Misiurewicz Point

Robert P. Munafo, 2011 Feb 27.

A point that has one or more rational external arguments.

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.

Sample MAXIMA session:

(%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	period	found by
0 + 0i	nucleus of R2a	1	f(1,0)
-1.0 + 0i	nucleus of R2.1/2a	2	f(2,0)
-2.0 + 0i	R2t	1	f(3,2)
-0.122561 + 0.744862i	nucleus of R2.1/3a	3	f(3,0)
-1.754878 + 0i	nucleus of R2F(1/2B1)S	3	f(3,0)
-0.228155 + 1.115142i	near R2F(1/3B1)t		f(4,3)
-1.543689 + 0i	R2.1/2t	2	f(4,3)
0 + i	near R2F(1/3B2)t	2	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.

revisions: 20070221 oldest on record; 20110227 add periods for some of the examples

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