Recurrence properties for linear dynamical systems: an approach via invariant measures
| Autor: | Sophie Grivaux, Antoni López-Martínez |
|---|---|
| Přispěvatelé: | Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Universitat Politècnica de València (UPV) |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
Frequently recurrent operators
Measure preserving systems Applied Mathematics General Mathematics 2000 MSC: 47A16 47A35 37A05 37B20 Dynamical Systems (math.DS) Functional Analysis (math.FA) 47A16 (Primary) 47A35 37A05 (Secondary) 37B20 Mathematics - Functional Analysis Recurrence Uniformly recurrent operators FOS: Mathematics Linear dynamics Mathematics - Dynamical Systems [MATH]Mathematics [math] Unimodular eigenvectors |
| Zdroj: | Journal de Mathématiques Pures et Appliquées Journal de Mathématiques Pures et Appliquées, 2022, 169, pp.155-188. ⟨10.1016/j.matpur.2022.11.011⟩ |
| ISSN: | 0021-7824 |
| DOI: | 10.1016/j.matpur.2022.11.011⟩ |
| Popis: | We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category. 35 pages |
| Databáze: | OpenAIRE |
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