This is clone of the original repsitory by Clinton Curry

TOC

• DrawLam.py

DrawLam.py

DrawLam.py is a Python library which makes it easy to draw laminations, including pullback laminations. It is capable of producing PNG and PDF output.

Prerequisites

This library calls upon PyCairo and gmpy for its dark deeds.

In ubuntu:

pip2 install gmpy --user

Usage

DrawLam.py is designed to be used in Python scripts. There are several sample scripts available in the "examples" directory in the source tree. The simplest possible script for drawing the lamination corresponding to the rabbit is given below;

lami3.py

• it takes ten iterative preimages of the (1/7, 2/7) leaf which do not cross the (1/7, 9/14) chord.
• The "True" associated with that leaf indicates that preimages are to be taken in the [1/7, 9/14) interval and the [9/14, 1/7) interval; "False" would give the opposite endpoint arrangement.

from DrawLam import DrawLam
from gmpy import mpq

L = DrawLam()
L.degree = 2
L.pullbackscheme = [(mpq(1,7), mpq(9,14), True)]
L.filename = "lami3.png"
L.start()
L.iterative_preimages((mpq(1,7), mpq(2,7)), 10)
L.writeout()

The resulta is lami3.png

examples

Original examples

Run from console

cd examples 
python lami1.py

and pdf file

Quadratic invariant laminations

Description and examples not by original author but by Adam Majewski ( only small modifications of some of the original examples)

Quadratic invariant laminations of unit disk and:

• coresponding Julia sets for complex quadratic polynomial ( to show the landing pattern of external rays and structure of Julia set )
• dynamics of angle doubling map

These images show laminations associated with period n components on the boundary of main cardioid.

To create images run from console:

cd regular_examples 
python lami2.py

Algorithm

• start with period p
• compute denominator d: $d = 2^p -1$

period 2

$ L_{1/3}$ : Lamination of unit circle which correspond to lamination of the dynamic plane for Basilica Julia set.

External rays 1/3 and 2/3

• on the parameter plane = angles of the wake 1/2
• on the dynamic plane land on the fixed point

Basilica lamination

The Basilica Julia set has only one kind of pinch point ( cut points): biaccesible = 2 external rays land on each cut point.

Image with major ( red dashed line) and minor leaf ( solid black line ) marked

It can be created with :

• lami_2_.py from regular_examples subdirectory

Code:

# lami_2.py
# python lami_2.py > lami2.txt
# As simply as possible, draw the lamination for period p

import sys
sys.path.append("..")
import DrawLam
from gmpy import mpq, version , gmp_version


# info 
print "Python version" 
print (sys.version) #parentheses necessary in python 3.  
ver = version()
print " gmpy version " 
print ver


gmp_ver = gmp_version() 
print "gmp version "
print gmp_ver


p = 2
print "period p = %d"% p
d = (2**p)-1
print "denominator d = %d"% d
depth = 10


L = DrawLam.DrawLam()
L.degree = 2
a1 = mpq(1,d)
a2 = mpq(1,d)+mpq(1,2)

MinorLeaf = (a1, mpq(2,d))
print "Minor Leaf :"
print MinorLeaf

MajorLeaf = (a1, a2) 
print "Major Leaf: "
print(MajorLeaf)


L.pullbackscheme = [(a1, a2, True)]
L.filename = "lami_"+str(p)+".png"


L.start()
print "draw preimages of minor leaf for depth %d"% depth
L.iterative_preimages(MinorLeaf, depth)
L.writeout()

text output:

Python version
2.7.15rc1 (default, Apr 15 2018, 21:51:34) 
[GCC 7.3.0]
 gmpy version 
1.17
gmp version 
6.1.1
period p = 2
denominator d = 3
Minor Leaf :
(mpq(1,3), mpq(2,3))
Major Leaf: 
(mpq(1,3), mpq(5,6))
Lamination data seems valid.
filetype:  png
draw preimages of minor leaf for depth 10
Writing file lami_2.png

See also :

• lami2.txt - new text output with all angles

Description by Will Smith in Thompson-Like Groups for Dendrite Julia Sets:

We see that the pinch points for the Basilica are points that have external rays at angles
that are rational numbers of the form $\frac{3k - 1}{3·2^n}$ and $\frac{3k + 1}{3·2^n }$ for some $k, n ∈ N$.

