Topological Entropy and Mixing Invariant Extremal Distributional Chaos
We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.
F-sensitivity and multi-sensitivity of hyperspatial dynamical systems
Abstract Let f be a continuous self-map defined on a compact metric space X and f ¯ be a continuous self-map naturally induced by f on the hyperspace K ( X ) of all nonempty compact subsets of X endowed with a Hausdorff metric. Firstly, a few sufficient and necessary conditions to ensure a dynamical system be F -sensitive or multi-sensitive are obtained. Then, the following results are proved: (1) If ( X , f ) is a non-minimal M-system, then ( K ( X ) , f ¯ ) has F s -sensitive pairs almost everywhere. (2) If ( K ( X ) , f ¯ ) or ( K ( Y ) , g ¯ ) is F -sensitive, then ( K ( X × Y ) , f × g ¯ ) is F -sensitive. (3) ( K ( X × Y ) , f × g ¯ ) is multi-sensitive if and only if ( K ( X ) , f ¯ ) or ( K ( Y ) , g ¯ ) is multi-sensitive, if and only if ( X , f ) or ( Y , g ) is multi-sensitive. Moreover, it is proved that f × g is multi-sensitive if and only if f or g is multi-sensitive. This is a positive answer to a question posed in R. Li and X. Zhou (2013) [18] .
Li-Yorke Sensitivity of Set-Valued Discrete Systems
Consider the surjective, continuous map and the continuous map of induced by , where is a compact metric space and is the space of all nonempty compact subsets of endowed with the Hausdorff metric. In this paper, we give a short proof that if is Li-Yoke sensitive, then is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of does not imply Li-Yorke sensitivity of .