Fractals/Iterations in the complex plane/jlamination

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Here doubling map is used to analyze dynamics of complex quadratic polynomials. It is dynamical system easier to analyze then complex quadratic map.


Periodic orbits of angles under doubling map[edit]

Note that here chord joining 2 points z1 and z2 on unit circle means that z_2 = z_1^2. It does not mean that these points are landing points of the same ray.

Some orbits do not cross :

but some do :

Orbit portraits[edit]

An orbit portrait can be in two forms:

  • list of lists of numbers (common fractions with even denominator)
  • image showing rays landing on periodic z points (= partition of dynamic plane)

Note that :

  • here chord joining 2 points z1 and z2 on unit circle means that these points are landing points of the same ray. It does not mean that z_2 = z_1^2.
  • An orbit portrait is a portrait of orbit, which is periodic under complex quadratic map.
  • The Julia set has many periodic orbits so it also hase many orbit portraits
  • An orbit portrait is combinatorial description of orbit
  • (Douady and Hubbard). Every repelling and parabolic periodic point of a quadratic polynomial fc is the landing point of an external ray with rational angle. Conversely, every external ray with rational angle lands either at a periodic or preperiodic point in J(fc ).[1]


Image[edit]

Image can be made in three forms :

  • image of dynamic plane with Julia set and external rays landing on periodic orbit
  • sketch of above image made in :
    • standard way : points of orbit are drawn inside unit circle and rays are made by lines joining angle ( point on unit circle) and point of orbit. It looks like sketch of above image
    • hyperbolic way : points are on unit circle and here chord joining 2 points z1 and z2 on unit circle means that these points are landing points of the same ray. It does not mean that z_2 = z_1^2. Chord is drawn using arc ( part of orthogonal circle ).

Lamination of Julia sets[edit]

"Laminations were introduced to the context of polynomial dynamics in the early 1980’s by Thurston"[2] Are used to show the landing pattern of external rays.



The lamination L gives :

  • a combinatorial description of the dynamics of quadratic map.[3] because action of doubling map on the unit circle is a model of action of complex polynomial on complex plane[4]
  • exact topological structure of Julia sets [5] = topological model for Julia set
  • the model of ray portraits. The external rays for angles in a lamination land at "cut points" of the Julia set / Mandelbrot set.

Note that here chord joining 2 points z1 and z2 on unit circle means that these points are landing points of the same ray. It does not mean that z_2 = z_1^2.


For a quadratic polynomials initial set has a form :[6]


\left \{    
\theta , \theta  +\frac{1}{2}
 \right \rbrace


?????

Definition[edit]

Laminations of the unit disk in the plane is a closed collection of chords (leaves, arcs ) inside the unit disk

quadratic laminations = those that remain invariant under the angle doubling map [7]

Notation[edit]

  • chord = leaf = continuous path on the unit disc identifying (connecting) two points on the unit circle
  • pullback = a pullback process = backward iteration

Properities of lamination[edit]

Lamination must satisfy the following rules :

  • leaves do not cross, although thay may share endpoints
  • lamination is forward and backward invariant (under doubling map)


Invariance of lamination[edit]

"Invariance of a lamination L in the unit disc means that:

  • whenever there is a leaf of L joining z_1 and z_2 , there is also a leaf of L joining z_1^2 and z_2^2
  • whenever there is a chord joining z_1 and z_2 , there are points \pm z_3^2 and \pm z_4^2 with z_3^2 = z_1 and z_4^2 = z_2 , and such that there are leaves of L joining z3 to z4 , and −z3 to −z4 ."[8]

Tools[edit]

Tools used to study dynamics of lamiantions :

  • Central Strip Lemma [9]

Drawing lamination[edit]

  • Drawlam : program for rendering laminations by Clinton P. Curry [10]. This program is licensed under a modified BSD-style license. It uses input file or reads from console.
  • Invariant lamination calculator Java applet by Danny Calegari. It computes the invariant lamination for a connected Julia set on the boundary of the Mandelbrot set with variable external angle. With Java src code
  • lamination by Danny Calegari. Cpp program for X11 using uses standard Xlib stuff. Source code is released under the terms of the GNU GPL. This program is a toy to do experiments with laminations of the circle. Represents it symbolically and pictorially. It needs only one input : the size of the lamination ( the number of endpoints of polygons). This set of endpoints is enumerated from 0 to size-1 in anticlockwise order. For each endpoint, the nextleaf points to the adjacent endpoint in the anticlockwise direction.

