Li-Yorke chaos of $C_{0}$-semigroups under the weak topology
DOI:
https://doi.org/10.5269/bspm.77686Resumen
In this paper, we introduce and investigate the concept of Li-Yorke chaos for $C_0$-semigroups in Banach spaces endowed with the weak topology. We establish that, unlike Li-Yorke chaos under the norm topology, weak Li-Yorke chaos can occur in finite-dimensional spaces. Several fundamental properties of such semigroups are presented. Furthermore, we derive necessary and sufficient conditions for a $C_0$-semigroup to exhibit Li-Yorke chaos in the weak topology.
Referencias
[1] M.N.J AL-Zubaidi, B. A. A. Ahmed, Some Properties of Li-Yorke Chaos. Mathematical Theory and Modeling Vol. 6.No.1, (2016).
[2] M.Amouch, A.Bachir, O.Benchiheb, S.Mecheri, Weakly recurrent operators. Mediterr. J. Math. 20:169,(2023).
[3] M.Amouch, A.Bachir, O.Benchiheb, S.Mecheri, Weakly Sequentially Recurrent Shifts Operators. Math Notes 114, 668–674,(2023).
[4] M. Amouch, H.Hassani, On super-recurrence of strongly continuous semigroup. Adv. Oper.Theory 7, 49,(2022).
[5] M. Amouch, H.Hassani, Weakly hypercyclic and weakly sequentially hypercyclic strongly continuous semigroups Preprint.
[6] F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Am. Math. Soc. 358 (11),(2006) 5083–5117.
[7] F.Bayart, É.Matheron, Dynamics of Linear Operators. vol. 179. Cambridge University Press, Cambridge,(2009).
[8] B. Beauzamy, Un opérateur, sur l'espace de Hilbert, dont tous les polynomes sont hypercycliques, C. R. Acad. Sci. Paris Sér. I Math. 303, 923–925,(1986).
[9] T.Bermudez, A.Bonilla, J.A.Conejero, A.Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Studia Math, 170(1), 57-75,(2005).
[10] G.D Birkhoff, Surface transformations and their dynamical applications. Acta Math. 43, 1-119,(1922).
[11] K. C. Chan and R. Sanders, A weakly hypercyclic operator that is not norm hypercyclic, J.Operator Theory 52, no. 1, 39–59,(2004).
[12] Conejero, J.A., Peris, A.: Hypercyclic translation C0-semigroups on complex sectors. Discrete Contin. Dyn. Syst. 25(4), 1195-1208,(2009).
[13] Y. Chong ,Z.Denghua Chaotic behavior of C0-semigroups of operators, Electronic Research Archive , vol.28(1), 331-336,(2020).
[14] De la Rosa, Manuel, Notes about weakly hypercyclic operators. Journal of Operator Theory 65(1),187-195,(2011).
[15] W.Desch, W.Schappacher, G.F.Webb, Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dynam. Systems 17, 1-27,(1997).
[16] K.G.Grosse-Erdmann, A.Peris, Linear Chaos. Springer, London (2011).
[17] M. Kostic, Li-Yorke chaotic properties of abstract differential equations of first order, Appl.Math. Comput. Sci. 1, 15–26,(2016).
[18] T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(10), 985-992,(1975).
[19] E.Lorenz,The Essence of Chaos, London: UCL Press (1993).
[20] R.A.Martinez-Avendano, and P. Rosenthal, An introduction to operators on the Hardy-Hilbert space. Springer Science and Business Media (2007).
[21] F. Martínez-Giménez, P. Oprocha and A.Peris, Distributional chaos for operators with full scrambled sets, Math. Z. 274, no. 1-2, 603–612,(2013).
[22] Müller, Vladimir, Orbits, weak orbits and local capacity of operators. Integral Equations and Operator Theory 41.2,(2001).
[23] Van Neerven, Jan, The asymptotic behaviour of semigroups of linear operators. Vol. 88. Springer Science and Business Media, 1996.
