geometry Mathematical Foundations and Applications, 2nd Edition Kenneth Falconer
* ↑ A tutorial on the visualization of forward orbits associated with Siegel disks in the quadratic Julia sets by G.Todd Miller
* ↑ Combined Methods of Depicting Julia Sets by Chris King
* ↑ More Fun With Irrational Internal Angles by Faber
* ↑ Irrationality of Fast Converging Series of Rational Numbers By Daniel Duverney
* ↑ NON-COMPUTABLE JULIA SETS by M. BRAVERMAN, M. YAMPOLSKY
* ↑ Mandelbrot Sequences and Orbits by Stefan Forcey
* ↑ Jay Hill : JuliaSetsOrbitsSiegelDisks.txt
* ↑ multiplier in wikipedia
* ↑ Golden ratio at wikipedia
* ↑ Golden Ratio Julia Set video by fractal
* ↑ wikipedia : Continued_fraction
* ↑ Combined Methods of Depicting Julia Sets by Chris King
* ↑ Golden mean Siegel disk by Curtis T McMullen
* ↑ Siegel disc by Jim Muth
* ↑ Siegel discs by Davoud Cheraghi
* ↑ a bettter Siegel disk program in Mathematica From Roger Bagula
* ↑ Xander's blog
* ↑ Galerie II : Dynamique holomorphe et analyse complexe by Arnaud Chéritat
* ↑ OLD AND NEW ON QUADRATIC SIEGEL DISKS by SAEED ZAKERI
* ↑ Rotation Sets and Polynomial Dynamics by Zakeri
* ↑ integer sequence id=A000045
* ↑ wolframalpha : continued+fraction[0;2,1,GoldenRatio]
* ↑ Infolding Siegel Disk by Arnaud Chéritat
* ↑ wolfram alpha : 9/13
* ↑ wolfram alpha : 38/55
* ↑ SIEGEL DISKS WITH SMOOTH BOUNDARIES ́ ARTUR AVILA, XAVIER BUFF, AND ARNAUD CHERITAT
* ↑ Some examples of quadratic polynomial Siegel disks by Davoud Cheraghi
* ↑ wolfram alpha : 2/7
* ↑ An Introduction To Small Divisors by S. Marmi
* Local connectivity of some Julia sets containing a circle with an irrational rotation. / Petersen, Carsten Lunde. In: Acta Mathematica, Vol. 177, No. 2, 1996, p. 163-224.
* Building blocks for quadratic Julia sets. Article in Transactions of the American Mathematical Society 351(3):1171-1201 · January 1999 DOI: 10.1090/S0002-9947-99-02346-6