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

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: 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 g k ( k = 1 , 2 , 3 , … ) which are topologically conjugate to g approximates f if trivial necessary conditions on periodic points are satisfied. orollary, 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.