Conceptions of Topological Transitivity on Symmetric Products

Abstract

Let 𝑋 be a topological space. For any 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; and for a given function 𝑓 ∶ 𝑋 → 𝑋 , we consider the induced functions F𝑛 (𝑓 ) ∶ F𝑛 (𝑋 ) → F𝑛 (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: 𝑓 ∈ M and F𝑛 (𝑓 ) ∈ M.