Construction of Symmetrical Filled-in Julia Sets on Spherical Surfaces

To continuously generate the images of filled-in Julia sets with attracting orbits of multi-periodicity on a spherical surface, a method of constructing a half spherical surface image on a discretional projective plane is presented. Firstly, the parameters are automatically selected to construct a spherical surface dynamic system whose Lyapunov exponent (L) is less than 0. Then, the iterating orbits of the points on the upper half spherical surface on a discretional projective plane are tracked with the mapping founded in the coordinates system whose origin is the spherical center by the coordinates transform. All the different attracting orbits are built in a link list. Finally, every point on the spherical surface is pointed a color according to the iterating number by which its iterating orbit goes to an attracting orbit in the list, and the filled-in Julia set image on the upper half spherical surface is constructed. The result shows that the method can be used to generate a great number of symmetric filled-in Julia set images on the sphere surface.