Shigehiro Ushiki = Second Julia Page


Stereo Viewer program for second Julia sets of regular polynomial maps in C2

A snap shot of the second Julia set for a skew product map.



Movies of second Julia sets of symmetric product maps in C2

Movies are stereo graphic

and in MPEG format(most movies are only for 4 seconds). Look at the right picture with your left eye and the left picture with your right eye. Fractal objects, "second Julia set" lives in complex two dimensional space, that is, in real four dimensional space. You are looking at the projection of the object to a real three dimensional space. The object is either rotating in the four dimensional space or changing their form according to the variation of the parameter. Rotation in the four-dimensional space is quite different from the rotation in the three dimensional space.

See "Second Julia sets of complex dynamical systems in C^2 -- computer visualization -- " for more explanation.


Movies from symmetric polynomial maps in C2

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.


Movies from regular polynomial maps in C2

Regular polynomial maps here are defined by
X = x^2 + a x + b y
Y = y^2 + c x + d y


Movies from regular polynomial maps in C2

with varied parameters.

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