In particular, the pinch point between the central interior region and the large region to the left of the central region has external rays at 1/3 and 2/3, and the pinch point between the central region and the large region to the right of the central region has external rays at 5/6 and 1/6. Also observe the intuitive reason that pinch points are so named; the removal of a pinch point will separate the Julia set into two or more pieces.

Period 3

$ L_{1/7}$ : Lamination of unit circle which correspond to lamination of the dynamic plane for Rabbit Julia set

External rays 1/7 and 2/7

• on the parameter plane = angles of the wake 1/3
• on the dynamic plane land on the fixed point

The Rabbit Julia set has only one kind of pinch point ( cut points) of order 3: triaccesible = 3 external rays land on each cut point.

Usual image :

It can be created with :

• lami3.py from examples subdirectory
• lami3.py from regular_examples subdirectory
• lami_3.py from regular_examples subdirectory

Image with major ( red dashed line) and minor leaf ( solid black line ) marked

It can be created with :

• lami2.py from examples subdirectory

Image with shaded gaps:

It can be created with :

• lami5.py from examples subdirectory

Here gap is a space between 3 connected which form curvilinear triangle. This gap correspond to the tri-accesible point ( cut point of order 3) on the dynamical plane

code from regular_examples/lami_3.py which draws regular_examples/lami_3.png image

# lami_3.py
# python lami_3.py
# As simply as possible, draw the lamination for period p

import sys
sys.path.append("..")
import DrawLam
from gmpy import mpq, version , gmp_version


# info 
print "Python version" 
print (sys.version) #parentheses necessary in python 3.  
ver = version()
print " gmpy version " 
print ver


gmp_ver = gmp_version()
print "gmp version "
print gmp_ver


p = 3
print "period p = %d"% p
d = (2**p)-1
print "denominator d = %d"% d
depth = 10


L = DrawLam.DrawLam()
L.degree = 2
a1 = mpq(1,d)
a2 = mpq(1,d)+mpq(1,2)

MinorLeaf = (a1, mpq(2,d))
print "Minor Leaf :"
print MinorLeaf

MajorLeaf = (a1, a2) 
print "Major Leaf: "
print(MajorLeaf)


L.pullbackscheme = [(a1, a2, True)]
L.filename = "lami_"+str(p)+".png"


L.start()
print "draw preimages of minor leaf for depth %d"% depth
L.iterative_preimages(MinorLeaf, depth)
L.writeout()

run :

python lami_3.py

text output:

Python version
2.7.15rc1 (default, Nov 12 2018, 14:31:15) 
[GCC 7.3.0]
 gmpy version 
1.17
gmp version 
6.1.1
period p = 3
denominator d = 7
Minor Leaf :
(mpq(1,7), mpq(2,7))
Major Leaf: 
(mpq(1,7), mpq(9,14))
Lamination data seems valid.
filetype:  png
draw preimages of minor leaf for depth 10
Writing file lami_3.png

Dendritic laminations

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

License

Copyright (c) 2012, Clinton Curry All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

• Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

• Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Acknowledgments

• Hat tip to anyone who's code was used
• Inspiration
• etc

technical notes

GitLab uses:

• the Redcarpet Ruby library for Markdown processing
• KaTeX to render math written with the LaTeX syntax, but only subset

Git

cd existing_folder
git init
git remote add origin [email protected]:adammajewski/drawlam.git
git add .
git commit -m "Initial commit"
git push -u origin master

local repo: :~/python/clintonc-drawlam