🚀 go-pugleaf

In article <5f1zj-c@rpi.edu> fokp@rpi.edu writes:
>The equation is:
>Zn+1=Z^2+C   where
>C is a complex number namely x+yi.
>The idea is to make a grid on the computer screen with the boundary
>of -2 to 2 for the x axis and -2 to 2 for the y axis.  Then you start
>calculating the set by putting every coordinate in the constant C and 
>Z.  You use the point as an initial value as well as a constant.

When calculating the Mandelbrot set, compute the orbit of  0  (the
critical point) for each value of  c .  Also, the  M  set lives in
parameter space with  Re(c)  along the horizontal axis and  Im(c) 
along the vertical axis.  The Julia set, on the other hand, is in
the complex plane with  Re(z)  and  Im(z)  along the horizontal
and vertical axes, respectively.  The parameter  c  is fixed in 
the case of Julia set.

>It is very similar to the Julia Set. In fact, Mandelbrot claimed that
>the Mandelbrot Set have contain every Julia Set.  

For certain values of  c , called _Misiurewicz points_, there is a
structural relationship between the  M  set and the corresponding
Julia set.  See Experiment 17.7 in Devaney's _A First Course in
Chaotic Dynamical Systems_ (Addison-Wesley, 1992) or section 7 in
Bodil Branner's excellent paper in _Chaos and Fractals:  The Math-
ematics Behind the Computer Graphics_ (AMS, 1989).  I believe the
definitive work on this was done by Tan Lei, but I don't have a
reference.

-- 
Tom Scavo
scavo@cie.uoregon.edu