Decompositions of Dynamical Systems Induced by the Koopman Operator

Abstract

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.

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References

  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).

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Correspondence to K. Küster.

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

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Küster, K. Decompositions of Dynamical Systems Induced by the Koopman Operator. Anal Math 47, 149–173 (2021). https://doi.org/10.1007/s10476-021-0068-8

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Key words and phrases

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

Mathematics Subject Classification

  • 37B05
  • 47A35
  • 47B33