Abstract
Let G be an abelian semigroup of matrices on \({\mathbb {K}}^{n}\) (\({\mathbb {K}}={\mathbb {C}}\) or \({\mathbb {R}}\)). We show that if G is hypercyclic, then it has no non-zero finite orbit. This result fails if we drop the assumption that G is abelian. As a consequence, if G is abelian, it is not chaotic. On the other hand, we show that G is not minimal for \(n\ge 3\), but it can be minimal for \(n=1\); for \({\mathbb {K}}={\mathbb {R}}\), the critical number is \(n=2\).
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Acknowledgements
This work was supported by the research unit: “Dynamical systems and their applications” (UR17ES21), Ministry of Higher Education and Scientific Research, Faculty of Science of Bizerte, Tunisia.
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Ayadi, A., Marzougui, H. Minimality and non-existence of non-zero finite orbits for abelian linear semigroups. Proc Math Sci 134, 7 (2024). https://doi.org/10.1007/s12044-023-00769-9
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DOI: https://doi.org/10.1007/s12044-023-00769-9