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Transactions of the American Mathematical Society

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On the topology of the set of completely unstable flows


Author: Zbigniew Nitecki
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0534115-8
Keywords: Completely unstable flow, $ \Omega $-explosion, plug, strong topology
Article copyright: © Copyright 1979 American Mathematical Society

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