Sequential shadowing implies spectral decomposition. (English) Zbl 1503.37032
Summary: We study chain recurrence for finitely generated group actions on metric spaces under the presence of the shadowing property. We introduce the sequential shadowing property for such actions and prove that this property implies the spectral decomposition property if the phase space is compact.
MSC:
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
References:
| [1] | Nobuo Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math. 6 (1983), no. 2, 329334. · Zbl 0537.58034 |
| [2] | N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems: Recent Ad-vances. North-Holland Mathematical Library, 52. Amsterdam: North-Holland Publishing Co., 1994. · Zbl 0798.54047 |
| [3] | Rufus Bowen, Equilibrium States and the Ergodic Theory of Anosov Dieo-morphisms. Lecture Notes in Mathematics, 470. Berlin-Heidelberg-New York: Springer, 1975. · Zbl 0308.28010 |
| [4] | Ali Barzanouni, Shadowing property on nitely generated group actions, J. Dyn. Syst. Geom. Theor. 12 (2014), no. 1, 6979. · Zbl 1376.37061 |
| [5] | Pramod Das and Tarun Das, Stable group actions on uniform spaces, Topology Proc. 56 (2020), 7183. · Zbl 1443.37016 |
| [6] | Abdul Gaar Khan, Pramod Das, and Tarun Das, GromovHausdor stability for group actions. Submitted. · Zbl 1435.37018 |
| [7] | Zbigniew Nitecki, Dierentiable Dynamics: An Introduction to the Orbit Struc-ture of Dieomorphisms. Cambridge, Mass.-London: The M.I.T. Press, 1971. · Zbl 0246.58012 |
| [8] | Alexey V. Osipov and Sergey B. Tikhomirov, Shadowing for actions of some nitely generated groups, Dyn. Syst. 29 (2014), no. 3, 337351. |
| [9] | S. Smale, Dierentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747817. |
| [10] | Khan) Department of Mathematics; Faculty of Mathematical Sciences; |
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