Fsensitivity and (F1, F2)sensitivity between dynamical systems and their induced hyperspace dynamical systems

The notions of F-sensitivity and (F1, F2)-sensitivity were introduced and studied by Wang et al. via Furstenberg families in [H.-Y. Wang, J.-C. Xiong, F. Tan, Discrete Dyn. Nat. Soc., 2010 (2010), 12 pages]. In this paper, the concepts of F-collective sensitivity (resp. (F1, F2)-collective sensitivity) and compact-type F-collective sensitivity (resp. compact-type (F1, F2)-collective sensitivity) are introduced as stronger forms of the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively, where F, F1 and F2 are Furstenberg families. It is proved that F-sensitivity (resp. (F1, F2)-sensitivity) of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the F-collective sensitivity (resp. (F1, F2)-collective sensitivity) of the original system; F-sensitivity (resp. (F1, F2)-sensitivity) of the induced hyperspace system defined on the space of all nonempty closed subsets (hit-or-miss topology) is equivalent to the compact-type F-collective sensitivity (resp. (F1, F2)-collective sensitivity) of the original HLCSC system. Moreover, it is shown that for a given dynamical system (E, d, f) and a given Furstenberg family F, if (E, d, f) is F-mixing, then it is F-collectively sensitive. Additionally, we prove that for a given dynamical system (E, d, f) and a given Furstenberg family F, (E, d, f) is F-mixing if and only if f*f*...*f is F-mixing for every n ≥ 2. Our results extend and improve some existing results.