From Osher Doctorow (user in mdoctorow@comcast.net network).
Let's examine the expression z^2 + c (where z^2 means z-squared) for z, c complex (c constant), that is to say z = x + iy, c = c1 + ic2, i = square root of -1. For the Mandelbrot set (M for short), we want /c/ < 2 where /c/ = sqrt(c1^2 + c2^2), sqrt( ) meaning square root of whatever's inside ( ). We'll rewrite this in terms of x and y as follows. 1) z^2 = (x + iy)(x + iy) = x^2 + iyx + ixy + (iy)^2 = [(x^2) - y^2] + 2ixy 2) z^2 + c = [x^2 - y^2 + c1] + i[2xy + c2] In Rare Event Theory (RET), the expression u + iv changes to real variables as follows. 3) u + iv --> u - v where the arrow indicates the result of the change or transformation. So from (2) bove, z^ + c changes as follows. 4) z^2 + c --> [x^2 - y^2 + c1] - [2xy + c2] Now we get a strange result. J. C. Sprott of Dept. Physics U. Wisconsin Madison, in "Mandelbrot Set Chaos," July 3, 1997, http://sprott.physics.wisc.edu/chaos/manchaos.htm computes the Mandelbrot set orbits in the c1-c2 plane, which he calls the a-b plane, respectively by the following where xn+1 means x with subscript n+1, etc. 5) xn+1 = xn^2 - yn^2 + a 6) yn+1 = 2xnyn + b These are precisely the right hand side expressions in the respective brackets in (4) with a = c1 and b = c2. Now, p1 is optimized at p1(x, y) = 1, where p1(x, y) = 1 + y - x. How do we write the right hand side of (4) in terms of p1? The right hand side of (4) has the form y - x for some y, x, so it is p1 - 1 for some p1. In fact, we have: 7) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1) - 1 Let's incorporate the -1 into c1 and call the result c1', so we can write (7) as follows. 8) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1') Now we can set p1 = 1, which says the following. 9) x^2 - y^2 -(2xy + c2) + (c1' + 1) = 1 and therefore we have the following. 10) x^2 - y^2 -(2xy + c2) + c1 = 0 or, expanding: 11) x^2 - y^2 - 2xy + (c1 - c2) = 0 Except for possibly degenerate cases, it is proved in analytic geometry that this is part of a hyperbola (by proper rotation of axes and translations). We therefore can study the Mandelbrot set M as a part of a hyperbola in Rare Event Theory, which simplifies it considerably. We have c1^2 + c2^2 < = 4 from before. Osher Doctorow