Monatshefte für Mathematik

, Volume 79, Issue 2, pp 81–92 | Cite as

Topological dynamics of transformations induced on the space of probability measures

  • Walter Bauer
  • Karl Sigmund
Article
Received:

Abstract

LetT be a continuous transformation of a compact metric spaceX. T induces in a natural way a transformationTM on the spaceM (X) of probability measures onX, and a transformationTK on the spaceK (X) of closed subsets ofX. This note investigates which of the topological properties ofT∶X→X (like distality, transitivity, mixing property etc. ...) are “inherited” byTM∶M (X)→M (X) andTK∶K (X)→K (X).

Keywords

Probability Measure Closed Subset Topological Property Topological Dynamic Continuous Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Billingsley, P.: Convergence of probability measures. Wiley. 1968.Google Scholar
  2. [2]
    Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc.153, 401–414 (1971).Google Scholar
  3. [3]
    Bowen, R.: Periodic points and measures for AxiomA diffeomorphisms. Trans. Amer. Math. Soc.154, 377–397 (1971).Google Scholar
  4. [4]
    Furstenberg, H.: Disjointness in ergodic theory. Math. Syst. Theory1, 1–50 (1967).Google Scholar
  5. [5]
    Goodwyn, L. W.: The product theorem for topological entropy. Trans. Amer. Math. Soc.158, 445–452 (1971).Google Scholar
  6. [6]
    Gottschalk, W., andG. Hedlund: Topological dynamics. Amer. Math. Soc. Coll. Publ. 34, 1955.Google Scholar
  7. [7]
    Kuratowski, M.: Topology, vol. II. Academic Press. 1968.Google Scholar
  8. [8]
    Oxtoby, J.: Ergodic sets. Bulletin AMS58, 116–136 (1952).Google Scholar
  9. [9]
    Parthasarathy, K. R.: Probability measures on metric spaces. Academic Press. 1967.Google Scholar
  10. [10]
    Ruelle, D.: Statistical mechanics on compact sets withZ ν-action satisfying expansiveness and specification. Bulletin AMS78, 988–991 (1972).Google Scholar
  11. [11]
    Sigmund, K.: Dynamical systems with the specification property. Trans. Amer. Math. Soc.190, 285–299 (1974).Google Scholar
  12. [12]
    Sigmund, K.: On the time evolution of statistical states for hyperbolic flows. Math. Z.138, 183–189 (1974).Google Scholar
  13. [13]
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Statistics36, 423–439 (1965).Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Walter Bauer
    • 1
  • Karl Sigmund
    • 2
  1. 1.I. Lehrkanzel für Mathematik der Universität SalzburgSalzburgAustria
  2. 2.Math. Institut der UniversitätWienAustria

Personalised recommendations