Pi and the Mandelbrot set

Newsgroups:  alt.fractals
From: [email protected] (Gerald Edgar)
Subject: Pi and the Mandelbrot set
Message-ID: <[email protected]>
Sender: [email protected]
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Organization: The Ohio State University, Dept. of Math.
Date: Fri, 27 Mar 1992 13.57.43 GMT
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David Petrie wrote:
  ] Here's a somewhat heuristic theoretical explanation of what's going on.
  ] atleast around the point (1/4,0).
  ]
  ] When we iterate the equation x := x^2 + 1/4+epsilon, starting at x = 0,
  ] x increases slowly to 1/2, and after it passes 1/2, it zooms off rapidly
  ] to  infinity. So the interesting behavior is when x = 1/2.  Let x = y+1/2.
  ] Then our equation reads  y := y^2 + y + epsilon, or using subscripts
  ] y_(n+l) = (y_n)^2 + y_n + epsilon.
  ]
  ] The y_n's are increasing smoothly and slowly, atleast near y = 0, so it
  ] is reasonable to consider y to be a function of the continuous variable n,
  ] and y_(n+1) - y_n is very close to y'(n)  (the derivative of y).
  ]
  ] So our equation now reads  y'(n) = y^2 + epsilon.  This has the solution
  ] y = a*tan(a*n+c) where a = Sqrt(epsilon).  The intitial point and end point
  ] of our iteration correspond to consecutive poles of the tangent function,
  ] giving a*n = pi, where a = Sqrt(epsilon) and n is the number of iterations
  ] to leave the set, exactly what Mr. Boll has found.
  ]
  ] A similar method seems to work around the point (-3/4,0), but I haven't
  ] completed the analysis (it seems to require a second degree diffeq).
  ]
  ] I did a little more experimentation, and found something really neat at
  ] the points (-1.25,epsilon).  There, n*epsilon/pi (n = #iterations) jumps
  ] around chaotically, but it is always very close to an integer or a half
  ] integer for epsilon very small.  Wierd!
  ]
  ]
  ] David Petry
  ]

A reference for this approach is the Guckenheimer & Holmes book,
"Nonlinear Oscillations ...", section 6.8.  I agree, it is "heuristic".

The point -3/4 can he dealt with similarly.  The function f(z) = z^2-3/4
has fixed point -1/2 with eigenvalue -1.  So we use f(f(z)) to get
eigenvalue +1.  Transform coordinates z = -1/2 + y so that we are
interested in what happens for y near 0.  Consider the
transformations z^2 - 3/4 + i*r, for small, positive, r.  After some
algebra, we get the differential equation:

     dy         4      3          2              2
    ----   =   y  - 2 y  + 2 I r y  - 2 I r y - r
     dt

(Here I = sqrt(-l).)  The time the trajectory spends near 0 (say between
-1 and 1) is the integral

       1
       /
       |                 1
       | ---------------------------------- dy
       |  4      3         2              2
      /  y  - 2 y  +2 I r y  - 2 I r y - r
     -l

Its asymptotics are now shown in the articles called
"Asymptotics of an integral" in sci.math.symbolic.  We conclude that
if n is the number of steps required for divergence, then
n is asymptotic to  Pi/(2*r).  Actually, this is the number of steps
for the composition f(f(z)), so the number of steps for f(z) itself
is double this, Pi/r.  So: Mr. Boll's asymptotic behavior is vindicated.

This is a heuristic argument, so there could be some work done to make it
rigorous . . .

Now, is -5/4 the next spot to study?  Or how about -2 ?
-----

 Date: Mon, 06 Oct 1997 14:36:12 -0700
 From: Jay R Hill <[email protected]>
 Newsgroups: sci.fractals
 Subject: Re: mandelbrot and pi?
 
 I took a look at several other points including -1.75. I found that 
 there are similar relationships with different exponents on n. The 
 trick is finding a reasonable path to follow.  At -0.75 the i direction
 is apparent.  At -1.75 a parabolic path worked, as I recall.  The path 
 has to follow nicely out through the filaments with out crossing them.
 The exponent depends on your parameterization of the path.
 
 Jay Hill   reply to [email protected]