A snap shot of the second Julia set for a skew product map.
See "Second Julia sets of complex dynamical systems in C^2 -- computer visualization -- " for more explanation.
Symmetric polynomial map was first studied by K.Uchimura.
This map is defined by
X = x^2 + c y
Y = y^2 + c x
considred in the two dimensional complex space. Itsu second Julia set is computed by taking a generic point and plotting many points of its backward images. Each point of C^2 pas four points as its backward image( except critical points). In the following movies,4^10 points are computed and projected into the real three dimensional space and plotted in a stereographic way.
Double Mandelbrot set.
Parameters from left half of the double Mandelbrot set.
Regular polynomial maps here are defined by
X = x^2 + a x + b y
Y = y^2 + c x + d y
Regular polynomial maps (x,y) to (X,Y) here are defined by
u = x^2 + a x + b y
v = y^2 + c x + d y
X = p u + q v
Y = r u + s v