On the Construction of Chaotic Dynamical Systems on the Box Fractal

Abstract

In this paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($B$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $B$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.