Convergence of dynamics and the Perron–Frobenius operator

Article
  • 4 Downloads

Abstract

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron–Frobenius operator. Our main result states that strong convergence of the powers of the Perron–Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007.Google Scholar
  2. [2]
    Y. Derriennic, Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires, Annales de l’Institut Henri Poincaré. Section B. Calcul des Probabilit és et Statistique. Nouvelle Série 12 (1976), 111–129.Google Scholar
  3. [3]
    J. Ding and A. Zhou, Statistical Properties of Deterministic Systems, Tsinghua University Texts, Springer-Verlag, Berlin; Tsinghua University Press, Beijing, 2009.Google Scholar
  4. [4]
    Y. Ding, The asymptotic behavior of Frobenius–Perron operator with local lower-bound function, Chaos, Solitons and Fractals 18 (2003), 311–319.Google Scholar
  5. [5]
    T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, Vol. 272, Springer, Cham, 2015.Google Scholar
  6. [6]
    S. R. Foguel, The ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, Vol. 21, Van Nostrand Reinhold Co., New York–Toronto, Ont.–London, 1969.Google Scholar
  7. [7]
    M. Gerlach and J. Glück, Lower bounds and the asymptotic behaviour of positive operator semigroups, Ergodic Theory and Dynamical Systems, to appear, DOI:10.1017/etds.2017.9.Google Scholar
  8. [8]
    A. Lasota, Statistical stability of deterministic systems, in Equadiff 82 (Würzburg, 1982), Lecture Notes in Mathematics, Vol. 1017, Springer, Berlin, 1983, pp. 386–419.Google Scholar
  9. [9]
    A. Lasota and J. A. Yorke, Exact dynamical systems and the Frobenius-Perron operator, Transactions of the American Mathematical Society 273 (1982), 375–384.Google Scholar
  10. [10]
    M. Lin, Mixing for Markov operators, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 19 (1971), 231–242.Google Scholar
  11. [11]
    R. D. Mauldin, Bimeasurable functions, Proceedings of the American Mathematical Society 83 (1981), 369–370.Google Scholar
  12. [12]
    R. Purves, Bimeasurable functions, Fundamenta Mathematicae 58 (1966), 149–157.Google Scholar
  13. [13]
    R. E. Rice, On mixing transformations, Aequationes Mathematicae 17 (1978), 104–108.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Personalised recommendations