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Disjoint Distributional Chaos in Fréchet Spaces

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Abstract

We introduce two different notions of disjoint distributional chaos for sequences of continuous linear operators in Fréchet spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results.

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References

  1. Albanese, A.A., Barrachina, X., Mangino, E.M., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12, 2069–2082 (2013)

    Article  MathSciNet  Google Scholar 

  2. Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criterion for chaos. Discret. Contin. Dyn. Syst. 12, 959–972 (2005)

    Article  Google Scholar 

  3. Barrachina, X.: Distributional Chaos of \(C_{0}\)-Semigroups of Operators. Universitat Politèchnica, València (2013). PhD. Thesis

    Google Scholar 

  4. Barrachina, X., Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Distributional chaos for the forward and backward control traffic model. Linear Algebra Appl. 479, 202–215 (2015)

    Article  MathSciNet  Google Scholar 

  5. Barrachina, X., Peris, A.: Distributionally chaotic translation semigroups. J. Differ. Equ. Appl. 18, 751–761 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bayart, F., Matheron, E.: Dynamics of Linear Operators, vol. 179. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  7. Bermúdez, T., Bonilla, A., Martinez-Gimenez, F., Peris, A.: Li–Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373, 83–93 (2011)

    Article  MathSciNet  Google Scholar 

  8. Bernal-González, L.: Disjoint hypercyclic operators. Stud. Math. 182, 113–131 (2007)

    Article  MathSciNet  Google Scholar 

  9. Bernal-González, L., Bonilla, A.: Order of growth of distributionally irregular entire functions for the differentiation operator. Complex Var. Elliptic Equ. 61, 1176–1186 (2016)

    Article  MathSciNet  Google Scholar 

  10. Bernardes Jr., N.C., Bonilla, A., Müler, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265, 2143–2163 (2013)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. Bès, J., Martin, Ö., Sanders, R.: Weighted shifts and disjoint hypercyclicity. J. Oper. Theory 72, 15–40 (2014)

    Article  MathSciNet  Google Scholar 

  13. Bès, J., Peris, A.: Disjointness in hypercyclicity. J. Math. Anal. Appl. 336, 297–315 (2007)

    Article  MathSciNet  Google Scholar 

  14. Bès, J., Martin, Ö., Peris, A., Shkarin, S.: Disjoint mixing operators. J. Funct. Anal. 263, 1283–1322 (2013)

    Article  MathSciNet  Google Scholar 

  15. Chen, C.-C., Kostić, M.: Disjoint topological transitivity for weighted translations on Orlicz spaces. Filomat, submitted. https://arxiv.org/pdf/1808.05800

  16. Chen, K.-Y.: Distributional chaos for weighted translation operators on groups, preprint. https://arxiv.org/pdf/1807.05191

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Duan, J., Fu, X.-C., Liu, P.-D., Manning, A.: A linear chaotic quantum harmonic oscillator. Appl. Math. Lett. 12, 15–19 (1999)

    Article  MathSciNet  Google Scholar 

  19. Fu, H., Tan, F.: On \(\lambda \)-power \(n\)-distributional chaos. Chin. Ann. Math. Ser. B 38(5), 1119–1130 (2017)

    Article  MathSciNet  Google Scholar 

  20. Fu, H.M., Xiong, J.C., Wang, H.Y.: The hierarchy of distributional chaos. Intern. J. Bifurc. Chaos Appl. Sci. Eng. 25(1), 1550001 (2015). https://doi.org/10.1142/S0218127415500017. (10 pages)

    Article  MathSciNet  MATH  Google Scholar 

  21. Godefroy, J., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)

    Article  MathSciNet  Google Scholar 

  22. Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Springer, London (2011)

    Book  Google Scholar 

  23. Kostić, M.: Abstract Volterra Integro-Differential Equations. CRC Press, Boca Raton (2015)

    Book  Google Scholar 

  24. Kostić, M.: Chaos for Linear Operators and Abstract Differential Equations. Nova Science Publishers Inc., New York (2020)

    Google Scholar 

  25. Liang, Y.-X., Zhou, Z.-H.: Disjoint supercyclic powers of weighted shifts on weighted sequence spaces. Turk. J. Math. 38, 1007–1022 (2014)

    Article  MathSciNet  Google Scholar 

  26. Luo, L., Hou, B.: Some remarks on distributional chaos for bounded linear operators. Turk. J. Math. 39, 251–258 (2015)

    Article  MathSciNet  Google Scholar 

  27. Martin, Ö.: Disjoint Hypercyclic and Supercyclic Composition Operators. PhD. Thesis, Bowling Green State University (2010)

  28. Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351, 607–615 (2009)

    Article  MathSciNet  Google Scholar 

  29. Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for operators with full scrambled sets. Math. Z. 274, 603–612 (2013)

    Article  MathSciNet  Google Scholar 

  30. Meise, R., Vogt, D.: Introduction to Functional Analysis, Translated from the German by M.S. Ramanujan and revised by the authors. Oxf. Grad. Texts Math., Clarendon Press, New York (1997)

  31. Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Am. Math. Soc. 369, 4977–4994 (2017)

    Article  MathSciNet  Google Scholar 

  32. Müller, V.: Orbits of operators and operator semigroups. Kyoto J. Math. 1737, 78–90 (2011)

    Google Scholar 

  33. Müller, V., Vršovský, J.: Orbits of linear operators tending to infinity. Rocky Mountain J. Math. 39, 219–230 (2009)

    Article  MathSciNet  Google Scholar 

  34. Oprocha, P.: Distributional chaos revisited. Trans. Am. Math. Soc. 361, 4901–4925 (2009)

    Article  MathSciNet  Google Scholar 

  35. Oprocha, P.: A quantum harmonic oscillator and strong chaos. J. Phys. A 39, 14559–14565 (2006)

    Article  MathSciNet  Google Scholar 

  36. Puig, Y.: A mixing operator \( T\) for which \((T, T^{2})\) is not disjoint transitive. Stud. Math. 237, 283–296 (2017)

    Article  Google Scholar 

  37. Salas, H.N.: Dual disjoint hypercyclic operators. J. Math. Anal. Appl. 374, 106–117 (2011)

    Article  MathSciNet  Google Scholar 

  38. 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)

    Article  MathSciNet  Google Scholar 

  39. Schweizer, B., Sklar, A., Smítal, J.: Distributional (and other) chaos and its measurement. Real. Anal. Exch. 27, 495–524 (2001/2002)

  40. Wu, X., Wang, L., Chen, G.: Weighted backward shift operators with invariant distributionally scrambled subsets. Ann. Fuct. Anal. 8, 199–210 (2017)

    Article  MathSciNet  Google Scholar 

  41. Yin, Z., He, S., Huang, Y.: On Li–Yorke and distributionally chaotic direct sum operators. Topol. Appl. 239, 35–45 (2018)

    Article  MathSciNet  Google Scholar 

  42. Yin, Z., Huang, Y.: Remarks on multiples of distributionally chaotic operators. Stud. Math. 243, 25–52 (2018)

    Article  MathSciNet  Google Scholar 

  43. Yin, Z., Yang, Q.: Distributionally \(n\)-chaotic dynamics for linear operators. Rev. Mat. Complut. 31, 111–129 (2018)

    Article  MathSciNet  Google Scholar 

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Correspondence to Marko Kostić.

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The author was partially supported by Grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

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Kostić, M. Disjoint Distributional Chaos in Fréchet Spaces. Results Math 75, 83 (2020). https://doi.org/10.1007/s00025-020-01210-7

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  • DOI: https://doi.org/10.1007/s00025-020-01210-7

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