This might be intriguing. An endless variety of Siegel disks can be constructed using the polynomials z^n + cz, n > 1 here cz determines a rotation; c = exp(2*pi*i*a) with a irrational on [0,1] gives Siegel disks. 0 is a fixed point (obviously) and the derivative at 0 is c. Just z -> cz rotates the whole plane by the angle a, but adding the "nonlinear seed" z^n crinkles the invariant circles and creates a superattractor at infinity. Siegel disks are the result. I got this from some web research on siegel disks I did recently. Now consider this: z^n + (az+b)/(cz+d), n > 1 We've replaced the rotation with a semilinear transformation. If ad-bc is nonzero, it combines a scaling, a rotation, and possibly inversion in the unit circle. If |ad-bc| = 1, there's no scaling involved. If |ad-bc| is exp(2*pi*i*x) with x irrational, the result should be Siegel-like in quality, with the unit circle distorted and crinkled into an invariant curve. However, the fate of 0 in this map is different. It's a fixed point if d != 0, but the derivative there is (ad-bc)/(d^2), so if d is large, 0 is quite strongly *attracting* rather than indifferent. Our siegel disk might well have a *hole* in it. OTOH, if d is 0, neither b nor c are 0 (we have assumed ad-bc != 0 remember) and so zero goes to 0^n + b/0, i.e. is a preimage of the superattracting fixed point at infinity. Again we have a "hole"... Something strange might be possible. Perhaps I'll investigate using fractint...Get more from the Web. FREE MSN Explorer download : http://explorer.msn.com