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The Specification Property for \(C_0\)-Semigroups

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Abstract

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, \(C_0\)-semigroups. In addition, we study the relationships of the specification property for \(C_0\)-semigroups (SgSP) with other dynamical properties: mixing, Devaney’s chaos, distributional chaos, and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black–Scholes equation.

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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. Aroza, J., Kalmes, T., Mangino, E.: Chaotic \(C_0\)-semigroups induced by semiflows in Lebesgue and Sobolev spaces. J. Math. Anal. Appl. 412, 77–98 (2014)

    Article  MathSciNet  Google Scholar 

  3. Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)

    Article  MathSciNet  Google Scholar 

  4. Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation-stability and chaos. Discrete Contin. Dyn. Syst. 29, 67–79 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Bartoll, S., Martínez-Giménez, F., Peris, A.: The specification property for backward shifts. J. Differ. Equ. Appl. 18, 599–605 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bartoll, S., Martínez-Giménez, F., Peris, A.: Operators with the specification property. J. Math. Anal. Appl. 436, 478–488 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bayart, F., Bermúdez, T.: Semigroups of chaotic operators. Bull. Lond. Math. Soc. 41, 823–830 (2009)

    Article  MathSciNet  Google Scholar 

  8. Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  10. Bayart, F., Ruzsa, I.Z.: Difference sets and frequently hypercyclic weighted shifts. Ergod. Theory Dyn. Syst. 35, 691–709 (2015)

    Article  MathSciNet  Google Scholar 

  11. Bermúdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Stud. Math. 170, 57–75 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li–Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35, 1723–1745 (2015)

    Article  MathSciNet  Google Scholar 

  14. 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 

  15. Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dyn. Syst. 27, 383–404 (2007)

    Article  MathSciNet  Google Scholar 

  16. Bowen, R.: Topological entropy and axiom \({\rm A}\). In: Global Analysis (Proc. Sympos. Pure Math., vol. XIV, Berkeley, Calif., 1968), pp. 23–41. Amer. Math. Soc., Providence (1970)

  17. Bowen, R.: Periodic orbits for hyperbolic flows. Am. J. Math. 94, 1–30 (1972)

    Article  MathSciNet  Google Scholar 

  18. Chakir, M., EL Mourchid, S.: Strong mixing Gaussian measures for chaotic semigroups. J. Math. Anal. Appl. 459, 778–788 (2018)

    Article  MathSciNet  Google Scholar 

  19. Conejero, J.A., Lizama, C., Murillo-Arcila, M., Peris, A.: Linear dynamics of semigroups generated by differential operators. Open Math. 15, 745–767 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Conejero, J.A., Müller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic \(C_0\)-semigroup. J. Funct. Anal. 244, 342–348 (2007)

    Article  MathSciNet  Google Scholar 

  21. Conejero, J.A., Peris, A.: Hypercyclic translation \(C_0\)-semigroups on complex sectors. Discrete Contin. Dyn. Syst. 25, 1195–1208 (2009)

    Article  MathSciNet  Google Scholar 

  22. Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behaviour of the hyperbolic heat transfer equation solutions. Int. J. Bifur. Chaos Appl. Sci. Eng. 20, 2943–2947 (2010)

    Article  Google Scholar 

  23. Costakis, G., Peris, A.: Hypercyclic semigroups and somewhere dense orbits. C. R. Math. Acad. Sci. Paris 335, 895–898 (2002)

    Article  MathSciNet  Google Scholar 

  24. Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17, 793–819 (1997)

    Article  MathSciNet  Google Scholar 

  25. Emamirad, H., Goldstein, G., Goldstein, J.A.: Chaotic solution for the Black–Scholes equation. Proc. Am. Math. Soc. 140, 2043–2052 (2012)

    Article  MathSciNet  Google Scholar 

  26. Goldstein, J.A., Mininni, R.M., Romanelli, S.: A new explicit formula for the solution of the Black–Merton–Scholes equation. In: Infinite Dimensional Stochastic Analysis, World Series Publ., pp. 226–235 (2008)

    Chapter  Google Scholar 

  27. Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Universitext, Springer-Verlag London Ltd., London (2011)

    Book  Google Scholar 

  28. Herzog, G.: On a universality of the heat equation. Math. Nachr. 188, 169–171 (1997)

    Article  MathSciNet  Google Scholar 

  29. Mangino, E.M., Peris, A.: Frequently hypercyclic semigroups. Stud. Math. 202, 227–242 (2011)

    Article  MathSciNet  Google Scholar 

  30. Mangino, E.M., Murillo-Arcila, M.: Frequently hypercyclic translation semigroups. Stud. Math. 227, 219–238 (2015)

    Article  MathSciNet  Google Scholar 

  31. Murillo-Arcila, M., Peris, A.: Strong mixing measures for \(C_0\)-semigroups. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109, 101–115 (2015)

    Article  Google Scholar 

  32. Oprocha, P.: Specification properties and dense distributional chaos. Discrete Contin. Dyn. Syst. 17, 821–833 (2007)

    Article  MathSciNet  Google Scholar 

  33. Rudnicki, R.: Chaoticity and invariant measures for a cell population model. J. Math. Anal. Appl. 339, 151–165 (2012)

    Article  MathSciNet  Google Scholar 

  34. Yin, Z., Wei, Y.: Recurrence and topological entropy of translation operators. J. Math. Anal. Appl. 460, 203–215 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second and third authors were also supported by Generalitat Valenciana, Projects PROMETEOII/2013/013 and PROMETEO/2017/102. We are indebted to the referee whose valuable comments produced an improvement in the presentation of the paper.

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  1. Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Edifici 8E, Acces F, 4a planta, 46022, València, Spain

    S. Bartoll, F. Martínez-Giménez, A. Peris & F. Rodenas

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  1. S. Bartoll
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  2. F. Martínez-Giménez
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  3. A. Peris
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Correspondence to A. Peris.

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Bartoll, S., Martínez-Giménez, F., Peris, A. et al. The Specification Property for \(C_0\)-Semigroups. Mediterr. J. Math. 16, 80 (2019). https://doi.org/10.1007/s00009-019-1353-7

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  • DOI: https://doi.org/10.1007/s00009-019-1353-7

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