# 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.

## Contents

# Periodic orbits of angles under doubling map[edit]

Note that here chord joining 2 points z1 and z2 on unit circle means that . 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 .
- 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 . 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 .

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

?????

## 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 and , there is also a leaf of L joining and
- whenever there is a chord joining and , there are points and with and , 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 for all parameters c in wake bounded by rays 1/3 and 2/3 there is repelling fixed point with orbit portrait :

## cut points of order 3[edit]

### period one orbit = fixed point[edit]

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

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

### period 2 orbit[edit]

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

with internal address 1-2-6.

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

where :

with orbit portrait :

### period 3 orbit[edit]

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

Orbit under quadratic map consists of 3 points :

orbit portrait associated with parabolic period 3 orbit is :^{[12]}

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 .

## cut points of order 5[edit]

### period one orbit = fixed point[edit]

# Questions[edit]

- How to compute orbit portraits ?
- How orbit portrait changes when I move inside Mandelbrot set ?
- CRITERION FOR RAYS LANDING TOGETHER by JINSONG ZENG

# See also[edit]

- lamination
- Lamination of Mandelbrot set
- Douady rabbit
- orbit portrait
- Roots and parabolic fixed points : external rays

# References[edit]

- ↑ Towards classification of laminations associated to quadratic polynomials A Dissertation, Presented by Carlos Cabrera
- ↑ [COMBINATORICS AND TOPOLOGY OF STRAIGHTENING MAPS I : COMPACTNESS AND BIJECTIVITY HIROYUKI INOU AND JAN KIWI]
- ↑ Cubic Critical Portraits and Polynomials with Wandering Gaps Authors: A. Blokh, C. Curry, L. Oversteegen
- ↑ Mayer, J. , 2010-08-05 "Pullback Laminations" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA
- ↑ THE SOLAR JULIA SETS OF BASIC QUADRATIC CREMER POLYNOMIALS by A. BLOKH, X. BUFF, A. CHERITAT, AND L. OVERSTEEGEN
- ↑ CUBIC CRITICAL PORTRAITS AND POLYNOMIALS WITH WANDERING GAPS by ALEXANDER BLOKH, CLINTON CURRY, AND LEX OVERSTEEGEN
- ↑ Houghton, Jeffrey. "Useful Tools in the Study of Laminations" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA, Aug 05, 2010
- ↑ MULTIPLE EQUIVALENT MATINGS WITH THE AEROPLANE POLYNOMIAL byMARY REES
- ↑ Houghton, Jeffrey. "Useful Tools in the Study of Laminations" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA, Aug 05, 2010
- ↑ Drawlam : program for rendering laminations by Clinton P. Curry
- ↑ Trees of visible components in the Mandelbrot set by Virpi K a u k o
- ↑ Boundaries of Bounded Fatou Components of Quadratic Maps Ross Flek and Linda Keen

- Lamination for z2 + i from gallery of images by Curtis T McMullen
- Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set by Karsten Keller. Book from series Lecture Notes in Mathematics

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