A topological shadowing theorem. (English) Zbl 1498.37026
Summary: We prove a topological version of the classical shadowing theorem in differentiable dynamics [A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Cambridge: Cambridge Univ. Press (1995; Zbl 0878.58020)]. We use it to unify some stability results in literature as the GH-topological, Hartman and Walters stability theorems [A. Arbieto and C. A. Morales Rojas, Discrete Contin. Dyn. Syst. 37, No. 7, 3531–3544 (2017; Zbl 1369.54024); J. Palis, Anais Acad. Brasil. Ci. 40, 263–266 (1968; Zbl 0184.17803); C. C. Pugh, Am. J. Math. 91, 363–367 (1969; Zbl 0197.20701); Proceedings of Conference, North Dakota State University, Fargo, N.D., 1977 (1978) (Berlin: Springer) pp. 231-244)].
MSC:
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37C50 | Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
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