Density properties of orbits for a hypercyclic operator on a Banach space

[2507.19752] Density properties of orbits for a hypercyclic operator on a Banach space Skip to main content Search Submit Donate Log in Mathematics > Functional Analysis arXiv:2507.19752 (math) [Submitted on 26 Jul 2025 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Density properties of orbits for a hypercyclic operator on a Banach space Authors: Jian Li , Xinsheng Wang , Jianjie Zhao View PDF HTML (experimental) Abstract: We study density properties of orbits for a hypercyclic operator $T$ on a separable Banach space $X$, and show that exactly one of the following four cases holds: (1) every vector in $X$ is asymptotic to zero with density one; (2) generic vectors in $X$ are distributionally irregular of type $1$; (3) generic vectors in $X$ are distributionally irregular of type $2\frac{1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in $X$ is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups. Subjects: Functional Analysis (math.FA) Cite as: arXiv:2507.19752 [math.FA] (or arXiv:2507.19752v2 [math.FA] for this version) https://doi.org/10.48550/arXiv.2507.19752 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Canad. Math. Bull. 69 (2026), no. 1, 61--76 Related DOI : https://doi.org/10.4153/S0008439525100908 Focus to learn more DOI(s) linking to related resources Submission history From: Jianjie Zhao [ view email ] [v1] Sat, 26 Jul 2025 02:49:12 UTC (13 KB) [v2] Fri, 28 Aug 2026 00:07:38 UTC (13 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2025-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

Related documents

• Density properties of orbits for a hypercyclic operator on a Banach space#682471
• Cyclic homology of algebras with one generator#490057
• Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces#1037362
• Normalized Orbit Frames and Essential Spectrum#1049996
• Existence and Regularity of Stable Resonant Spectral Submanifolds and Linearization Maps#1012436