Fractals/Iterations in the complex plane/jlamination

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

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 :

• 
  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

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 ).

partition of the dynamic plane by dynamic rays

The partition is formed by the dynamic rays at angles $\theta /2$ and $(\theta +1)/2$, 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: $\theta /2$ and $(\theta +1)/2$ (the two inverse images of $\theta$ 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

• 
  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

"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

The 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

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

• Drawlam : program for rendering laminations by Clinton P. Curry. This program is licensed under a modified BSD-style license. It uses input file or reads from console.
  • bitbucket official repo
  • gitlab unofficial repo
• 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

• c = i
• 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 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"
• 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 ${\frac {k}{12*2n}}$ 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 ${\frac {k}{7*2n}}$ 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:

• Lamination for z2 + i from galley by Curtis T McMullen

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 :

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

• 
  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 ${\frac {1}{3}}$ associated with basilica Julia set $f_{c}(z)=z^{2}-1$
• 
  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 ${\frac {3k-1}{3*2^{n}}},{\frac {3k+1}{3*2^{n}}}$ 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 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.

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

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 :

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

• 
  Paritition of dynamic plane : Douady rabbit Julia set and external rays landing on fixed point $\alpha _{c}\,$
• 
  Orbit portrait made in hyperbolic way. Paritition of the circle by period 3 orbit ( angles under doubling map)
• 
  Lamination $L_{\frac {1}{7}}$ associated with rabbit Julia set

period 2 orbit

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

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

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 :

${\mathcal {P}}({\mathcal {O}})=\left\{{\mathcal {A}}_{1},{\mathcal {A}}_{2}\right\rbrace =\left\{\left({\frac {22}{63}},{\frac {25}{63}},{\frac {37}{63}}\right),\left({\frac {44}{63}},{\frac {50}{63}},{\frac {11}{63}}\right),\right\rbrace$

• 
  Paritition of parameter plane by rays landing on root point with internal adress 1-2-6
• 
  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

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 .

• 
  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

period one orbit = fixed point

• 
  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

• 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
• 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

• lamination
• Lamination of Mandelbrot set
• Douady rabbit
• orbit portrait
• Roots and parabolic fixed points : external rays
• tessallation of the (unit) circle, tiling
• non-crossing circle paritition
• spiral of spirals by benice
• self-similar groups

References

• 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