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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E. Akin and J. Wiseman, Chain recurrence for general spaces, arXiv:1707.09601 (2017).
J. Auslander and P. Seibert, Prolongations and stability in dynamical systems, Ann. Inst. Fourier, 2 (1964), 237–268.
N. P. Bhatia and G. P. Szegő, Stability Theory of Dynamical Systems, Springer (1970).
C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics, 38, Amer. Math. Soc. (Providence, R.I., 1978).
J. de Vries, Topological Dynamical Systems: An introduction to the Dynamics of Continuous Mappings, De Gruyter (2014).
C. Ding, Chain prolongation and chain stability, Nonlinear Anal., 68 (2008), 2719–2726.
J. Dugundji, Topology, Allyn and Bacon (1966).
N. Edeko, On the isomorphism problem for non-minimal transformations with discrete spectrum, Discrete Contin. Dyn. Syst., 39 (2019), 6001–6021.
T. Eisner, B. Farkas, M. Haase, and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Texts in Mathematics, vol. 272, Springer (2015).
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.
W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc., 50 (1944), 915–919.
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.
D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin., 36 (1995), 585–597.
M. S. Osborne, Hausdorffization and such, Amer. Math. Monthly, 121 (2014), 727–733.
K. E. Petersen, Ergodic Theory, Cambridge university Press (1989).
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.
T. Ura, Sur le courant extérieur à une région invariante, Funkcial. Ekvac., 2 (1959), 105–143.
B. van Munster, The Hausdorff quotient, Thesis, Universiteit Leiden (2014).
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
Key words and phrases
- Koopman operator
- fixed space
- topological ergodicity
- Lyapunov stability
Mathematics Subject Classification