Fractal diffusion patterns of periodic points in the Mandelbrot set

The Mandelbrot set is one of the most profound study objects in the fractal field. However, the characteristics of internal structures of the Mandelbrot set are not fully studied, which is an obstacle to further understanding of its convergence mechanism. Here, we demonstrate that if the threshold is not taken when calculating the periodic number in the iteration process, the internal structures formed by the collection of periodic points will show fractal diffusion patterns from the theoretical area to the target area. Even when the threshold is taken, these diffusion characteristics remain until the threshold is larger than a certain value. This critical value of the threshold can be regarded as a transition gate of the internal structure from an unstable state to a stable state. The findings of this study indicate that the fractal patterns of the Mandelbrot set can not only exist in the outer area but also the interior area of the Mandelbrot set.