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

Abstract

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].