Abstract
The main aim of this paper is to consider various notions of (dense) \(m_{n}\)-distributional chaos of type s and (dense) reiterative \(m_{n}\)-distributional chaos of type s for general sequences of linear not necessarily continuous operators in Fréchet spaces. Here, \((m_{n})\) is an increasing sequence in \([1,\infty )\) satisfying \(\liminf _{n\rightarrow \infty }\frac{m_{n}}{n}>0\) and s could be \(0,1,2,2+,2\frac{1}{2},3,1+,2-,2_{Bd},2_{Bd}+\). We investigate \(m_{n}\)-distributionally chaotic properties and reiteratively \(m_{n}\)-distributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Fréchet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.
Similar content being viewed by others
References
Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. 2012, 457019 (2012). https://doi.org/10.1155/2012/457019
Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 1798. Cambridge University Press, Cambridge (2009)
Bayart, F., Ruzsa, I.Z.: Difference sets and frequently hypercyclic weighted shifts. Ergod. Theory Dyn. Syst. 35, 691–709 (2015)
Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland, Amsterdam (1988)
Bermúdez, T., Bonilla, A., Martinez-Gimenez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373, 83–93 (2011)
Bernardes Jr., N.C., Bonilla, A., Müler, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265, 2143–2163 (2013)
Bernardes Jr., N.C., Bonilla, A., Peris, A., Wu, X.: Distributional chaos for operators on Banach spaces. J. Math. Anal. Appl. 459, 797–821 (2018)
Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li-Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35, 1723–1745 (2015)
Bernardes Jr., N.C., Bonilla, A., Peris, A.: Mean Li-Yorke chaos in Banach spaces. J. Funct. Anal. 3, 1426 (2020)
Bonilla, A., Kostić, M.: Reiterative distributional chaos on Banach spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 29(14), 1950201 (2019). https://doi.org/10.1142/S0218127419502018
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)
Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17, 1–27 (1997)
Downarowicz, T.: Positive topological entropy implies chaos DC2. Proc. Am. Math. Soc. 142, 137–149 (2013)
Godefroy, J., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)
Grivaux, S., Matheron, É., Menet, Q.: Linear dynamical systems on Hilbert spaces: typical properties and explicit examples. Memoirs Am. Math. Soc. (in press)
Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Springer, London (2011)
Ji, L., Weber, A.: Dynamics of the heat semigroup on symmetric spaces. Ergod. Theory Dyn. Syst 30, 457–468 (2010)
Kostić, M.: Generalized Semigroups and Cosine Functions. Mathematical Institute SANU, Belgrade (2011)
Kostić, M.: Abstract Volterra Integro-Differential Equations. CRC Press, Boca Raton, Fl (2015)
Kostić, M.: Chaos for Linear Operators and Abstract Differential Equations. Nova Science Publishers Inc., New York (2020)
Kostić, M.: Distributionally chaotic properties of abstract fractional differential equations. Novi Sad J. Math. 45, 201–213 (2015)
Kostić, M.: Li-Yorke chaotic properties of abstract differential equations of first order. Appl. Math. Comput. Sci. 1, 15–26 (2016)
Kostić, M.: \({{\cal{F}}}\)-Hypercyclic operators on Fréchet spaces. Publ. Inst. Math. Nouv. Sér 106, 1–18 (2019)
Kostić, M.: Disjoint distributional chaos in Fréchet spaces. preprint ar**v:1812.03824
Kostić, M.: Disjoint reiterative \(m_{n}\)-distributional chaos. Novi Sad J. Math. (2019). https://doi.org/10.30755/NSJOM.09449
Kostić, M.: Disjoint Li-Yorke chaos in Fréchet spaces. Electron. J. Math. Anal. Appl. 8, 248–272 (2020)
Kostić, M., Velinov, D.: Reiterative \((m_{n})\)-distributional chaos for binary relations over metric spaces. Mat. Bilten 43, 5–25 (2019)
Luo, L., Hou, B.: Some remarks on distributional chaos for bounded linear operators. Turk. J. Math. 39, 251–258 (2015)
Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Am. Math. Soc. 369, 4977–4994 (2017)
Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351, 607–615 (2009)
Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for operators with full scrambled sets. Math. Z. 274, 603–612 (2013)
Wu, X.: Maximal distributional chaos of weighted shift operators on Köthe sequence spaces. Czech. Math. J. 64, 105–114 (2014)
Wu, X., Chen, G., Zhu, P.: Invariance of chaos from backward shift on the Köthe sequence space. Nonlinearity 27, 271 (2014). https://doi.org/10.1088/0951-7715/27/2/271
Wu, X., Wang, L., Chen, G.: Weighted backward shift operators with invariant distributionally scrambled subsets. Ann. Fuct. Anal. 8, 199–210 (2017)
Wu, X., Zhu, P.: Li-Yorke chaos of backward shift operators on Köthe sequence spaces. Topol. Appl. 160, 924–929 (2013)
**ong, J.C., Fu, H.M., Wang, H.Y.: A class of Furstenberg families and their applications to chaotic dynamics. Sci. China Math. 57, 823–836 (2014)
Yin, Z., He, S., Huang, Y.: On Li-Yorke and distributionally chaotic direct sum operators. Topol. Appl. 239, 35–45 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mohammad Sal Moslehian.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is partially supported by Grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.
Rights and permissions
About this article
Cite this article
Kostić, M. Reiterative \(m_{n}\)-Distributional Chaos of Type s in Fréchet Spaces. Bull. Malays. Math. Sci. Soc. 43, 3963–4005 (2020). https://doi.org/10.1007/s40840-020-00906-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-00906-x
Keywords
- \(m_{n}\)-distributional chaos of type s
- Reiterative \(m_{n}\)-distributional chaos of type s
- \(\lambda \)-distributional chaos of type s
- Reiterative \(\lambda \)-distributional chaos of type s
- Fréchet spaces