Dynamical analysis of some special shift maps on discrete Sierpinski triangle

Abstract

There is a natural relationship between fractals and chaos. In addition, different chaotic dynamical systems can be defined depending on the structure of the relevant fractal. In this paper, we first define a family of dynamical systems on the discrete Sierpinski triangle by using the composition of the shift map and the elements of S3, which is the group of symmetries of the equilateral triangle, via the code representations of the points. Moreover, we give a general formula to construct different dynamical systems on this fractal. Thus, we obtain a family of dynamical systems which are Devaney chaotic. Finally, we classify these dynamical systems in term of topological conjugacy.