# images

* rays landing on the alpha fixed point
* biaccesible
* triaccesible
* rays landing on the critical point and value
* 1/4
* 1/6
* 9/56
* 129/16256

# Theory

Let $\Psi _{c}\,$ be the map ( Conformal map, isomorphism) from the complement (exterior) of the closed unit disk ${\overline {\mathbb {D} }}$ to the complement of the filled Julia set $\ K_{c}$.
    $\Psi _{c}:{\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\to {\hat {\mathbb {C} }}\setminus K_{c}$
where ${\hat {\mathbb {C} }}$ denotes the Riemann sphere (extended complex plane.
Let $\Psi _{c}=\Phi _{c}^{-1}\,$ denote
* the inverse Boettcher map
* the Riemann map
The external ray of angle $\theta \,$ noted as ${\mathcal {R}}_{\theta }^{K}$is:
* the image under $\Psi _{c}\,$ of straight lines ${\mathcal {R}}_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\}$
    ${\mathcal {R}}_{\theta }^{K}=\Psi _{c}({\mathcal {R}}_{\theta })$
* set of points of exterior of filled-in Julia set with the same external angle $\theta$
    ${\mathcal {R}}_{\theta }^{K}=\{z\in {\hat {\mathbb {C} }}\setminus K_{c}:\arg(\Phi _{c}(z))=\theta \}$

# Important questions
## What are the unique rays?

* "The ray, t, which lands on the critical point (or on the root point of the bulb containing the critical point) of a Julia set is unique to that Julia set, so is a way of identifying the Julia set. So I call the julia set Jc by the name t. In the rest, I can only deal with julia sets where t=p/q is rational."
* Critically preperiodic polynomials are typically parameterized by the angle $\theta$ of the external ray landing at the critical value

## How to aproximate external rays?

* binary decomposition method for Julia set ( BDM/J)
* field lines of the scalar field ( potential)
* SAM = Stripe Average Method is basically a cheap way to calculate an angle.
* boundaries of BDM
* SAM
In BDM images the external rays of angles (measured in turns):
$angle=(k/2^{n})~~{\mbox{mod }}~1\,$
can be seen as borders of subsets.

## when two external dynamic rays land at the same point

Questions:
* Q1: Given a point in the Julia set, how many rays are there landing at this point?
* Q2: Given two rays $R_{\theta _{1}}$ and $R_{\theta _{2}}$, under what conditions do they land at the same point?
** When external rays land at the same point
** The landing equivalence = Which rays land at the same point?
** numerical criterion = they have the same ( numerically ) landing point
* Q3: Which rays support the same Fatou component?
Theorem 1.2 Let f be a polynomial of degree d ≥ 2 with locally connected Julia set. If two angles $\theta _{1}$ and $\theta _{2}$ have the same itinerary with respect to a critical portrait, then the landing points of external rays $R_{\theta _{1}}$ and $R_{\theta _{2}}$ either coincide or belong to the boundary of a Fatou domain, which is eventually iterated onto a Siegel disk.

## How to find angle of the external ray that land on the critical values z= c ?

That depends on the situation.
* There are combinatorial methods when the parameter c is a Misiurewicz point, for example.
* Most often you construct the angle first and then determine the parameter.
* But if you want to read an angle from the Julia set or from the Hubbard tree, you can proceed according to Demo 4 pages 1 & 2 from program Mandel

## Constructing the spine of filled Julia set

Algorithm for constructiong the spine is described by A. Douady
* join $-\beta \,$ and $\beta \,$,
* (to do )

## What means to draw external ray ?

It means:
* calculate (approximate) some points of ray
* join points by line segments
This will give an approximation of ray ${\mathcal {R}}$.

