Let X be a topological space. For any positive integer n , we consider the n -fold symmetric product of X , ℱ n ( X ), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : X → X , we consider the induced functions ℱ n ( ƒ ): ℱ n ( X ) → ℱ n ( X ). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ + -transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++ , semi-open and irreducible. In this paper we study the relationship between the following statements: ƒ ∈ M and ℱ n ( ƒ ) ∈ M .

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)) .