On the complexity of upper frequently hypercyclic vectors
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912454351519744 |
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| author | Glab, Szymon Leonetti, Paolo |
| author_facet | Glab, Szymon Leonetti, Paolo |
| contents | Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \in X$ such that $\{n \in ω: T^nx \in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{δσδ}$-set which is either empty or contains a dense $G_δ$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{δσ}$-set.
Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_δ$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{UFHC}(T)$ is not a $F_{σδ}$-set (hence not $G_δ$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the complexity of upper frequently hypercyclic vectors Glab, Szymon Leonetti, Paolo Functional Analysis Classical Analysis and ODEs Dynamical Systems General Topology Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \in X$ such that $\{n \in ω: T^nx \in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{δσδ}$-set which is either empty or contains a dense $G_δ$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{δσ}$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_δ$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{UFHC}(T)$ is not a $F_{σδ}$-set (hence not $G_δ$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology. |
| title | On the complexity of upper frequently hypercyclic vectors |
| topic | Functional Analysis Classical Analysis and ODEs Dynamical Systems General Topology |
| url | https://arxiv.org/abs/2506.22341 |