The generalized M-J sets for bicomplex numbers

摘要

We explained the theory about bicomplex numbers, discussed the precondition of that addition and multiplication are closed in bicomplex number mapping of constructing generalized Mandelbrot-Julia sets (abbreviated to M-J sets), and listed out the definition and constructing arithmetic of the generalized Mandelbrot-Julia sets in bicomplex numbers system. And we studied the connectedness of the generalized M-J sets, the feature of the generalized Tetrabrot, and the relationship between the generalized M sets and its corresponding generalized J sets for bicomplex numbers in theory. Using the generalized M-J sets for bicomplex numbers constructed on computer, the author not only studied the relationship between the generalized Tetrabrot sets and its corresponding generalized J sets, but also studied their fractal feature, finding that: (1) the bigger the value of the escape time is, the more similar the 3-D generalized J sets and its corresponding 2-D J sets are; (2) the generalized Tetrabrot set contains a great deal information of constructing its corresponding 3-D generalized J sets; (3) both the generalized Tetrabrot sets and its corresponding cross section make a feature of axis symmetry; and (4) the bigger the value of the escape time is, the more similar the cross section and the generalized Tetrabrot sets are.