File:Basilica lamination.png
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Contents
Summary
[edit]DescriptionBasilica lamination.png |
English: quadratic invariant lamination associated with basilica Julia set . "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 :
Input file
[edit]old one
[edit]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
[edit]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
[edit]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 . Here one of the fixed points is a landing point of two rays 1/3 and 2/3. These are periodic rays and period of rays is 2. Point is a landing point of two rays 1/6 and 5/6. These are preperiodic rays.
Major leaf : (1/3 ; 5/6)
Minor leaves :
compare with
[edit]-
Basilica jUlia set and external rays
-
quadratic invariant lamination associated with rabbit Julia set
-
Topological model of Mandelbrot set using Lavaurs algorith up to period 12
references
[edit]- ↑ mathoverflow.net question: can-an-almost-injective-function-exist-between-compact-connected-metric-space
- ↑ Rational maps represented by both rabbit and aeroplane matings by Freddie R. Exall
- ↑ math.stackexchange.com - questions : quasiconformal “automorphism” groups of julia sets
Licensing
[edit]- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 10:48, 12 December 2010 | 1,000 × 1,000 (28 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=quadratic invariant lamination <math>L_{\frac{1}{3}}</math> associated withj basilica Julia set <math>f_c(z) = z^2 -1</math>}} |Source=Made with use of program drawlam by Clinton P. Curry |Author=[[User:Adam ma |
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