On supercyclicity for abelian semigroups of matrices on \(\mathbb{R}^n\). (English) Zbl 1508.47012
Oper. Matrices 12, No. 3, 855-865 (2018); corrigendum ibid. 15, No. 2, 777-781 (2021).
Summary: We give a complete characterization of supercylicity for abelian semigroups of matrices on \(\mathbb{R}^n\), \(n \geqslant 1\). We solve the problem of determining the minimal number of matrices over \(\mathbb{R}\) which form a supercyclic abelian semigroup on \(\mathbb{R}^n\). In particular, we show that no abelian semigroup generated by \([\frac{n-1}{2}]\) matrices on \(\mathbb{R}\) can be supercyclic. (\([\cdot]\) denotes the integer part.) This answers a question raised by the second author in [H. Marzougui, Monatsh. Math. 175, No. 3, 401–410 (2014; Zbl 1318.47015)]. Furthermore, we show that supercyclicity and \(\mathbb{R}_+\)-supercyclicity are equivalent.
Keywords:
supercyclic; hypercyclic; semigroups of matrices; dense orbit; somewhere dense; positive supercyclicityCitations:
Zbl 1318.47015References:
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