[24] Xinxing Wu,Li–Yorke chaos of translation semigroups. Journal of Difference Equations and Applications,(2013)
[25] X.Zhang, G. R.Chen, Linear Li–Yorke chaos in a finite-dimensional space with weak topology, Int.J. Bifurcation and Chaos 31,No. 14, Article ID 2150219, 11 p. (2021).
[2] M.Amouch, A.Bachir, O.Benchiheb, S.Mecheri, Weakly recurrent operators. Mediterr. J. Math. 20:169,(2023).
[3] M.Amouch, A.Bachir, O.Benchiheb, S.Mecheri, Weakly Sequentially Recurrent Shifts Operators. Math Notes 114, 668–674,(2023).
[4] M. Amouch, H.Hassani, On super-recurrence of strongly continuous semigroup. Adv. Oper.Theory 7, 49,(2022).
[5] M. Amouch, H.Hassani, Weakly hypercyclic and weakly sequentially hypercyclic strongly continuous semigroups Preprint.
[6] F. Bayart, S. Grivaux, Frequently hypercyclic operators, Trans. Am. Math. Soc. 358 (11),(2006) 5083–5117.
[7] F.Bayart, É.Matheron, Dynamics of Linear Operators. vol. 179. Cambridge University Press, Cambridge,(2009).
[8] B. Beauzamy, Un opérateur, sur l'espace de Hilbert, dont tous les polynomes sont hypercycliques, C. R. Acad. Sci. Paris Sér. I Math. 303, 923–925,(1986).
[9] T.Bermudez, A.Bonilla, J.A.Conejero, A.Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Studia Math, 170(1), 57-75,(2005).
[10] G.D Birkhoff, Surface transformations and their dynamical applications. Acta Math. 43, 1-119,(1922).
[11] K. C. Chan and R. Sanders, A weakly hypercyclic operator that is not norm hypercyclic, J.Operator Theory 52, no. 1, 39–59,(2004).
[12] Conejero, J.A., Peris, A.: Hypercyclic translation C0-semigroups on complex sectors. Discrete Contin. Dyn. Syst. 25(4), 1195-1208,(2009).
[13] Y. Chong ,Z.Denghua Chaotic behavior of C0-semigroups of operators, Electronic Research Archive , vol.28(1), 331-336,(2020).
[14] De la Rosa, Manuel, Notes about weakly hypercyclic operators. Journal of Operator Theory 65(1),187-195,(2011).
[15] W.Desch, W.Schappacher, G.F.Webb, Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dynam. Systems 17, 1-27,(1997).
[16] K.G.Grosse-Erdmann, A.Peris, Linear Chaos. Springer, London (2011).
[17] M. Kostic, Li-Yorke chaotic properties of abstract differential equations of first order, Appl.Math. Comput. Sci. 1, 15–26,(2016).
[18] T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(10), 985-992,(1975).
[19] E.Lorenz,The Essence of Chaos, London: UCL Press (1993).
[20] R.A.Martinez-Avendano, and P. Rosenthal, An introduction to operators on the Hardy-Hilbert space. Springer Science and Business Media (2007).
[21] F. Martínez-Giménez, P. Oprocha and A.Peris, Distributional chaos for operators with full scrambled sets, Math. Z. 274, no. 1-2, 603–612,(2013).
[22] Müller, Vladimir, Orbits, weak orbits and local capacity of operators. Integral Equations and Operator Theory 41.2,(2001).
[23] Van Neerven, Jan, The asymptotic behaviour of semigroups of linear operators. Vol. 88. Springer Science and Business Media, 1996.
[24] Xinxing Wu,Li–Yorke chaos of translation semigroups. Journal of Difference Equations and Applications,(2013)
[25] X.Zhang, G. R.Chen, Linear Li–Yorke chaos in a finite-dimensional space with weak topology, Int.J. Bifurcation and Chaos 31,No. 14, Article ID 2150219, 11 p. (2021).
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2025-09-22
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