A topological dynamical system on the Cantor set approximates its factors and its natural extension

Abstract

Using topological conjugacies, a continuous mapping from the Cantor set onto itself approximates its factors that are continuous surjective mappings on the Cantor set. Using topological conjugacies, a continuous mapping from the Cantor set onto itself and its natural extension approximate to each other. As a corollary, we shall show that a sofic subshift that is homeomorphic to the Cantor set is approximated by some subshifts of finite type. Furthermore, extending the former result in Shimomura (in press) [4], we get the following result:Let f and g be continuous mappings from the Cantor set onto itself. Suppose that f is chain mixing and g is aperiodic. Then, a sequence of continuous mappings gk(k=1,2,3,…) which are topologically conjugate to g approximates f if trivial necessary conditions on periodic points are satisfied.As a corollary, in the set of all chain mixing topological dynamical systems on the Cantor set, the topological conjugacy class of any topological dynamical system without periodic point is dense.