# On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

Costakis G, Hadjiloucas D, Manoussos A (2010)

Journal of Mathematical Analysis and Applications 365(1): 229-237.

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In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper Subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on R-n is n + 1, thus complementing a recent result due to Feldman. (C) 2009 Elsevier Inc. All rights reserved.

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Costakis G, Hadjiloucas D, Manoussos A. On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple.

*Journal of Mathematical Analysis and Applications*. 2010;365(1):229-237.Costakis, G., Hadjiloucas, D., & Manoussos, A. (2010). On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple.

*Journal of Mathematical Analysis and Applications*,*365*(1), 229-237. doi:10.1016/j.jmaa.2009.10.020Costakis, G., Hadjiloucas, D., and Manoussos, A. (2010). On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple.

*Journal of Mathematical Analysis and Applications*365, 229-237.Costakis, G., Hadjiloucas, D., & Manoussos, A., 2010. On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple.

*Journal of Mathematical Analysis and Applications*, 365(1), p 229-237. G. Costakis, D. Hadjiloucas, and A. Manoussos, “On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple”,

*Journal of Mathematical Analysis and Applications*, vol. 365, 2010, pp. 229-237. Costakis, G., Hadjiloucas, D., Manoussos, A.: On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple. Journal of Mathematical Analysis and Applications. 365, 229-237 (2010).

Costakis, G., Hadjiloucas, D., and Manoussos, Antonios. “On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple”.

*Journal of Mathematical Analysis and Applications*365.1 (2010): 229-237.
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arXiv 0904.3142