Dynamic properties for the induced maps on symmetric product suspensions of a topological space

Abstract

Given a nondegenerate compact perfect and Hausdorff topological space X, n ∈ N and a function f : X → X, we consider the n-fold symmetric product of X, Fn(X), and the induced function Fn(f) : Fn(X) → Fn(X). In this paper, if n ≥ 2, we begin the study of the n-fold symmetric product suspension of the topological space X, SFn(X). We study the relationships between the following statements: (1) f ∈ M, (2) Fn(f) ∈ M, and (3) SFn(f) ∈ M, where M is one of the following classes of maps: almost transitive, exact, mixing, transitive, totally transitive, strongly transitive, exactly Devaney chaotic, orbit-transitive, an F-system, scattering, T T++, Touhey, backward minimal, totally minimal, Property P, strong property P or twosided transitive.