Local Distributional Chaos

  1. Francisco Balibrea 1
  2. Lenka Rucká 2
  1. 1 Universidad de Murcia
    info

    Universidad de Murcia

    Murcia, España

    ROR https://ror.org/03p3aeb86

  2. 2 Silesian University in Opava
    info

    Silesian University in Opava

    Opava, República Checa

    ROR https://ror.org/02w4f2z17

Revista:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Año de publicación: 2022

Volumen: 21

Número: 4

Tipo: Artículo

DIALNET GOOGLE SCHOLAR

Otras publicaciones en: Qualitative theory of dynamical systems

Resumen

Distributional chaos was introduced in Schweizer and Smítal (Trans Am Math Soc 344:737–754, 1994) for continuous maps of the interval, as chaotic behavior based on development of distances between the orbits of points in the system. In Balibrea et al. (Chaos Solitons Fractals 23(5):1581–1583, 2005), this phenomenon was generalized to continuous maps of compact metric space and was distinguished into three different forms, chaos DC1, DC2 and DC3. In Loranty and Pawlak (Chaos 29:013104, 2019), the local idea of such behavior is studied, which leads to the definition of distributionally chaotic points (DC-points). It is also proved in Loranty and Pawlak (2019), that for interval maps, positive topological entropy implies existence of DC1-point. In this paper this result for interval maps is strengthened; it is proved that positive topological entropy implies existence of an uncountable set of DC1-points, and moreover this set can be chosen perfect. In greater dimensions than one, we deal with triangular systems on I 2. In this case the relationship between topological entropy and different types of distributional chaos is not clearly understood and several different results are possible. In the paper we use an example of map F given by Kolyada (Ergod Theory Dyn Syst 12:749–768, 1992) to prove that the corresponding two dimensional system (I 2, F) has positive topological entropy but without containing DC2-points, proving that there is no concentration of DC2-chaos.

Referencias bibliográficas

  • 1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)
  • 2. Balibrea, F., Smítal, J., Štefánková, M.: The three versions of distributional chaos. Chaos Solitons Fractals 23(5), 1581–1583 (2005)
  • 3. Bernardes, N.C., Bonilla, A., Peris, A., Wu, X.: Distributional chaos for operators on Banach spaces. J. Math. Anal. Appl. 459(2), 797–821 (2018)
  • 4. Downarowicz, T.: Positive topological entropy implies chaos DC2. Proc. Am. Math. Soc. 142(1), 137–149 (2014)
  • 5. Holmgren, R.A.: A First Course in Discrete Dynamical Systems, 2nd edn. Springer Universitext (1996)
  • 6. Kolyada, S.F.: On dynamics of triangular maps of the square. Ergod. Theory Dyn. Syst. 12, 749–768 (1992)
  • 7. Kurkova, V.: Sharkovsky’s program for classification of triangular maps is almost completed. Nonlinear Anal. 73, 1663–1669 (2010)
  • 8. Liao, G., Fan, Q.: Minimal subshifts which display Schweizer-Smítal chaos and have zero topological entropy. Sci. China 41, 33–38 (1998)
  • 9. Loranty, A., Pawlak, R.: On the local aspects of distributional chaos. Chaos 29, 013104 (2019)
  • 10. Pikula, R.: On some notions of chaos in dimension zero. Colloq. Math. 107, 167–177 (2007)
  • 11. Ruette, S.: Chaos on the interval, Volume 67 of University Lecture Series. AMS (2017)
  • 12. Schweizer, B., Smítal, J.: Measure of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Am. Math. Soc. 344, 737–754 (1994)
  • 13. Smítal, J., Štefánková, M.: Distributional chaos for triangular maps. Chaos Solitons Fractals 21, 1125– 1128 (2004)
Fuente de los datos: Dialnet