Topological semi-conjugacy between dynamical systems induced on the space of probability measures and subshifts of finite type, and chaos

This paper establishes topological semi-conjugacy between a symbolic dynamical system (\Sigma_{A},\sigma_{A}) and the system (M(X),T_{M}) induced on the space of probability measures by a compact topological dynamical system (X, T ) . Some conditions on an original system (X,T) are obtained for the induced system (M(X),T_M) to have a subsystem topologically semi-conjugate to (\Sigma_{A},\sigma_{A}) , and some relations between an original system (X,T) and the induced system (M(X),T_{M}) concerning semi-conjugacy to (\Sigma_{A},\sigma_{A}) are given. By using these results, several criteria of Li–Yorke chaos and Devaney chaos for the induced system (M(X),T_{M}) are established. Also, a characterization of the \mathcal{F} -mixing dynamical system (X,T) with Furstenberg family \mathcal{F} is obtained by means of the system (M(X),T_{M}) , which gives an answer to the question posed by Fu and Xing [Chaos Solitons Fractals (2012), 439–443].