Decompositions of Dynamical Systems Induced by the Koopman Operator


For a topological dynamical system we characterize the decomposition of the state space induced by the fixed space of the corresponding Koopman operator. For this purpose, we introduce a hierarchy of generalized orbits and obtain the finest decomposition of the state space into absolutely Lyapunov stable sets. Analogously to the measure-preserving case, this yields that the system is topologically ergodic if and only if the fixed space of its Koopman operator is one-dimensional.

This is a preview of subscription content, access via your institution.


  1. [1]

    E. Akin and J. Wiseman, Chain recurrence for general spaces, arXiv:1707.09601 (2017).

  2. [2]

    J. Auslander and P. Seibert, Prolongations and stability in dynamical systems, Ann. Inst. Fourier, 2 (1964), 237–268.

    MathSciNet  Article  Google Scholar 

  3. [3]

    N. P. Bhatia and G. P. Szegő, Stability Theory of Dynamical Systems, Springer (1970).

  4. [4]

    C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, Amer. Math. Soc. (Providence, R.I., 1978).

    Google Scholar 

  5. [5]

    J. de Vries, Topological Dynamical Systems: An introduction to the Dynamics of Continuous Mappings, De Gruyter (2014).

  6. [6]

    C. Ding, Chain prolongation and chain stability, Nonlinear Anal., 68 (2008), 2719–2726.

    MathSciNet  Article  Google Scholar 

  7. [7]

    J. Dugundji, Topology, Allyn and Bacon (1966).

  8. [8]

    N. Edeko, On the isomorphism problem for non-minimal transformations with discrete spectrum, Discrete Contin. Dyn. Syst., 39 (2019), 6001–6021.

    MathSciNet  Article  Google Scholar 

  9. [9]

    T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, vol. 272, Springer (2015).

  10. [10]

    S. Frick, K. E. Petersen, and S. Shields, Dynamical properties of some adic systems with arbitrary orderings, Ergodic Theory Dynam. Systems, 37 (2017), 2131–2162.

    MathSciNet  Article  Google Scholar 

  11. [11]

    W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc., 50 (1944), 915–919.

    MathSciNet  Article  Google Scholar 

  12. [12]

    G. M. Kelly, A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on, Bull. Austral. Math. Soc., 22 (1980), 1–83.

    MathSciNet  Article  Google Scholar 

  13. [13]

    D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585–597.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    M. S. Osborne, Hausdorffization and such, Amer. Math. Monthly, 121 (2014), 727–733.

    MathSciNet  Article  Google Scholar 

  15. [15]

    K. E. Petersen, Ergodic Theory, Cambridge university Press (1989).

  16. [16]

    T. Shimomura, On a structure of discrete dynamical systems from the view point of chain components and some applications, Japan. J. Math., 15 (1989), 99–126.

    MathSciNet  Article  Google Scholar 

  17. [17]

    T. Ura, Sur le courant extérieur à une région invariante, Funkcial. Ekvac., 2 (1959), 105–143.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    B. van Munster, The Hausdorff quotient, Thesis, Universiteit Leiden (2014).

Download references

Author information



Corresponding author

Correspondence to K. Küster.

Additional information

The author’s work was supported by Evangelisches Studienwerk Villigst.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Küster, K. Decompositions of Dynamical Systems Induced by the Koopman Operator. Anal Math 47, 149–173 (2021).

Download citation

Key words and phrases

  • Koopman operator
  • fixed space
  • topological ergodicity
  • Hausdorffization
  • Lyapunov stability

Mathematics Subject Classification

  • 37B05
  • 47A35
  • 47B33