I have changed in main.cc :

#include <math.h>
#include <iostream> // I have removed .h
#include <stdlib.h>
#include "graphics.cc"
using namespace std; // added because : main.cc:101: error: ‘cout’ was not declared in this scope

and then in program directory :

make
./lamiantion

Examples[edit]

cut points of order 2[edit]

period one orbit = fixed point[edit]

For complex quadratic polynomials f_c(z) for all parameters c in wake bounded by rays 1/3 and 2/3 there is repelling fixed point with orbit portrait :


{\mathcal P}({\mathcal O}) = \left \{   {\mathcal A}_1  \right \rbrace = 
\left \{    
\left(\frac{1}{3},\frac{2}{3} \right)       
 \right \rbrace

cut points of order 3[edit]

period one orbit = fixed point[edit]

Orbit under quadratic map consists of one ( fixed point) :


{\mathcal O} = \left \{ z_1   \right \rbrace = \left \{ \alpha_c \right \rbrace

This point is a landing point of 3 external rays and has orbit portrait :


{\mathcal P}({\mathcal O}) = \left \{   {\mathcal A}_1  \right \rbrace = 
\left \{    
\left(\frac{1}{7},\frac{2}{7},\frac{4}{7} \right)       
 \right \rbrace

period 2 orbit[edit]

c is a root point of Mandelbrot set between period 2 and 6 components :[11]

c= -1 + \frac{1}{4} e^{2\pi i \frac{2}{3}} \in \partial M

with internal address 1-2-6.

Six periodic cycle of rays is landing on two-periodic parabolic orbit :


{\mathcal O} = \left \{ z_{2,1} , z_{2,2}   \right \rbrace

where :


z_{2,1} =  -\frac{1}{2} + \frac{1}{2} \sqrt{1 -  e^{2\pi i \frac{2}{3}}}


z_{2,2} =  -\frac{1}{2} - \frac{1}{2} \sqrt{1 -  e^{2\pi i \frac{2}{3}}}

with orbit portrait :


{\mathcal P}({\mathcal O}) = 
\left \{   {\mathcal A}_1, {\mathcal A}_2 \right \rbrace = 

\left \{    
\left(\frac{22}{63},\frac{25}{63},\frac{37}{63} \right) ,        
 \left(\frac{44}{63},\frac{50}{63},\frac{11}{63} \right) ,
\right \rbrace

period 3 orbit[edit]

Parameter c is a center of period 9 hyperbolic component of Mandelbrot set

c= -0.03111+0.79111*i

Orbit under quadratic map consists of 3 points :


{\mathcal O} = \left \{ z_{3,1} , z_{3,2}, z_{3,1}   \right \rbrace



orbit portrait associated with parabolic period 3 orbit {\mathcal O} is :[12]

{\mathcal P}({\mathcal O}) = \left \{   {\mathcal A}_1, {\mathcal A}_2, {\mathcal A}_3  \right \rbrace= 

\left \{    
\left(\frac{74}{511},\frac{81}{511},\frac{137}{511} \right) ,        
 \left(\frac{148}{511},\frac{162}{511},\frac{274}{511} \right) ,
 \left(\frac{296}{511},\frac{324}{511},\frac{37}{511} \right) 
\right \rbrace

Valence = 3 rays per orbit point ( = each point is a landing point of 3 external rays )

Rays for above angles land on points of that orbit .

Questions[edit]

  • How to comput orbit portraits ?
  • How orbit portrait changes when I move inside Mandelbrot set ?


See also[edit]

References[edit]

Volume 1732, 2000, DOI: 10.1007/BFb0103999. Springer-Verlag, Berlin-Heidelberg-NewYork 2000