🚀 go-pugleaf

In article <C3Aqwx.B2B@world.std.com> mrob@world.std.com (Robert P Munafo) writes:

> 
>The pixel-counting method is intended to count the area of the union of
>the mu-molecules.  It does not address the area (if any) of the filaments.
>These mu-molecules are the totally-connected "hyperbolic components", i.e.
>the disks and cardioids.  (Refer to Mandelbrot, _The Fractal Geometry of
>Nature_, p. 183).
> 
>It is clear that each of these mu-atoms, in isolation, is simply a disc or a
>cardioid.  The area of these can be reliably estimated via pixel counting. 
>It is also clear that the mu-molecules are self-similar.  Remember, we're
>talking about a mu-molecule only, exclusive of the filaments and other
>mu-molecules in the region.  Its area is just as easy to approximate via
>pixel-counting as the area of the Koch snowflake curve, or the area of the
>Julia set for -1+0i .

The hyperbolic components of M are not disks or cardioids, 
except for the period 1 and 2 case. 

One problem with pixel counting is that as you look at components of 
higher period, there are more and more of them. You have no idea how many 
are falling between pixels at any resolution, and there is no reason to 
think that the limiting value of the pixel counting is the area of the 
union of these components (even if we ignore the fact that some 
pixels will remain bounded even though they are outside of M). 

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Yuval Fisher                                 e-mail: yfisher@ucsd.edu
Institute for Non-Linear Science 0402                               
University of California, San Diego                                  
La Jolla, CA 92093                                                  
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