# Drawing dynamic external ray

Algorithms
* Newton method
* backwards iteration
* inverse Boettcher map
* trace an external ray ( curve) for angle t in turns

## backwards iteration

This method has been used by several people and proved by Thierry Bousch.
Code in c++ by Wolf Jung can be found in procedure QmnPlane::backray in file qmnplane.cpp ( see source code of the program Mandel ).
* Ray for periodic angle ( simplest case )
It will be explained by an example :
First choose external angle $t\,$ (in turns). External angle for periodic ray is a rational number.
    $t={\frac {1}{3}}\,$
Compute period of external angle under doubling map.
Because "1/3 doubled gives 2/3 and 2/3 doubled gives 4/3, which is congruent to 1/3"
    $1\equiv 2{\pmod {3}}.\,$
or
    ${\frac {2}{3}}\equiv {\frac {1}{3}}{\pmod {1}}.\,$
so external angle $t={\frac {1}{3}}\,$ has period 2 under doubling map.
Start with 2 points near infinity (in conjugate plane):
on ray 1/3 is a point $w=R*e^{2\pi i/3}\,$
on ray 2/3 is a point $w=R*e^{4\pi i/3}\,$.
Near infinity $z=w\,$ so one can swith to dynamical plane ( Boettcher conjugation )
Backward iteration (with proper chose from two possibilities) of point on ray 1/3 goes to ray 2/3, back to 1/3 and so on.
In C it is :
or in Maxima CAS :
One has to divide set of points into 2 subsets ( 2 rays). Draw one of these 2 sets and join the points. It will be an approximation of ray.
* Ray for preperiodic angle ( to do )
Run it :
And output :
Point on the ray is moving backwards:
* very fast when it it far from Julia set
* very slow near Julia set ( after 50 iterations distance between points = 0 pixels )
See example computing ( here pixel size = 0.003 ) :
One can choose points which differ by pixel size :
One can see that moving from pixel 12 to 11 near alfa needs 27 000 iterations. Computing points up to 1 pixel near alfa needs : 2m1.236s

## Drawing dynamic external ray using inverse Boettcher map by Curtis McMullen

This method is based on C program by Curtis McMullen and its Pascal version by Matjaz Erat
There are 2 planes here :
* w-plane ( or f0 plane )
* z-plane ( dynamic plane of fc plane )
Method consist of 3 big steps :
* compute some w-points of external ray of circle for angle $\theta \,$ and various radii (rasterisation)
    $w_{0}=R*e^{2\pi it}\,$ where $1<R<\infty \,$
* map w-points to z-point using inverse Boettcher map $\Psi _{c}=\Phi _{c}^{-1}\,$
    $z_{0}=\Psi _{c}(w_{0})=f_{c}^{-n}(w_{0}^{2^{n}})\,$
* draw z-points ( and connect them using segments ( line segment is a part of a line that is bounded by two distinct end points )
First and last steps are easy, but second is not so needs more explanation.

### Rasterisation

For given external ray in $f_{0}\,$ plane each point of ray has :
* constant value $t\,$ ( external angle in turns )
* variable radius $R\,$
so $w\,$ points of ray are parametrised by radius $R\,$ and can be computed using exponential form of complex numbers :
    $w=F_{t}(R)=R*e^{2*\Pi *i*t}\,$
One can go along ray using linear scale :
It gives some w points with equal distance between them.
Another method is to use nonlinera scale.
To do it we introduce floating point exponent $r\,$ such that :
    $R=2^{r}\,$
and
    $w=G_{t}(r)=2^{r}*e^{2*\Pi *i*t}\,$
To compute some w points of external ray in $f_{0}\,$ plane for angle $t\,$ use such Maxima code :
In this method distance between points is not equal but inversely proportional to distance to boundary of filled Julia set.
It is good because here ray has greater curvature so curve will be more smooth.

