# Article

 Title: The nonexistence of universal metric flows (English) Author: Geschke, Stefan Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 59 Issue: 4 Year: 2018 Pages: 487-493 Summary lang: English . Category: math . Summary: We consider dynamical systems of the form $(X,f)$ where $X$ is a compact metric space and $f\colon X\to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $\omega$-limit sets, answering a question by Will Brian. (English) Keyword: universal metric dynamical system Keyword: minimal dynamical system MSC: 37B05 MSC: 37B10 MSC: 54H20 idZBL: Zbl 06997364 idMR: MR3914714 DOI: 10.14712/1213-7243.2015.264 . Date available: 2018-12-28T15:11:22Z Last updated: 2020-01-05 Stable URL: http://hdl.handle.net/10338.dmlcz/147552 . Reference: [1] Anderson R. D.: On raising flows and mappings.Bull. Amer. Math. Soc. 69 (1963), no. 2, 259–264. MR 0144324, 10.1090/S0002-9904-1963-10945-3 Reference: [2] Balcar B., Błaszczyk A.: On minimal dynamical systems on Boolean algebras.Comment. Math. Univ. Carolin. 31 (1990), no. 1, 7–11. MR 1056164 Reference: [3] Beleznay F., Foreman M.: The collection of distal flows is not Borel.Amer. J. Math. 117 (1995), no. 1, 203–239. MR 1314463, 10.2307/2375041 Reference: [4] Ben Yaacov I., Melleray J., Tsankov T.: Metrizable universal minimal flows of Polish groups have a comeagre orbit.Geom. Funct. Anal. 27 (2017), no. 1, 67–77. MR 3613453, 10.1007/s00039-017-0398-7 Reference: [5] Bowen R.: $\omega$-limit sets for axiom $A$ diffeomorphism.J. Differential Equations 18 (1975), 333–339. MR 0413181, 10.1016/0022-0396(75)90065-0 Reference: [6] Brian W.: Is there a universal $\omega$-limit set?.available at mathoverflow.net/questions/ 209634. Reference: [7] Ellis R.: Lectures on Topological Dynamics.W. A. Benjamin, New York, 1969. Zbl 0193.51502, MR 0267561 Reference: [8] Furstenberg H.: The structure of distal flows.Amer. J. Math. 83 (1963), 477–515. MR 0157368, 10.2307/2373137 Reference: [9] Morse M., Hedlund G. A.: Symbolic dynamics II. Sturmian trajectories.Amer. J. Math. 62 (1940), no. 1, 1–42. MR 0000745, 10.2307/2371431 Reference: [10] Turek S.: A note on universal minimal dynamical systems.Comment. Math. Univ. Carolin. 32 (1991), no. 4, 781–783. MR 1159826 .

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