Information fractal dimension of Random Permutation Set

Random permutation set (RPS) is a recently introduced set based on the Dempster-Shafer evidence theory, which considers all possible permutations of the elements within a given set. The information dimension is a significant fractal dimension which plays a vital role in the information theory. Nevertheless, how to develop the information dimension of a specific permutation mass function in RPS remains an unresolved problem. To solve this problem, we propose a new dimension named information fractal dimension of Random Permutation Set. Moreover, several properties of the proposed dimension are explored and numerical examples are provided to illustrate its effectiveness. The research discovers an interesting property related to the permutation mass function corresponding to the maximum RPS entropy: its information dimension is 2, which is equivalent to the fractal dimension of Brownian motion and Peano curve.