🚀 go-pugleaf

In article <1993Aug27.120648.11566@mnemosyne.cs.du.edu>, lmitchel@nyx.cs.du.edu (lloyd mitchell) writes:
|> It seems to me that most of the points on the spike are periodic, but the
|> windows are very small and the periods are very high.  I say this based on
|> a hunch from looking at the spike; we've all seen periodic windows but I
|> don't know of any chaotic windows.  

The trouble is the chaotic windows are dirty, with periodic specs in them.

|> The M-points (M stands for a mathematician's name that I can't remember), 
[ Misiurewicz ? ]
|> where the orbit is periodic after a few iterations (-2, -1.543689..., 
|> etc.) are (I think) countable, and are isolated.  

There is an infinite number, they are 'everywhere'.

|> It would seem that truly chaotic points may be, also.

I see the periodic zones as ever decreasing in density as I zoom in.
Consider the interval near -1.75 where little Mandelbroties all line up
to 'enter' the cusp.  There is a sequence of 'self-similar' cardioid
patterns, each with a period 3 greater. In each pattern, there seems to
be a bisecting pattern.  Here is the data:
period     center        approx size
  5  -1.6254137251233038  0.0020294
  8  -1.7110794700131522  0.0000864
 11  -1.7320062728696561  0.0000109
 14  -1.7397176014514633  0.0000026
 17  -1.7433378329345028  0.0000009
 20  -1.7453196244460596  0.00000038
 23  -1.7465233432992069  0.00000020
 26  -1.7473107209745575  0.00000012
 29  -1.7478549581283644  0.00000007
 32  -1.7482474524728249  0.00000005

In between are smaller ones with about ??% of the length.  Notice
the length goes down as the square of the distance to -1.75.  The 
periodic zones are getting less per centage of the length.  (Is this
conclusion correct?)

Take a look with fractint at these little Mandelbroties, especially at
the 'tree' above them (in the +i direction).  I was surprised to find
a pattern that we like to make with iterated function systems fractals.
The pattern is

                                                        


                  _|    |          \_
                    \___|   |   ___/ 
                    /    \  |  /   \_
                  _/      \ | /    | 
                   |       \|/  
                            | 
                            |
                            | 
                            | 


|> 
|> Related question:  the Feigenbaum point is at the very tip of the 
|> cascade of 1-period, 2-period, 4-, 8-, 16-, etc., components.  Does 
|> this mean that chaotic points are only found at the tips of such 
|> doubling cascades?
|> 
|> Kerry Mitchell

There has to be a problem with this, we would have to have each 
Feigenbaum point butt up to a cusp of the next.  Now each  little 
Mandelbrotie has at least a 26 degree angle from its spike (equavelent 
to -2) to the structure at its top (equavelent to near +i) [greater angle
if we count from the Feigenbaum point (equavelent to -1.401155)] and if you 
count the tree like structure I mention above - There is not room to 
jam it into the cusp, which approaches zero width.  I'd think there
would be a continuous chaotic window.  But no matter where you examine
it, there are always small periodic zones.  Most interesting.

--Jay 'Not tonight honey, I'm caught on a chaotic filement' Hill

{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}