Distributional chaos for weighted translation operators on groups. (English) Zbl 07848705
Summary: In this paper, we study distributional chaos for weighted translations on locally compact groups. We give a sufficient condition for such operators to be distributionally chaotic and construct an example of distributionally chaotic weighted translations by way of the sufficient condition. In particular, we prove the existence of distributional chaos and Li-Yorke chaos for weighted translations operators with aperiodic elements. Furthermore, we also investigate the set of distributionally irregular vectors (DIV) of weighted translations through the cone and equivalence classes. When the field is that of complex numbers, we uncover several properties on certain subsets of DIV, including their connectedness and correspondences with some measurable subsets in locally compact groups.
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
Keywords:
distributional chaos; Li-Yorke chaos; irregular vector; weighted translation; locally compact groupReferences:
| [1] | Abakumov, E.; Kuznetsova, Y., Density of translates in weighted \(L^p\) spaces on locally compact groups, Monatshefte Math., 183, 397-413, 2017 · Zbl 1420.47003 |
| [2] | Azimi, M. R.; Akbarbaglu, I., Hypercyclicity of weighted translations on Orlicz spaces, Oper. Matrices, 12, 27-37, 2018 · Zbl 1462.47004 |
| [3] | Bayart, F.; Matheron, É., Dynamics of Linear Operators, Cambridge Tracts in Math., vol. 179, 2009, Cambridge University Press: Cambridge University Press Cambridge · Zbl 1187.47001 |
| [4] | Bermúdez, T.; Bonilla, A.; Martínez-Giménez, F.; Peris, A., Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373, 83-93, 2011 · Zbl 1214.47012 |
| [5] | Bernardes, N. C.; Bonilla, A.; Müler, V.; Peris, A., Distributional chaos for linear operators, J. Funct. Anal., 265, 2143-2163, 2013 · Zbl 1302.47014 |
| [6] | Bernardes, N. C.; Bonilla, A.; Peris, A.; Wu, X., Distributional chaos for operators on Banach spaces, J. Math. Anal. Appl., 459, 797-821, 2018 · Zbl 1482.47009 |
| [7] | Bernardes, N. C.; Darji, U. B.; Pires, B., Li-Yorke chaos for composition operators on \(L^p\)-spaces, Monatshefte Math., 191, 13-35, 2020 · Zbl 1485.47012 |
| [8] | Chen, C-C., Disjoint hypercyclic weighted translations on groups, Banach J. Math. Anal., 11, 459-476, 2017 · Zbl 1420.47004 |
| [9] | Chen, C-C.; Chu, C-H., Hypercyclic weighted translations on groups, Proc. Am. Math. Soc., 139, 2839-2846, 2011 · Zbl 1221.47017 |
| [10] | Chen, C-C.; Tabatabaie, S. M., Chaotic operators on hypergroups, Oper. Matrices, 12, 143-156, 2018 · Zbl 1462.47006 |
| [11] | Chen, C-C.; Chen, K-Y.; Oztop, S.; Tabatabaie, S. M., Chaotic translations on weighted Orlicz spaces, Ann. Pol. Math., 122, 129-142, 2019 · Zbl 07087822 |
| [12] | Chen, K-Y., On aperiodicity and hypercyclic weighted translation operators, J. Math. Anal. Appl., 462, 1669-1678, 2018 · Zbl 1481.47007 |
| [13] | Conejero, J. A.; Kostić, M.; Miana, P. J.; Murillo-Arcila, M., Distributionally chaotic families of operators on Fréchet spaces, Commun. Pure Appl. Anal., 15, 1915-1939, 2016 · Zbl 1394.47013 |
| [14] | Grosse-Erdmann, K.-G.; Peris, A., Linear Chaos, Universitext, 2011, Springer · Zbl 1246.47004 |
| [15] | Martínez-Giménez, F.; Oprocha, P.; Peris, A., Distributional chaos for backward shifts, J. Math. Anal. Appl., 351, 607-615, 2009 · Zbl 1157.47008 |
| [16] | Martínez-Giménez, F.; Oprocha, P.; Peris, A., Distributional chaos for operators with full scrambled sets, Math. Z., 274, 603-612, 2013 · Zbl 1279.47017 |
| [17] | Oprocha, P., A quantum harmonic oscillator and strong chaos, J. Phys. A, 39, 14559-14565, 2006 · Zbl 1107.81026 |
| [18] | Schweizer, B.; Smítal, J., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Am. Math. Soc., 344, 737-754, 1994 · Zbl 0812.58062 |
| [19] | Struble, R. A., Metrics in locally compact groups, Compos. Math., 28, 217-222, 1974 · Zbl 0288.22010 |
| [20] | Wu, X.; Wang, L.; Chen, G., Weighted backward shift operators with invariant distributionally scrambled subsets, Ann. Funct. Anal., 8, 199-210, 2017 · Zbl 1373.47027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
