Mixing properties of set-valued maps on hyperspaces via Furstenberg families

Let X be a metric space and f be a continuous self-map of X  . On K(X)K(X), the set of all nonempty compact subsets of X, f   induces a continuous map f¯ naturally by letting f¯(K)=f(K) for K∈K(X)K∈K(X). In this paper Furstenberg families are heavily used to investigate the relationships between mixing properties of f   and those of f¯. Consequently, several general conclusions are developed, which extent the results of Banks [Chaos for induced hyperspace maps, Chaos Solitons & Fractals 2005; 25(3):681–685.], Kwietniak and Oprocha [Topological entropy and chaos for induced hyperspace maps, Chaos Solitons & Fractals 2007; 33:76–86.].