Fractals/Iterations in the complex plane/jlamination
Lamination is a tool ( model) for investigating dynamics of polynomials.^{[1]} 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
editPeriodic orbits of angles under doubling map
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 :

Period 2 orbit ( angles under doubling map)

Period 3 orbit ( angles under doubling map)

Period 4 orbit ( angles under doubling map)

Period 5 orbit ( angles under doubling map)

Period 6 orbit ( angles under doubling map)
but some do :

Period 6 orbit of 11/63 under doubling map

Period 9 orbit of 74/511 under doubling map
Orbit portraits
editAn 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 ).^{[2]}
Image
editImage 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 ).
partition of the dynamic plane by dynamic rays
editThe partition is formed by the dynamic rays at angles and , which land together at the critical point.
The paritition:
 the partition used in the definition of the kneading sequence: divide open unit disc ( or Circle group ) S1 into two parts: and (the two inverse images of under angle doubling);
 the open part containing the angle 0 is labeled 0
 the other open part is labeled 1
 the boundary gets the label ⋆
 a corresponding partition of the dynamic plane by dynamic rays, shown here for the example of a Misiurewicz polynomial^{[3]}

1/4

1/6

9/56

129/16256
How to find angle of the dynamic external ray that land on the critical value z = c ?
Lamination of dynamical plane
edit"Laminations were introduced to the context of polynomial dynamics in the early 1980’s by Thurston"^{[4]} Are used to show the landing pattern of external rays.
The lamination L gives :
 a combinatorial description of the dynamics of quadratic map.^{[5]} because action of doubling map on the unit circle is a model of action of complex polynomial on complex plane^{[6]}
 exact topological structure of Julia sets ^{[7]} = 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 :^{[8]}
?????
Definition
editThe lamination consists of:
 the closed unit disk D2
 a hyperbolic arc (or leaf ) connecting pair of points ( angles in turns) on the boundary circle. The external rays ( on the dynamical plane) for these angles land at the same point
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 ^{[9]}
Notation
edit 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 ^{[10]}
 closest in length to 1/2^{[11]}
 Minor leaf is the image of Major Leaf
 critical chord: a critical chord is a chord of , whose endpoints map to the same point under ^{[12]}^{[13]}
 Major leaf is:
 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 .^{[14]}
Properities of lamination
editLamination 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 ."^{[15]}
Tools
editTools used to study dynamics of lamiantions :
 Central Strip Lemma ^{[16]}
Drawing lamination
edit Drawlam : program for rendering laminations by Clinton P. Curry.^{[17]} This program is licensed under a modified BSDstyle 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 size1 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
editThe Dendrite Lamination
edit c = i
 The point located at
 is called
 the main triple point
 the fixed point because it is fixed under the mapping
 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.
 is called
 the central pool point has external rays at 1/12 and 7/12
 no triple point will ever be mapped to a pool under φ, and vice versa" ^{[18]}
 we are only concerned with pinch points that are triple points or pools.
 A point in the dendrite is called a pool if it is the landing point for two external rays, both of whose angles are of the form for some k, n ∈ N, where k ≡ 1 mod 6.
 A point in the dendrite is called a triple point if its removal separates the dendrite into three connected components. Such a point is the landing point for three external rays, whose angles all have of the form for some k, n ∈ N, where k is congruent to 1, 2 or 4, mod 7.
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
editperiod one orbit = fixed point ( Basilica lamination)
editFor 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 :

External rays landing on fixed point alpha

Paritition of the circle by 2 rays landing on period 1 orbit ( angles under doubling map)

quadratic invariant lamination associated with basilica Julia set

Rays landing on cut points
"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 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 . Here one of the fixed points 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 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 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
 external 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.
cut points of order 3
editperiod one orbit = fixed point
editOrbit under quadratic map consists of one ( fixed point) :
This point is a landing point of 3 external rays and has orbit portrait :

Paritition of dynamic plane : Douady rabbit Julia set and external rays landing on fixed point

Orbit portrait made in hyperbolic way. Paritition of the circle by period 3 orbit ( angles under doubling map)

Lamination associated with rabbit Julia set
period 2 orbit
editc is a root point of Mandelbrot set between period 2 and 6 components :^{[19]}
with internal address 126.
Six periodic cycle of rays is landing on twoperiodic parabolic orbit :
where :
with orbit portrait :

Paritition of parameter plane by rays landing on root point with internal adress 126

Paritition of dynamic plane by rays landing on parabolic orbit

Paritition of the circle by period 6 orbit ( angles under doubling map)
period 3 orbit
editParameter 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 :^{[20]}
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 .

Paritition of dynamic plane : Julia set and external rays landing on period 3 orbit

Orbit of 74/511 under doubling map
cut points of order 5
editperiod one orbit = fixed point
edit
cycle 5 superattracting orbit = c is a center of period 5 component orbit : {5/31 , 10/31 , 20/31 , 9/31 , 18/31}

orbit : {5/31 , 10/31 , 20/31 , 9/31 , 18/31}, crossing

orbit of 1/31 under doubling map, non crossing, = {1/31 , 2/31 , 4/31 , 8/31 , 16/31}. Parameter c will be on the parabolic point with internal angle 2/5
Questions
edit How to compute orbit portraits ?
 How orbit portrait changes when I move inside Mandelbrot set ?
 Inside a connected component of the interior of the Mandelbrot set the lamination is the same^{[21]}
 CRITERION FOR RAYS LANDING TOGETHER
 CRITERION FOR RAYS LANDING TOGETHER by JINSONG ZENG on the dynamical plane
 on the parameter plane ( angles of the wake and conjugate angle )
See also
edit lamination
 Lamination of Mandelbrot set
 Douady rabbit
 orbit portrait
 Roots and parabolic fixed points : external rays
 tessallation of the (unit) circle, tiling
 noncrossing circle paritition
 spiral of spirals by benice
 selfsimilar groups
References
edit ↑ LAMINATIONS IN THE LANGUAGE OF LEAVES by ALEXANDER M. BLOKH, DEBRA MIMBS, LEX G. OVERSTEEGEN, AND KIRSTEN I. S. VALKENBURG
 ↑ Towards classification of laminations associated to quadratic polynomials A Dissertation, Presented by Carlos Cabrera
 ↑ Rational parameter rays of the Mandelbrot set by Dierk Schleicher
 ↑ 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. , 20100805 "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
 ↑ TOPOLOGICAL ENTROPY OF QUADRATIC POLYNOMIALS AND DIMENSION OF SECTIONS OF THE MANDELBROT SET by GIULIO TIOZZO
 ↑ The Lavaurs Algorithm for Thurston’s Quadratic Minor Lamination by John C. Mayer
 ↑ Laminational models for some spaces of polynomials of any degree by Alexander Blokh, Lex Oversteegen, Ross Ptacek, Vladlen Timorin
 ↑ Laminations of the Unit Disk and Cubic Julia Sets by John C. Mayer
 ↑ Smart Criticality for Cubic Laminations by Alexander Blokh, Lex Oversteegen, Ross Ptacek , Vladlen Timorin
 ↑ 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 : python program for rendering laminations by Clinton P. Curry
 ↑ ThompsonLike Groups for Dendrite Julia Sets by Will Smith
 ↑ 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
 ↑ math.stackexchange question: arethelaminationsofjuliasetsofthesameperiodthesame
 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. SpringerVerlag, BerlinHeidelbergNewYork 2000