Dynamics of the generalized 3x + 1 function determined by its fractal images

Two different complex maps were obtained by generalizing 3x + 1 function to the complex plane, and fractal images for these two complex maps were constructed by using escape time, stopping time and total stopping time arithmetic. The dynamics of the generalized 3x + 1 function based on the structural characteristics of the fractal images was studied. We found that: (1) the size and structure of the stable regions, stopping regions, total stopping regions, and divergent regions for the three types of fractal images depend on convergence rate of the map on the x and y axes. (2) The black stable regions constructed, respectively, by escape time and total stopping time are almost overlapped, demonstrating that 3x + 1 function converged steadily. (3) All of the three fractal images are symmetric to the real axis. The structures on the neighborhood of positive integer number are symmetric to a perpendicular line, which is corresponding to the point or its nearby points on the x axis. And the structures have complicated fractal structure characteristics. These findings indicate that the generalized 3x + 1 function on integer number and its neighborhood contains plentiful information in the complex plane.