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
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 . 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 :
(1/3 ; 2/3)[2][3] is the "characteristic leaf". These rays land on the fixed point .
(1/6 ; 5/6)
compare with
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
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{{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