| Misiurewicz Point |
Robert P. Munafo, 2007 February 21.
Certain branch points are Misiurewicz points (including all "obvious" branch points in the filaments) and certain terminal points are also Misiurewicz points.
A fairly simple technique in maxima can be used to get the exact
coordinates of some of the simpler nuclei and terminal Misiurewicz
points. It works by solving equations that set one lemniscate equal
to another:
(%i1) L(n) := if n=0 then 0 else L(n-1)^2+c;
2
(%o1) L(n) := if n = 0 then 0 else L (n - 1) + c
(%i2) s(a,b) := solve([L(a)=L(b)],[c]);
(%o2) s(a, b) := solve([L(a) = L(b)], [c])
(%i3) f(a,b) := float(rectform(s(a,b)));
(%o3) f(a, b) := float(rectform(s(a, b)))
(%i4) f(1,0);
(%o4) [c = 0.0]
(%i5) f(4,3);
(%o5) [c = - 1.115142508039938 %i - .2281554936539612,
c = 1.115142508039938 %i - .2281554936539612, c = - 1.543689012692077,
c = - 2.0, c = 0.0]
Here are the points it finds:
| coords | description | found by |
| 0 + 0i | nucleus of R2a | f(1,0) |
| -1.0 + 0i | nucleus of R2.1/2a | f(2,0) |
| -2.0 + 0i | R2t | f(3,2) |
| -0.122561 + 0.744862i | nucleus of R2.1/3a | f(3,0) |
| -1.754878 + 0i | nucleus of R2F(1/2B1)S | f(3,0) |
| -0.228155 + 1.115142i | near R2F(1/3B1)t | f(4,3) |
| -1.543689 + 0i | R2.1/2t | f(4,3) |
| 0 + i | near R2F(1/3B2)t | f(4,2) |
| 0.419643 + 0.606291i | near R2F(1/4B2)t | f(5,3) |
| -1.839287 + 0i | west of R2F(1/2B1)S | f(5,3) |
The idea of solving for Lemniscate pairs other than the cases L(N)=L(0) comes from Adam Majewski.
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