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

The notions of F-sensitivity and (F-1, F-2)-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. (F-1, F-2)-collective sensitivity) and compact-type F-collective sensitivity (resp. compact-type (F-1, F-2)-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, F-1 and F-2 are Furstenberg families. It is proved that F-sensitivity (resp. (F-1, F-2)-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. (F-1, F-2)-collective sensitivity) of the original system; F-sensitivity (resp. (F-1, F-2)-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. (F-1, F-2)-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 x f x . . . x f is F-mixing for every n >= 2. Our results extend and improve some existing results. (C) 2017 All rights reserved.