Li-Yorke chaos for composition operators on \(L^p\)-spaces. (English) Zbl 1485.47012
Let \((X,\mathcal{B},\mu)\) be a measure space and \(f:X\to X\) be measurable such that \(T_f:\phi\to \phi\circ f\) is a bounded linear operator on \(L^p(X,\mathcal{B},\mu)\) with \(1\leq p<\infty\). The main result of this paper is a characterization of Li-Yorke chaos for such composition operators \(T_f\) on \(L^p(X,\mathcal{B},\mu)\). Several consequences are obtained when \(\mu\) and/or the symbol \(f\) have some extra properties. As a consequence, a characterization of Li-Yorke chaos in the particular case of weighted shifts on \(\ell^p(\mathbb{Z})\) is obtained. The article closes with several particular examples and counterexamples proving the optimality of some results.
Reviewer: Romuald Ernst (Calais)
References:
| [1] | Bayart, F.; Darji, Ub; Pires, B., Topological transitivity and mixing of composition operators, J. Math. Anal. Appl., 465, 1, 125-139 (2018) · Zbl 1509.47029 · doi:10.1016/j.jmaa.2018.04.063 |
| [2] | Bayart, F.; Matheron, É., Dynamics of Linear Operators (2009), Cambridge: Cambridge University Press, Cambridge · Zbl 1187.47001 |
| [3] | Bermúdez, T.; Bonilla, A.; Martínez-Giménez, F.; Peris, A., Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373, 1, 83-93 (2011) · Zbl 1214.47012 · doi:10.1016/j.jmaa.2010.06.011 |
| [4] | Bernardes, Nc Jr; Bonilla, A.; Müller, V.; Peris, A., Distributional chaos for linear operators, J. Funct. Anal., 265, 9, 2143-2163 (2013) · Zbl 1302.47014 · doi:10.1016/j.jfa.2013.06.019 |
| [5] | Bernardes, Nc Jr; Bonilla, A.; Müller, V.; Peris, A., Li-Yorke chaos in linear dynamics, Ergod. Theory Dyn. Syst., 35, 6, 1723-1745 (2015) · Zbl 1352.37100 · doi:10.1017/etds.2014.20 |
| [6] | Bernardes, Nc Jr; Cirilo, Pr; Darji, Ub; Messaoudi, A.; Pujals, Er, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl., 461, 1, 796-816 (2018) · Zbl 1385.37041 · doi:10.1016/j.jmaa.2017.11.059 |
| [7] | Bès, J.; Menet, Q.; Peris, A.; Puig, Y., Recurrence properties of hypercyclic operators, Math. Ann., 366, 1, 545-572 (2016) · Zbl 1448.47015 · doi:10.1007/s00208-015-1336-3 |
| [8] | Grivaux, S., Matheron, É., Menet, Q.: Linear dynamical systems on Hilbert spaces: typical properties and explicity examples. Preprint, arXiv:1703.01854 · Zbl 1479.47002 |
| [9] | Grosse-Erdmann, K-G; Peris Manguillot, A., Linear Chaos (2011), London: Springer, London · Zbl 1246.47004 |
| [10] | Hajian, Ab; Kakutani, S., Weakly wandering sets and invariant measures, Trans. Am. Math. Soc., 110, 136-151 (1964) · Zbl 0122.29804 · doi:10.1090/S0002-9947-1964-0154961-1 |
| [11] | Li, Ty; Yorke, Ja, Period three implies chaos, Am. Math. Mon., 82, 10, 985-992 (1975) · Zbl 0351.92021 · doi:10.1080/00029890.1975.11994008 |
| [12] | Menet, Q., Linear chaos and frequent hypercyclicity, Trans. Am. Math. Soc., 369, 7, 4977-4994 (2017) · Zbl 1454.47014 · doi:10.1090/tran/6808 |
| [13] | Prǎjiturǎ, Gt, Irregular vectors of Hilbert space operators, J. Math. Anal. Appl., 354, 2, 689-697 (2009) · Zbl 1166.47015 · doi:10.1016/j.jmaa.2009.01.034 |
| [14] | Rudin, W., Functional Analysis (1991), New York: McGraw-Hill Inc, New York · Zbl 0867.46001 |
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.
