On sensitivity and transitivity of a dynamical system and its induced dynamical systems

Let (X,f) be a dynamical system, i.e. X is a compact metric space and f is a continuous self-map on X and let K(X), M(X) and F(X) be the sets of all non-empty compact subsets of X, Borel probability measures on X and upper semi-continuous fuzzy sets on X, respectively. Then K(X), M(X) and F(X) are metric spaces under Hausdorff metric, prohorov metric and level-wise metric, respectively. Therefore, (X,f) naturally induces three new systems (K(X),f¯), (M(X),fˆ) and (F(X),f~). In this article, we investigate the connection of (r,s)-sensitivity, (r,s)-asymptotic sensitivity, (r,s)-Li-Yorke sensitivity and Δ-transitivity of (X,f) and its induced systems (K(X),f¯), (M(X),fˆ) and (F(X),f~) and we obtain some desired results. For instance, we prove that