Dynamical Behaviours of Some Horseshoe-like Maps on Cantor Dust C × C

Abstract

Dynamical systems are one of the important research topics that have applications in many disciplines. In particular, investigating whether a dynamical system is chaotic gives us important information about the dynamical system. In this paper, some dynamical systems are defined on Cantor dust C × C by inspiring of the construction of Smale’s horseshoe map. We also express these dynamical systems via ternary numbers and compute their periodic points. Then we show that they are chaotic in the sense of Devaney. Finally, we compare these dynamical systems according to topologically equivalent or not.