### Mapping

Mapping points from $f_{0}\,$-plane to $f_{c}\,$-plane consist of 4 minor steps :
* forward iteration in $f_{0}\,$ plane
    $w_{n}=w_{0}^{2^{n}}=R^{2^{n}}*e^{2\pi i2^{n}t}\,$
until $w_{n}\,$ is near infinity
    $abs(w_{n})>R_{limit}\,$
* switching plane ( from $f_{0}\,$ to $f_{c}\,$)
    $z_{n}=w_{n}\,$
( because here, near infinity : $w=z\,$)
* backward iteration in $f_{c}\,$ plane the same ( $n\,$) number of iterations
* last point $z_{0}\,$ is on our external ray
1,2 and 4 minor steps are easy. Third is not.
    $z_{n-1}=f_{c}^{-1}(z_{n})={\sqrt {z_{n}-c}}\,$
Backward iteration uses square root of complex number. It is 2-valued functions so backward iteration gives binary tree.
One can't choose good path in such tree without extre informations. To solve it we will use 2 things :
* equicontinuity of basin of attraction of infinity
* conjugacy between $f_{0}\,$ and $f_{c}\,$ planes

### Equicontinuity of basin of attraction of infinity

Basin of attraction of infinity ( complement of filled-in Julia set) contains all points which tends to infinity under forward iteration.
$A_{f_{c}}(\infty )\ {\overset {\underset {\mathrm {def} }}{=}}\ \{z\in \mathbb {C} :f_{c}^{(k)}(z)\to \infty \ as\ k\to \infty \}.$
Infinity is superattracting fixed point and orbits of all points have similar behaviour. In other words orbits of 2 points are assumed to stay close if they are close at the beginning.
It is equicontinuity ( compare with normality).
In $f_{c}\,$ plane one can use forward orbit of previous point of ray for computing backward orbit of next point.

### Detailed version of algorithm

* compute first point of ray (start near infinity ang go toward Julia set )
    $w=2^{r}*e^{2\Pi it}\,$ where $r=r_{max}\,$
here one can easily switch planes :
    $z=w\,$
It is our first z-point of ray.
* compute next z-point of ray
** compute next w-point of ray for $r=r*caution\,$
** compute forward iteration of 2 points : previous z-point and actual w-point. Save z-orbit and last w-point
** switch planes and use last w-point as a starting point :$z_{n}=w_{last}\,$
** backward iteration of new $z_{n}\,$ toward new $z_{0}\,$ using forward orbit of previous z point
** $z_{0}\,$ is our next z point of our ray
* and so on ( next points ) until $r<r_{Min}\,$
Maxima CAS src code

# See also

* parameter external ray
* External rays precision - how important it is

# References

* HAUSDORFF LIMITS OF EXTERNAL RAYS: THE TOPOLOGICAL PICTURE by CARSTEN LUNDE PETERSEN AND SAEED ZAKERI
* ↑ Theory of Matings from Cornell University
* ↑ Criterion for rays landing together by Jinsong Zeng ( arXiv:1503.05931 math.DS
* ↑ THE CLASSIFICATION OF CRITICALLY PREPERIODIC POLYNOMIALS AS DYNAMICAL SYSTEMS by BEN BIELEFELD, YUVAL FISHER, AND JOHN HUBBARD
* ↑ On Post Critically Finite Polynomials Part One: Critical Portraits by Alfredo Poirier
* ↑ A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Ga, USA, 1986.
* ↑ Family of Invariant Cantor Sets as orbits of Differential Equations II: Julia Sets by Yi-Chiuan Chen, Tomoki Kawahira, Juan-Ming Yuan
* ↑ An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
* ↑ Thierry Bousch : De combien tournent les rayons externes? Manuscrit non publié, 1995
* ↑ Program Mandel by Wolf Jung
* ↑ Explanation by Wolf Jung
* ↑ Modular arithmetic in wikipedia
* ↑ Square root of complex number gives 2 values so one has to choose only one. For details see Wolf Jung page
* ↑ c program by Curtis McMullen (quad.c in Julia.tar.gz)
* ↑ Quadratische Polynome by Matjaz Erat
* ↑ wikipedia : Complex_quadratic_polynomial / planes / Dynamical_plane
* ↑ wikipedia : Line segment