Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 534115
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F10, 34D10

Retrieve articles in all journals with MSC: 58F10, 34D10

Additional Information

Keywords: Completely unstable flow, $ \Omega $-explosion, plug, strong topology
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society