Chaotic Dynamical Systems on Symbolic Spaces

Abstract

Chaotic dynamical systems, preferably on a Cantor-like space with some arithmetic operations are considered as good pseudo-random number generators. There are many definitions of chaos, of which Devaney-chaos and positive topological entropy seem to be the strongest. These two together imply several other kinds of chaos. For data hiding schemes, systems with more types of chaotic features are considered to be better. Let A = f0;1;··· ; p−1g. We define some continuous maps on AZ using addition with a carry, in combination with the shift map. We get some dynamical systems that are conjugate to a power of the shift map, or have positive entropy. In one case we can give bounds for the topological entropy. We also obtain one system with positive entropy, which is also Devaney-chaotic: i.e., it is transitive, sensitive and has a dense set of periodic points.