# Fractals/Iterations in the complex plane/jlamination

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 mapEdit

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 portraitsEdit

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]

## ImageEdit

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 setsEdit

"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]

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

?????

## DefinitionEdit

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]

## NotationEdit

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

## Properities of laminationEdit

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 laminationEdit

"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]

## ToolsEdit

Tools used to study dynamics of lamiantions :

• Central Strip Lemma [9]

## Drawing laminationEdit

• 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


# ExamplesEdit

## cut points of order 2Edit

### period one orbit = fixed pointEdit

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 3Edit

### period one orbit = fixed pointEdit

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 orbitEdit

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

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

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 orbitEdit

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 .

# QuestionsEdit

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