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...