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Disjoint hypercyclicity, Sidon sets and weakly mixing operators

Part of: Sequences and sets General theory of linear operators Topological dynamics

Published online by Cambridge University Press:  22 August 2023

RODRIGO CARDECCIA Open the ORCID record for RODRIGO CARDECCIA [Opens in a new window]
RODRIGO CARDECCIA*
Affiliation:
Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A. and CONICET, Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, Argentina
*
e-mail: rodrigo.cardeccia@ib.edu.ar
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Abstract

We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

Keywords

Sidon setsdisjoint hypercyclicitynon-weakly mixing operators

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 37B99: None of the above, but in this section
Secondary: 11B99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 5 , May 2024 , pp. 1315 - 1329
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Bayart, F. and Matheron, E.. Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250(2) (2007), 426–441.10.1016/j.jfa.2007.05.001CrossRefGoogle Scholar
Bayart, F. and Matheron, E.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics, 179). Cambridge University Press, Cambridge, 2009.10.1017/CBO9780511581113CrossRefGoogle Scholar
Bayart, F. and Matheron, E.. (Non-)weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier (Grenoble) 59(1) (2009), 1–35.10.5802/aif.2425CrossRefGoogle Scholar
Bernal-González, L.. Disjoint hypercyclic operators. Studia Math. 182(2) (2007), 113–131.10.4064/sm182-2-2CrossRefGoogle Scholar
Bès, J., Martin, O., Peris, A. and Shkarin, S.. Disjoint mixing operators. J. Funct. Anal. 263(5) (2012), 1283–1322.10.1016/j.jfa.2012.05.018CrossRefGoogle Scholar
Bès, J., Menet, Q., Peris, A. and Puig, Y.. Recurrence properties of hypercyclic operators. Math. Ann. 366(1–2) (2016), 545–572.10.1007/s00208-015-1336-3CrossRefGoogle Scholar
Bès, J., Menet, Q., Peris, A. and Puig, Y.. Strong transitivity properties for operators. J. Differential Equations 266(2–3) (2019), 1313–1337.10.1016/j.jde.2018.07.076CrossRefGoogle Scholar
Bès, J. and Peris, A.. Hereditarily hypercyclic operators. J. Funct. Anal. 167(1) (1999), 94–112.10.1006/jfan.1999.3437CrossRefGoogle Scholar
Bès, J. and Peris, A.. Disjointness in hypercyclicity. J. Math. Anal. Appl. 336(1) (2007), 297–315.10.1016/j.jmaa.2007.02.043CrossRefGoogle Scholar
Bonilla, A. and Grosse-Erdmann, K.-G.. Upper frequent hypercyclicity and related notions. Rev. Mat. Complut. 31(3) (2018), 673–711.10.1007/s13163-018-0260-yCrossRefGoogle Scholar
Bonilla, A., Grosse-Erdmann, K.-G., López-Martínez, A. and Peris, A.. Frequently recurrent operators. J. Funct. Anal. 283(12) (2022), 109713.10.1016/j.jfa.2022.109713CrossRefGoogle Scholar
Cardeccia, R. and Muro, S.. Arithmetic progressions and chaos in linear dynamics. Integral Equations Operator Theory 94 (2022), 11.10.1007/s00020-022-02687-3CrossRefGoogle Scholar
Cardeccia, R. and Muro, S.. Multiple recurrence and hypercyclicity. Math. Scand. 128 (2022), 133256.10.7146/math.scand.a-133256CrossRefGoogle Scholar
Cilleruelo, J., Sidon sets in ${\mathbb{N}}^d$ . J. Combin. Theory Ser. A 117(7) (2010), 857–871.10.1016/j.jcta.2009.12.003CrossRefGoogle Scholar
De La Rosa, M. and Read, C.. A hypercyclic operator whose direct sum $T\oplus T$ is not hypercyclic. J. Operator Theory 61(2) (2009), 369–380.Google Scholar
Desch, W. and Schappacher, W.. On products of hypercyclic semigroups. Semigroup Forum 71(2) (2005), 301–311.10.1007/s00233-005-0523-zCrossRefGoogle Scholar
Erdős, P.. Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. 5(2) (1994), 261–269.Google Scholar
Erdős, P. and Turán, P.. On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. (2) 16 (1941), 212–215.10.1112/jlms/s1-16.4.212CrossRefGoogle Scholar
Ernst, R., Esser, C. and Menet, Q.. $\mathbf{\mathcal{U}}$ -frequent hypercyclicity notions and related weighted densities. Israel J. Math. 241(2) (2021), 817–848.10.1007/s11856-021-2115-3CrossRefGoogle Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981. M. B. Porter Lectures.10.1515/9781400855162CrossRefGoogle Scholar
Grosse-Erdmann, K.-G. and Peris Manguillot, A.. Linear Chaos. Springer, Berlin, 2011.10.1007/978-1-4471-2170-1CrossRefGoogle Scholar
Herrero, D. A.. Hypercyclic operators and chaos. J. Operator Theory 28(1) (1992), 93–103.Google Scholar
Lindström, B.. An inequality for ${B}_2$ -sequences. J. Combin. Theory 6 (1969), 211–212.10.1016/S0021-9800(69)80124-9CrossRefGoogle Scholar
Moothathu, T. K. S.. Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138.10.4064/cm120-1-9CrossRefGoogle Scholar
Sanders, R. and Shkarin, S.. Existence of disjoint weakly mixing operators that fail to satisfy the disjoint hypercyclicity criterion. J. Math. Anal. Appl. 417(2) (2014), 834–855.10.1016/j.jmaa.2014.03.063CrossRefGoogle Scholar
Shkarin, S.. A short proof of existence of disjoint hypercyclic operators. J. Math. Anal. Appl. 367(2) (2010), 713–715.10.1016/j.jmaa.2010.01.005CrossRefGoogle Scholar
Singer, J.. A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43(3) (1938), 377–385.10.1090/S0002-9947-1938-1501951-4CrossRefGoogle Scholar