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Frequent universality criterion and densities

Part of: General theory of linear operators Topological dynamics

Published online by Cambridge University Press:  25 November 2019

R. ERNST  and
A. MOUZE
R. ERNST
Affiliation:
Romuald Ernst, LMPA, Centre Universitaire de la Mi-Voix, Maison de la Recherche Blaise-Pascal, 50 rue Ferdinand Buisson, BP 699, 62228 Calais Cedex, France email ernst.r@math.cnrs.fr
A. MOUZE
Affiliation:
Augustin Mouze, Laboratoire Paul Painlevé, UMR 8524, Cité Scientifique, 59650Villeneuve d’Ascq, France email Augustin.Mouze@math.univ-lille1.fr
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Abstract

We improve a recent result by giving the optimal conclusion both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of $A$-densities, where $A$ refers to some weighted densities sharper than the natural lower density. Moreover, we construct an operator which is logarithmically frequently hypercyclic but not frequently hypercyclic.

Keywords

frequent universalityweighted densitieshypercyclicity

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 37B50: Multi-dimensional shifts of finite type, tiling dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 846 - 868
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Copyright
© Cambridge University Press, 2019

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References

Bayart, F. and Grivaux, S.. Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358(11) (2006), 5083–5117.CrossRefGoogle Scholar
Bayart, F. and Grivaux, S.. Invariant Gaussian measures for operators on banach spaces and linear dynamics. Proc. Lond. Math. Soc. (3) 94(1) (2007), 181–210.CrossRefGoogle Scholar
Bayart, F. and Matheron, É.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics) . Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Bayart, F. and Ruzsa, I. Z.. Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys. 35(3) (2015), 691–709.CrossRefGoogle Scholar
Bès, J., Menet, Q., Peris, A. and Puig, Y.. Recurrence properties of hypercyclic operators. Math. Ann. 366(1–2) (2016), 545–572.CrossRefGoogle Scholar
Bonilla, A. and Grosse-Erdmann, K.-G.. Frequently hypercyclic operators and vectors. Ergod. Th. & Dynam. Sys. 27(2) (2007), 383–404.CrossRefGoogle Scholar
Bonilla, A. and Grosse-Erdmann, K.-G.. Erratum: Frequently hypercyclic operators and vectors. Ergod. Th. & Dynam. Sys. 29(6) (2009), 1993–1994.CrossRefGoogle Scholar
Ernst, R. and Mouze, A.. A quantitative interpretation of the frequent hypercyclicity criterion. Ergod. Th. & Dynam. Sys. 39(4) (2019), 898–924.CrossRefGoogle Scholar
Freedman, A. R. and Sember, J. J.. Densities and summability. Pacific J. Math. 95(2) (1981), 293–305.CrossRefGoogle Scholar
Grivaux, S.. Personal communication (2018).Google Scholar
Jardas, C. and Sarapa, N.. A summability method in some strong laws of large numbers. Math. Commun. 2(2) (1997), 107–124.Google Scholar
Murillo-Arcila, M. and Peris, A.. Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398(2) (2013), 462–465.CrossRefGoogle Scholar