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 $F_n\left(X\right)$ with the Vietoris topology. For $n>1$ and $K\in F_n\left(X\right)$ the subset $F_n(K,X)$ of $F_n\left(X\right)$ is defined as the collection of elements of $F_n\left(X\right)$ containing K. We consider the quotient hyperspace $F_n^K\left(X\right)=F_n\left(X\right)⧸F_n(K,X)$, which is obtained from $F_n\left(X\right)$ by shrinking $F_n(K,X)$ to one point set. Furthermore, we consider the induced maps $F_n\left(f\right):F_n\left(X\right)\to F_n\left(X\right)$ and $F_n^K\left(f\right):F_n^K\left(X\right)\to F_n^K\left(X\right)$. In this paper, we introduce the dynamical system $\left(F_n^K\right(X),F_n^K(f\left)\right)$ and we study relationships between the conditions $f\in M$, $F_n\left(f\right)\in M$ and $F_n^K\left(f\right)\in M$, 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.