Abstract
Having a finite number of topological spaces Xi and functions fi : Xi → Xi , and considering one of the following classes of functions: exact, transitive, strongly transitive, totally transitive, orbit-transitive, strictly orbittransitive, ω-transitive, mixing, weakly mixing, mild mixing, chaotic, exactly Devaney chaotic, minimal, backward minimal, totally minimal, T T++ , scattering, Touhey or an F -system, in this paper, we study dynamical behaviors of the systems Xi, fi , ∏ Xi, ∏ fi , Fn ∏ Xi , Fn ∏ fi , and Fn Xi , Fn fi .
Highlights
Given a topological space X and a positive integer n, we consider the n -fold symmetric product of X, Fn(X), consisting of all nonempty subsets of X with at most n points [7]
Degirmenci and Kocak [8] considered two metric spaces, X and Y, and two functions f : X → X and g : Y → Y and they analyzed the relationship between f, g and f × g when any of them is a chaotic function
Let X1, . . . , Xm be topological spaces and for each i ∈ {1, . . . , m} let fi section, we present some topological and dynamical properties of the space basic results that we need to know about the function
Summary
Given a topological space X and a positive integer n , we consider the n -fold symmetric product of X , Fn(X) , consisting of all nonempty subsets of X with at most n points [7]. Degirmenci and Kocak [8] considered two metric spaces, X and Y , and two functions f : X → X and g : Y → Y (not necessarily continuous) and they analyzed the relationship between f , g and f × g when any of them is a chaotic function. They proved the following result: if f is continuous and chaotic, and g is chaotic and mixing (not necessarily continuous), f × g is chaotic. We are going to answer similar questions that we can find in [6, 8, 11, 13, 20, 21], considering topological spaces and functions not necessarily continuous
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