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2010•arXiv (Cornell University)Open access

Hypercyclic Abelian Semigroups of Matrices on Rn

Adlene Ayadi, Habib Marzougui

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Abstract

We give a complete characterization of existence of dense orbit for any abelian semigroup of matrices on R^{n}. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on R^{n}. In particular, we show that no abelian semigroup generated by [(n+1)/2] matrices on Rn can be hyper-cyclic. ([ ] denotes the integer part).

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What this paper is about

We give a complete characterization of existence of dense orbit for any abelian semigroup of matrices on R^{n}. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on R^{n}. In particular, we show that no abelian semigroup generated by [(n+1)/2] matrices on Rn can be hyper-cyclic. ([ ] denotes the integer part).

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Available abstract

We give a complete characterization of existence of dense orbit for any abelian semigroup of matrices on R^{n}. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on R^{n}. In particular, we show that no abelian semigroup generated by [(n+1)/2] matrices on Rn can be hyper-cyclic. ([ ] denotes the integer part).

Key concepts: Abelian group, Semigroup, Mathematics, Characterization (materials science), Integer (computer science), Pure mathematics, Orbit (dynamics), Finitely-generated abelian group

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