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Parallelizability revisited


Author: Otomar Hájek
Journal: Proc. Amer. Math. Soc. 27 (1971), 77-84
MSC: Primary 54.82; Secondary 34.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0271925-7
MathSciNet review: 0271925
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Abstract: A classical theorem (Antosiewicz and Dugundji) states that a dynamical system on a locally compact separable metric space is parallelizable if and only if it is dispersive. In this paper it is shown that separability may be omitted, and, under a further condition, local compactness weakened to local Lindelöfness. The crucial step consists in a purely topological characterization of complete instability.


References [Enhancements On Off] (What's this?)

  • [1] N. P. Bhatia and O. Hájek, Theory of dynamical systems. I, IV, IFDAM Technical Notes BN-599, BM-606, University of Maryland, 1969. MR 0324143 (48:2495)
  • [2] N. Bourbaki, General topology, Hermann, Paris; Addison-Wesley, Reading, Mass., 1966. MR 34 #5044a.
  • [3] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [4] J. Dugundji and H. A. Antosiewicz, Parallelizable flows and Lyapunov's second method, Ann. of Math. (2) 73 (1961), 543-555. MR 23 #A395. MR 0123064 (23:A395)
  • [5] O. Hájek, Dynamical systems in the plane, Academic Press, New York and London, 1968. MR 39 #1767. MR 0240418 (39:1767)
  • [6] W. Huebsch, On the covering homotopy theorem, Ann. of Math. (2) 61 (1955), 555-563. MR 19, 974. MR 0091470 (19:974d)
  • [7] L. Markus, Parallel dynamical systems, Topology 8 (1969), 47-57. MR 38 #2806. MR 0234489 (38:2806)
  • [8] V. V. Nemyckiĭ, Topological problems of the theory of dynamical systems, Uspehi Mat. Nauk 4 (1949), no. 6 (34), 91-153; English transl., Amer. Math. Soc. Transl. (1) 5 (1962), 414-497. MR 11, 526. MR 0033983 (11:526c)
  • [9] V. V. Nemyckiĭ, and V. V. Stepanov, Qualitative theory of differential equations, 2nd ed., GITTL, Moscow, 1949; English transl., Princeton, Math. Series, vol. 22, Princeton Univ. Press, Princeton, N. J., 1960. MR 22 #12258. MR 0121520 (22:12258)
  • [10] P. Seibert, Stability in dynamical systems, Proc. NATO Advanced Study Inst. Stability Problems of Solutions of Differential Equations (Padua, 1965) Edizioni Orderisi, Gubbio, 1966. MR 0399595 (53:3438)
  • [11] N. Steenrod, The topology of fibre bundles, Princeton Math. Series, vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. MR 0039258 (12:522b)
  • [12] T. Ura, Local isomorphisms and local parallelizability in dynamical systems theory, Math. Systems Theory 3 (1969), 1-16. MR #489. MR 0247220 (40:489)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271925-7
Keywords: Dynamical systems, parallelizability, global sections, completely unstable systems, dispersive systems, local sections, wandering points, fiber bundles, cross-sections, paracompact locally Lindelöf spaces
Article copyright: © Copyright 1971 American Mathematical Society

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