Recurrence properties of hypercyclic operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2014
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| _version_ | 1866917606138576896 |
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| author | Bès, Juan Menet, Quentin Peris, Alfredo de Dios, Yunied Puig |
| author_facet | Bès, Juan Menet, Quentin Peris, Alfredo de Dios, Yunied Puig |
| contents | We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1410_1349 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Recurrence properties of hypercyclic operators Bès, Juan Menet, Quentin Peris, Alfredo de Dios, Yunied Puig Functional Analysis Dynamical Systems 47A16 We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators. |
| title | Recurrence properties of hypercyclic operators |
| topic | Functional Analysis Dynamical Systems 47A16 |
| url | https://arxiv.org/abs/1410.1349 |