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Density properties of orbits for a hypercyclic operator on a Banach space

Part of: General theory of linear operators Topological dynamics

Published online by Cambridge University Press:  18 July 2025

Jian Li Open the ORCID record for Jian Li [Opens in a new window] ,
Xinsheng Wang Open the ORCID record for Xinsheng Wang [Opens in a new window]  and
Jianjie Zhao Open the ORCID record for Jianjie Zhao [Opens in a new window]
Jian Li
Affiliation:
https://ror.org/01a099706 Institute for Mathematical Sciences and Artificial Intelligence & Department of Mathematics, Shantou University , Shantou, 515821, Guangdong, China e-mail: lijian09@mail.ustc.edu.cn
Xinsheng Wang
Affiliation:
Department of Mathematics, https://ror.org/01a099706 Shantou University , Shantou, 515821, Guangdong, China e-mail: wangxs@stu.edu.cn
Jianjie Zhao*
Affiliation:
School of Mathematics, https://ror.org/014v1mr15 Hangzhou Normal University , Hangzhou, 311121, Zhejiang, China
*
e-mail: zjianjie@hznu.edu.cn
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Abstract

We study density properties of orbits for a hypercyclic operator T on a separable Banach space X, and show that exactly one of the following four cases holds: (1) every vector in X is asymptotic to zero with density one; (2) generic vectors in X are distributionally irregular of type $1$; (3) generic vectors in X are distributionally irregular of type $2\frac {1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in X is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell ^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups.

Keywords

Hypercyclicityvector asymptotic to zero with density onedistributionally irregular vectorvector divergent to infinity with density oneC0-semigroup

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

J. Li was supported in part by NSF of China (12222110). X. Wang was partially supported by STU Scientific Research Initiation Grant (SRIG, No. NTF22020) and NSF of China (12301230). J. Zhao was supported by NSF of China (12301226).

References

Albanese, A. A., Barrachina, X., Mangino, E. M., and Peris, A., Distributional chaos for strongly continuous semigroups of operators . Commun. Pure Appl. Anal. 12(2013), no. 5, 2069–2082. https://doi.org/10.3934/cpaa.2013.12.2069. MR3015670 $\uparrow$ 11,13.Google Scholar
Barrachina, X. and Peris, A., Distributionally chaotic translation semigroups . J. Differ. Equ. Appl. 18(2012), no. 4, 751–761. https://doi.org/10.1080/10236198.2011.625945. MR2905295 $\uparrow$ 13.Google Scholar
Bayart, F. and Matheron, É., Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009. MR2533318 $\uparrow$ 1.Google Scholar
Bayart, F. and Ruzsa, I. Z., Difference sets and frequently hypercyclic weighted shifts . Ergodic Theory Dyn. Syst. 35(2015), no. 3, 691–709. https://doi.org/10.1017/etds.2013.77. MR3334899 $\uparrow$ 9.Google Scholar
Bermúdez, T., Bonilla, A., Martínez-Giménez, F., and Peris, A., Li-Yorke and distributionally chaotic operators . J. Math. Anal. Appl. 373(2011), no. 1, 83–93. https://doi.org/10.1016/j.jmaa.2010.06.011. MR2684459 $\uparrow$ 2,8.Google Scholar
da Costa Bernardes, N. Jr., Bonilla, A., Müller, V., and Peris, A., Distributional chaos for linear operators . J. Funct. Anal. 265(2013), no. 9, 2143–2163. https://doi.org/10.1016/j.jfa.2013.06.019. MR3084499 $\uparrow$ 5.Google Scholar
da Costa Bernardes, N. Jr., Bonilla, A., Peris, A., and Wu, X., Distributional chaos for operators on Banach spaces . J. Math. Anal. Appl. 459(2018), no. 2, 797–821. https://doi.org/10.1016/j.jmaa.2017.11.005. MR3732556 $\uparrow$ 2,11.Google Scholar
Conejero, J. A., Müller, V., and Peris, A., Hypercyclic behaviour of operators in a hypercyclic ${C}_0$ -semigroup . J. Funct. Anal. 244(2007), no. 1, 342–348. https://doi.org/10.1016/j.jfa.2006.12.008. MR2294487 $\uparrow$ 11.Google Scholar
Desch, W., Schappacher, W., and Webb, G. F., Hypercyclic and chaotic semigroups of linear operators . Ergodic Theory Dyn. Syst. 17(1997), no. 4, 793–819. https://doi.org/10.1017/S0143385797084976. MR1468101 $\uparrow$ 10.Google Scholar
Fomin, S. V., On dynamical systems with a purely point spectrum . Doklady Akad Nauk SSSR (N.S.) 77(1951), 29–32 (Russian). MR0043397 $\uparrow$ 3.Google Scholar
Grivaux, S. and Matheron, É., Invariant measures for frequently hypercyclic operators . Adv. Math. 265(2014), 371–427. https://doi.org/10.1016/j.aim.2014.08.002. MR3255465 $\uparrow$ 6,9.Google Scholar
Grosse-Erdmann, K.-G. and Peris, A., Linear chaos, Universitext, Springer, London, 2011. MR2919812 $\uparrow$ 1, 9, 10, 11, 14.Google Scholar
Jiang, Z. and Li, J., Chaos for endomorphisms of completely metrizable groups and linear operators on Fréchet spaces . J. Math. Anal. Appl. 543(2025), no. 2, Paper No. 129033. 51pp. https://doi.org/10.1016/j.jmaa.2024.129033. MR4824645 $\uparrow$ 3, 5.Google Scholar
Li, J., Trichotomy for the orbits of a hypercyclic operator on a Banach space . Proc. Amer. Math. Soc. 152(2024), no. 12, 5207–5217. https://doi.org/10.1090/proc/16974. $\uparrow$ 2,10.Google Scholar
Li, J., Tu, S., and Ye, X., Mean equicontinuity and mean sensitivity . Ergodic Theory Dyn. Syst. 35(2015), no. 8, 2587–2612. https://doi.org/10.1017/etds.2014.41. MR3456608 $\uparrow$ 3.Google Scholar
Martínez-Giménez, F., Oprocha, P., and Peris, A., Distributional chaos for operators with full scrambled sets . Math. Z. 274(2013), nos. 1–2, 603–612. https://doi.org/10.1007/s00209-012-1087-8. MR3054346 $\uparrow$ 8.Google Scholar
Menet, Q., Linear chaos and frequent hypercyclicity . Trans. Amer. Math. Soc. 369(2017), no. 7, 4977–4994. https://doi.org/10.1090/tran/6808. MR3632557 $\uparrow$ 10.Google Scholar