Summary

DescriptionBasilica lamination.png	English: quadratic invariant lamination ${\displaystyle L_{\frac {1}{3}}}$ associated with basilica Julia set ${\displaystyle f_{c}(z)=z^{2}-1}$. "The quotient of the unit circle by a certain equivalence relation, which is encoded by the following picture, called a lamination" Lasse Rempe-Gillen[1]
Source	Made with use of program drawlam by Clinton P. Curry
Author	Adam majewski

See also :

• QUASISYMMETRIES OF THE BASILICA AND THE THOMPSON GROUP by MIKHAIL LYUBICH AND SERGEI MERENKOV

Input file

old one

It is a input file for Drawlam : program for rendering laminations by Clinton P. Curry http://www.math.uab.edu/~curry/programming.html

use it (in program directory):

./drawlam < basilica.in

it will make a file : basilica.png

basilica.png
2 
10 
1000 1000 
-1.25 1.25 -1.25 1.25 

1/3 5/6 1

1 
1/3 2/3

comment : 
save it as a file : 
 basilica.in
and use : 
 ./drawlam < basilica.in

new

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

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 3k−13⋅2n\frac{3k - 1}{3·2^n}3⋅2n3k−1​ and 3k+13⋅2n\frac{3k + 1}{3·2^n }3⋅2n3k+1​ for some k,n∈Nk, n ∈ Nk,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.

comment

Basilica Julia set = Julia set of the polynomial P(z) = z^2 − 1

"There is a cycle of two periodic Fatou components: One contains the critical point z = 0, the other the critical value z=-1 (which in turn is mapped back to zero). These are connected via a fixed point, which is commonly denoted ${\displaystyle \alpha }$. Here one of the fixed points ${\displaystyle z=\alpha ={\frac {1-{\sqrt {5}}}{2}}}$ is a landing point of two rays 1/3 and 2/3. These are periodic rays and period of rays is 2. Point ${\displaystyle z=-\alpha }$ is a landing point of two rays 1/6 and 5/6. These are preperiodic rays.

Major leaf : (1/3 ; 5/6)

Minor leaves :

• (1/3 ; 2/3)^[2][3] is the "characteristic leaf". These rays land on the fixed point ${\displaystyle \alpha }$.
• (1/6 ; 5/6)

compare with

• 
• 
  Basilica jUlia set and external rays
• 
  quadratic invariant lamination ${\displaystyle L_{\frac {1}{7}}}$ associated with rabbit Julia set
• 
  Topological model of Mandelbrot set using Lavaurs algorith up to period 12

references

1. ↑ mathoverflow.net question: can-an-almost-injective-function-exist-between-compact-connected-metric-space
2. ↑ Rational maps represented by both rabbit and aeroplane matings by Freddie R. Exall
3. ↑ math.stackexchange.com - questions : quasiconformal “automorphism” groups of julia sets

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