Hostname: page-component-54dcc4c588-sdd8f Total loading time: 0 Render date: 2025-10-06T07:07:23.406Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  18 July 2025

Jian Li
Affiliation:
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, Shantou University , Shantou, 515821, Guangdong, China e-mail: wangxs@stu.edu.cn
Jianjie Zhao*
Affiliation:
School of Mathematics, Hangzhou Normal University , Hangzhou, 311121, Zhejiang, China

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.

Information

Type
Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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, 20692082. https://doi.org/10.3934/cpaa.2013.12.2069. MR3015670 $\uparrow$ 11,13.CrossRefGoogle Scholar
Barrachina, X. and Peris, A., Distributionally chaotic translation semigroups . J. Differ. Equ. Appl. 18(2012), no. 4, 751761. https://doi.org/10.1080/10236198.2011.625945. MR2905295 $\uparrow$ 13.CrossRefGoogle Scholar
Bayart, F. and Matheron, É., Dynamics of linear operators, Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009. MR2533318 $\uparrow$ 1.10.1017/CBO9780511581113CrossRefGoogle Scholar
Bayart, F. and Ruzsa, I. Z., Difference sets and frequently hypercyclic weighted shifts . Ergodic Theory Dyn. Syst. 35(2015), no. 3, 691709. https://doi.org/10.1017/etds.2013.77. MR3334899 $\uparrow$ 9.CrossRefGoogle 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, 8393. https://doi.org/10.1016/j.jmaa.2010.06.011. MR2684459 $\uparrow$ 2,8.CrossRefGoogle 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, 21432163. https://doi.org/10.1016/j.jfa.2013.06.019. MR3084499 $\uparrow$ 5.CrossRefGoogle 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, 797821. https://doi.org/10.1016/j.jmaa.2017.11.005. MR3732556 $\uparrow$ 2,11.CrossRefGoogle 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, 342348. https://doi.org/10.1016/j.jfa.2006.12.008. MR2294487 $\uparrow$ 11.CrossRefGoogle Scholar
Desch, W., Schappacher, W., and Webb, G. F., Hypercyclic and chaotic semigroups of linear operators . Ergodic Theory Dyn. Syst. 17(1997), no. 4, 793819. https://doi.org/10.1017/S0143385797084976. MR1468101 $\uparrow$ 10.CrossRefGoogle Scholar
Fomin, S. V., On dynamical systems with a purely point spectrum . Doklady Akad Nauk SSSR (N.S.) 77(1951), 2932 (Russian). MR0043397 $\uparrow$ 3.Google Scholar
Grivaux, S. and Matheron, É., Invariant measures for frequently hypercyclic operators . Adv. Math. 265(2014), 371427. https://doi.org/10.1016/j.aim.2014.08.002. MR3255465 $\uparrow$ 6,9.CrossRefGoogle Scholar
Grosse-Erdmann, K.-G. and Peris, A., Linear chaos, Universitext, Springer, London, 2011. MR2919812 $\uparrow$ 1, 9, 10, 11, 14.10.1007/978-1-4471-2170-1CrossRefGoogle 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.CrossRefGoogle Scholar
Li, J., Trichotomy for the orbits of a hypercyclic operator on a Banach space . Proc. Amer. Math. Soc. 152(2024), no. 12, 52075217. 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, 25872612. https://doi.org/10.1017/etds.2014.41. MR3456608 $\uparrow$ 3.CrossRefGoogle Scholar
Martínez-Giménez, F., Oprocha, P., and Peris, A., Distributional chaos for operators with full scrambled sets . Math. Z. 274(2013), nos. 1–2, 603612. https://doi.org/10.1007/s00209-012-1087-8. MR3054346 $\uparrow$ 8.CrossRefGoogle Scholar
Menet, Q., Linear chaos and frequent hypercyclicity . Trans. Amer. Math. Soc. 369(2017), no. 7, 49774994. https://doi.org/10.1090/tran/6808. MR3632557 $\uparrow$ 10.CrossRefGoogle Scholar