Visualizing generalized 3x + 1 function dynamics based on fractal

摘要

We generalize 3x + 1 function to the complex plane, gain two different complex maps, and construct fractal images for this two complex maps using escape time, stopping time and total stopping time arithmetic respectively, study the dynamics for generalized 3x + 1 function on the base of the structure characteristics of the fractal images. We find that: The sizes and structures of the stable regions, stopping regions, total stopping regions, divergent regions for the three types of fractal images depend on convergence rate of the map on the x- and y-axis. The black stable regions constructed by escape time and total stopping time are almost similar, which show that 3x + 1 function converged steadily. All of the three types of fractal images are symmetric about the real axis. The structures on the neighborhood of positive integer number are symmetric about a perpendicular line. The perpendicular line is corresponding to the point or its nearby points on the x-axis. And the structures have complicated fractal structure characteristics. These indicate that generalized 3x + 1 function on the neighborhood of integer number contain plentiful information in the complex plane, which need research further.