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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.10455 |
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| _version_ | 1866908405698920448 |
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| author | Zeng, Hongbo |
| author_facet | Zeng, Hongbo |
| contents | Given a nondegenerate compact perfect and Hausdorff topological space $X$,$n\in \mathbb{N}$ and a function $f:X\rightarrow X$, we consider the $n$-fold symmetric product of $X$, $F_n(X)$ and the induced function $F_n(f):F_n(X)\rightarrow F_n(X)$. If $n\geq2$, we consider the $n$-fold symmetric product suspension of $X$, $SF_n(X)$ and the induced function$SF_n(f):SF_n(X)\rightarrow SF_n(X)$. In this paper, we study the relationships between the following statements: (1) $f\in \mathcal{M}$,(2) $F_n(f)\in \mathcal{M}$, and (3)$SF_n(f)\in \mathcal{M}$, where $\mathcal{M}$ is one of the following classes of map: sensitive, cofinitely sensitive, multi-sensitive, Z-transitive, quasi-periodic, accessible, indecomposable, multi-transitive, $\bigtriangleup$-transitive, $\bigtriangleup$-mixing, Martelli's chaos, Transitive, $F$-system, $TT_{++}$, Touhey, two-sided transitive, fully exact, strongly transitive. These results improve and extend some existing ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sensitivity and transitivity for the induced maps on symmetric product suspensions of a topological space Zeng, Hongbo Dynamical Systems Given a nondegenerate compact perfect and Hausdorff topological space $X$,$n\in \mathbb{N}$ and a function $f:X\rightarrow X$, we consider the $n$-fold symmetric product of $X$, $F_n(X)$ and the induced function $F_n(f):F_n(X)\rightarrow F_n(X)$. If $n\geq2$, we consider the $n$-fold symmetric product suspension of $X$, $SF_n(X)$ and the induced function$SF_n(f):SF_n(X)\rightarrow SF_n(X)$. In this paper, we study the relationships between the following statements: (1) $f\in \mathcal{M}$,(2) $F_n(f)\in \mathcal{M}$, and (3)$SF_n(f)\in \mathcal{M}$, where $\mathcal{M}$ is one of the following classes of map: sensitive, cofinitely sensitive, multi-sensitive, Z-transitive, quasi-periodic, accessible, indecomposable, multi-transitive, $\bigtriangleup$-transitive, $\bigtriangleup$-mixing, Martelli's chaos, Transitive, $F$-system, $TT_{++}$, Touhey, two-sided transitive, fully exact, strongly transitive. These results improve and extend some existing ones. |
| title | Sensitivity and transitivity for the induced maps on symmetric product suspensions of a topological space |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.10455 |