# Fractals/Iterations in the complex plane/jlamination Editor's note This book is still under development. Please help us

Lamination is a tool ( model) for investigating dynamics of polynomials. 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

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

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 ).

## Image

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

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

The lamination L gives :

• a combinatorial description of the dynamics of quadratic map. because action of doubling map on the unit circle is a model of action of complex polynomial on complex plane
• exact topological structure of Julia sets  = 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 :

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

?????

## Definition

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 

## Notation

• $\sigma _{d}$  is a map, in case of d=2 it is period doubling map
• chord = leaf = continuous path on the unit disc identifying (connecting) two points on the unit circle
• Major leaf is:
• A leaf of maximal length in a lamination 
• closest in length to 1/2
• Minor leaf is the image of Major Leaf
• critical chord: a $\sigma _{d}$ -critical chord is a chord of $D$ , whose endpoints map to the same point under $\sigma _{d}$ 
• pullback = a pullback process = backward iteration
• Minor tags of dendritic quadratic polynomials = Let pc = z^2 + c be a dendritic quadratic polynomial; the convex hull Gc of all points a ∈ S^1 such that φ(a) = c is called the minor tag of pc .

## Properities of lamination

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

"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 ."

## Tools

Tools used to study dynamics of lamiantions :

• Central Strip Lemma 

## Drawing lamination

• 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

### The Dendrite Lamination

• The point located at $z={\frac {1-{\sqrt {1-4i}}}{2}}$
• is called
• the main triple point
• the fixed point because it is fixed under the mapping $f(z)=z^{2}+i$
• has external rays at 1/7, 2/7 and at 4/7 which rather than just touching down directly, form an infinite logarithmic spiral around the point before reaching it.
• the central pool has external rays at 1/12 and 7/12
• no triple point will ever be mapped to a pool under φ, and vice versa" 
• we are only concerned with pinch points that are triple points or pools.

Algorithm:

• "We begin with the unit circle, and, as before, add arcs connecting any two points on the circle for which the external rays land at the same point, if that point is either a triple point or a pool.
• Thus, we connect the points 1/7, 2/7 and 4/7 in a triangle, and we connect the points 1/12 and 7/12 in an arc.
• We continue in this manner, drawing more triangles for the triple points, and more arcs for the pools"

Images:

### cut points of order 2

#### period one orbit = fixed point ( Basilica lamination)

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 :

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

"the Basilica has only one kind of pinch point, and for which there are gaps between arcs in the lamination" Will Smith

Algorithm:

• We begin with the unit circle,
• add arc connecting 1/3 and 2/3 ( minor leaf = angles of the wake containing period 2 component of the Mandelbrot set)
• 1/6 and 5/6 and each other pair of rational numbers with the form (3k−1)/(3·2^n) and (3k+1)/(3·2^n) for some integer k, n
• when we have finished, we have produced the invariant lamination for the Basilica

Preriodic points: period one ( repelling = in the Julia set)

• fixed point $\alpha$ . Here one of the fixed points $z=\alpha ={\frac {1-{\sqrt {5}}}{2}}$  is a landing point of two external rays 1/3 and 2/3. These are periodic rays ( preperiod = 0 and period = 2). Note that period of landing point is not equal to period of ray that lands on it
• Point $z=-\alpha$  is a landing point of two rays 1/6 and 5/6. These are preperiodic rays: preperiod =1, period = 2

period 2 ( superattracting = centers of components) These pointa are centers of 2 main components. Their preimages are centers of other components

• the critical point z = 0
• the other the critical value z = -1

z = 0.000000000000000 +0.521555030187677*i has preperiod 3 and period 1. It is the landing point of

• internal ray 1/4 of component with center z=0
• external rays 5/24 (001p10) and 7/24 which have preperiod = 3 and period = 2.

z = 0.000000000000000 -0.521555030187677 i has preperiod 3 and period 1. It is the landing point of

• internal ray 3/4
• extarnal rays 17/24 or 101p10 and 19/24 which have preperiod = 3 and period = 2.

z = 0.334146940762091 +0.378310439392182 i has preperiod 5 and period 1. It is the landing point of

• internal ray 1/8
• extarnal rays The angle 17/96 or 00101p10 and 19/96 have preperiod = 5 and period = 2.

$\Psi (x)={\begin{cases}2x-k&{\text{if }}x\in [k+1/3,k+2/3],k\in \mathbb {Z} \\(x+k+1)/2&{\text{if }}x\in (k-1/3,k+1/3),k\in \mathbb {Z} \end{cases}}$

### cut points of order 3

#### period one orbit = fixed point

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

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

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

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 orbit

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 :

${\mathcal {P}}({\mathcal {O}})=\left\{{\mathcal {A}}_{1},{\mathcal {A}}_{2},{\mathcal {A}}_{3}\right\rbrace =\left\{\left({\frac {74}{511}},{\frac {81}{511}},{\frac {137}{511}}\right),\left({\frac {148}{511}},{\frac {162}{511}},{\frac {274}{511}}\right),\left({\frac {296}{511}},{\frac {324}{511}},{\frac {37}{511}}\right)\right\rbrace$

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 .