Dynamic properties of the dynamical system (FnK(X), FnK(f))

Abstract

Let (X, f) be a dynamical system, where X is a nondegenerate continuum and f is a map. For any positive integer n, we consider the hyperspace Fn(X) with the Vietoris topology. For n > 1 and K ∈ Fn(X) the subset Fn(K, X) of Fn(X) is defined as the collection of elements of Fn(X) containing K. We consider the quotient hyperspace FnK(X) = Fn(X)ÁFn(K, X), which is obtained from Fn(X) by shrinking Fn(K, X) to one point set. Furthermore, we consider the induced maps Fn(f) : Fn(X) → Fn(X) and FnK(f) : FnK(X) → FnK(X). In this paper, we introduce the dynamical system (FnK(X), FnK(f)) and we study relationships between the conditions f ∈ M, Fn(f) ∈ M and FnK(f) ∈ M, where M is one of the following classes of maps: transitive, mixing, weakly mixing, totally transitive, exact, exact in the sense of Akin-Auslander-Nagar, strongly transitive in the sense of Akin-Auslander-Nagar, exact transitive, fully exact, strongly exact transitive, strongly product transitive, orbit-transitive, Devaney chaotic, irreducible, T T++, strongly transitive and very strongly transitive.