Set-Valued Chaos in Linear Dynamics

Abstract

We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator T : X → X on a topological vector space X, and the natural hyperspace extensions T and T of T to the spaces K(X) of compact subsets of X and C(X) of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T , T and T . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in [1] and [30].

Chaos on Fuzzy Dynamical Systems

“…The main results in this section are the equivalence between the Devaney chaos of f in K(X) and of f in F (X) and, as a consequence, the equivalence of Devaney chaos for a continuous linear operator T on a metrizable and complete locally convex space X, for its Zadeh extension T defined on the space of normal fuzzy sets F (X) and for the induced hyperspace map T on K(X). This extends previous results of D. Jardón, I. Sánchez, and M. Sanchis about the transitivity in fuzzy metric spaces [4] (see also [20]) and another result of N. Bernardes, A. Peris, and F. Rodenas [2] about the linear Devaney chaos of locally convex spaces.…”

“…On the other hand, it was shown in [2] (Theorem 2.2), in the setting of the dynamics of a continuous linear operator T on a complete locally convex space X, the equivalence of Devaney's chaos of T on X and of T on K(X).…”

“…The first study about the connection between the dynamical properties of the dynamical system generated by the map f and the induced system generated by f on the hyperspace was given by Bauer and Sigmund [1] in 1975. Since this work, the subject of hyperspace dynamical systems has attracted the attention of many researchers (see for instance [2,3] and the references therein).…”

Mean Li-Yorke chaos in Banach spaces

“…For operators on Banach spaces, DC1 and DC2 are always equivalent [13,Theorem 2], and imply Li-Yorke chaos. Li-Yorke chaos and distributional chaos for linear operators have been studied in [6,8,10,11,12,13,14,27,28,32,33,38,40,41,42], for instance.…”

Weak stability of non-autonomous discrete dynamical systems

“…It is natural to ask the following question: What is the relation between dynamical properties of the original and set-valued systems? The study of the dynamics of the induced system has been extensively studied and many elegant results have been obtained [15,16,17, and the references therein].…”