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Keywords:
universal metric dynamical system; minimal dynamical system
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.
References:
[1] Anderson R. D.: On raising flows and mappings. Bull. Amer. Math. Soc. 69 (1963), no. 2, 259–264. DOI 10.1090/S0002-9904-1963-10945-3 | MR 0144324
[2] Balcar B., Błaszczyk A.: On minimal dynamical systems on Boolean algebras. Comment. Math. Univ. Carolin. 31 (1990), no. 1, 7–11. MR 1056164
[3] Beleznay F., Foreman M.: The collection of distal flows is not Borel. Amer. J. Math. 117 (1995), no. 1, 203–239. DOI 10.2307/2375041 | MR 1314463
[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. DOI 10.1007/s00039-017-0398-7 | MR 3613453
[5] Bowen R.: $\omega$-limit sets for axiom $A$ diffeomorphism. J. Differential Equations 18 (1975), 333–339. DOI 10.1016/0022-0396(75)90065-0 | MR 0413181
[6] Brian W.: Is there a universal $\omega$-limit set?. available at mathoverflow.net/questions/ 209634.
[7] Ellis R.: Lectures on Topological Dynamics. W. A. Benjamin, New York, 1969. MR 0267561 | Zbl 0193.51502
[8] Furstenberg H.: The structure of distal flows. Amer. J. Math. 83 (1963), 477–515. DOI 10.2307/2373137 | MR 0157368
[9] Morse M., Hedlund G. A.: Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), no. 1, 1–42. DOI 10.2307/2371431 | MR 0000745
[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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