Sensitivity and transitivity for the induced maps on symmetric product suspensions of a topological space

Hongbo Zeng

paper

Sensitivity and transitivity for the induced maps on symmetric product suspensions of a topological space

arXiv:2506.10455

Abstract

Given a nondegenerate compact perfect and Hausdorff topological space $X$ , $n \in N$ and a function $f : X \to X$ , we consider the $n$ -fold symmetric product of $X$ , $F_{n} (X)$ and the induced function $F_{n} (f) : F_{n} (X) \to F_{n} (X)$ . If $n \geq 2$ , we consider the $n$ -fold symmetric product suspension of $X$ , $S F_{n} (X)$ and the induced function $S F_{n} (f) : S F_{n} (X) \to S F_{n} (X)$ . In this paper, we study the relationships between the following statements: (1) $f \in M$ ,(2) $F_{n} (f) \in M$ , and (3) $S F_{n} (f) \in M$ , where $M$ is one of the following classes of map: sensitive, cofinitely sensitive, multi-sensitive, Z-transitive, quasi-periodic, accessible, indecomposable, multi-transitive, $△$ -transitive, $△$ -mixing, Martelli's chaos, Transitive, $F$ -system, $T T_{++}$ , Touhey, two-sided transitive, fully exact, strongly transitive. These results improve and extend some existing ones.