On the topology of the set of completely unstable flows
HTML articles powered by AMS MathViewer
- by Zbigniew Nitecki PDF
- Trans. Amer. Math. Soc. 252 (1979), 147-162 Request permission
Abstract:
We show that: (1) on any open manifold other than the line or plane, there exist nonsingular flows with $\Omega \ne \emptyset$ which can be perturbed, in the strong ${C^r}$ topology (any r), to flows with $\Omega \ne \emptyset$, and that (2) on certain open 3-manifolds there exist flows with $\Omega \ne \emptyset$ which cannot be approximated, in the strong ${{\mathcal {C}}^1}$ topology, by flows satisfying both $\Omega \ne \emptyset$ and no ${{\mathcal {C}}^1}$ $\Omega$-explosions. These examples give partial negative answers to the conjecture of Takens and White, that the completely unstable flows with the strong ${{\mathcal {C}}^r}$ topology equal the closure of their interior.References
- Morris W. Hirsch, On imbedding differentiable manifolds in euclidean space, Ann. of Math. (2) 73 (1961), 566–571. MR 124915, DOI 10.2307/1970318
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
- S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 303–366. MR 0334281
- Zbigniew Nitecki, Explosions in completely unstable flows. I. Preventing explosions, Trans. Amer. Math. Soc. 245 (1978), 43–61. MR 511399, DOI 10.1090/S0002-9947-1978-0511399-2
- Zbigniew Nitecki, Explosions in completely unstable flows. II. Some examples, Trans. Amer. Math. Soc. 245 (1978), 63–88. MR 511400, DOI 10.1090/S0002-9947-1978-0511400-6
- Zbigniew Nitecki, Bifurcation from completely unstable flows on the cylinder, Bifurcation theory and applications in scientific disciplines (Papers, Conf., New York, 1977) Ann. New York Acad. Sci., vol. 316, New York Acad. Sci., New York, 1979, pp. 86–107. MR 556825
- Peter Percell, Presentations of $3$-manifolds arising from vector fields, Trans. Amer. Math. Soc. 221 (1976), no. 2, 361–377. MR 407857, DOI 10.1090/S0002-9947-1976-0407857-X
- Peter B. Percell and F. Wesley Wilson Jr., Plugging flows, Trans. Amer. Math. Soc. 233 (1977), 93–103. MR 448441, DOI 10.1090/S0002-9947-1977-0448441-2
- Charles C. Pugh, Russell B. Walker, and F. Wesley Wilson Jr., On Morse-Smale approximations–a counterexample, J. Differential Equations 23 (1977), no. 1, 173–182. MR 436218, DOI 10.1016/0022-0396(77)90140-1
- Floris Takens and Warren White, Vector fields with no nonwandering points, Amer. J. Math. 98 (1976), no. 2, 415–425. MR 418163, DOI 10.2307/2373895
- F. Wesley Wilson Jr., On the minimal sets of non-singular vector fields, Ann. of Math. (2) 84 (1966), 529–536. MR 202155, DOI 10.2307/1970458
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 252 (1979), 147-162
- MSC: Primary 58F10; Secondary 34D10
- DOI: https://doi.org/10.1090/S0002-9947-1979-0534115-8
- MathSciNet review: 534115