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Vector fields with topological stability


Authors: Kazumine Moriyasu, Kazuhiro Sakai and Naoya Sumi
Journal: Trans. Amer. Math. Soc. 353 (2001), 3391-3408
MSC (2000): Primary 37C10, 37C15, 37C75, 37D20, 37D50; Secondary 37B99, 54H20
DOI: https://doi.org/10.1090/S0002-9947-01-02748-9
Published electronically: April 9, 2001
MathSciNet review: 1828611
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Abstract:

In this paper, we give a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it is proved that the $C^1$ interior of the set of all topologically stable $C^1$ vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.


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  • 1. P. Fleming and M. Hurley, A converse topological stability theorem for flows on surfaces, J. Diff. Eqs. 53 (1984), 172-191. MR 87j:58056
  • 2. J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301-308. MR 44:1042
  • 3. S. Gan, Another proof for $C^1$ stability conjecture for flows, Sci. China Ser. A, 41 (1998), 1076-1082. MR 2000a:37006
  • 4. S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows, Annals of Math. 145 (1997), 81-137; Correction, Annals of Math. 150 (1999), 353-356. MR 98b:58096; MR 2000h:37029
  • 5. M. Hurley, Consequences of topological stability, J. Diff. Eqs. 54 (1984), 60-72. MR 85m:58105
  • 6. M. Hurley, Bistable vector fields are Axiom A, Bull. Austral. Math. Soc. 51 (1995), 83-86. MR 95j:58092
  • 7. S. T. Liao, The qualitative theory of differential dynamical systems, Science Press, 1996. MR 98g:58041
  • 8. K. Moriyasu, The topological stability of diffeomorphisms, Nagoya Math. J. 123 (1991), 91-102. MR 92g:58067
  • 9. J. Palis and W. de Melo, Geometric Theory of Dynamical systems, An Introduction, Springer, 1982. MR 84a:58004
  • 10. C. Pugh and C. Robinson, The $C^1$ closing lemma, including Hamiltonians, Ergod. Th. and Dynam. Sys. 3 (1983), 261-313. MR 85m:58106
  • 11. C. Robinson, Structural stability of $C^1$ flows, Dynamical systems-Warwick 1974 (ed. by A. Manning) , Lecture Notes in Math. 468, Springer, 1975, 262-277. MR 58:31251
  • 12. C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), 425-437. MR 58:13200
  • 13. K. Sakai, Diffeomorphisms with persistency, Proc. Amer. Math. Soc. 124 (1996), 2249-2254. MR 96i:58134
  • 14. K. Sakai, Topologically stable flows on surfaces, Far East J. Appl. Math. 1 (1997), 133-143.
  • 15. L. Wen, Combined two stabilities imply Axiom A for vector fields, Bull. Austral. Math. Soc. 48 (1993), 23-30. MR 94e:58073
  • 16. L. Wen, On the $C^1$ stability conjecture for flows, J. Diff. Eqs. 129 (1996), 334-357. MR 97j:58082
  • 17. L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Amer. Math. Soc. 352 (2000), 5213-5230. MR 2001b:37024

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Additional Information

Kazumine Moriyasu
Affiliation: Department of Mathematics, Tokushima University, Tokushima 770-8502, Japan
Email: moriyasu@ias.tokushima-u.ac.jp

Kazuhiro Sakai
Affiliation: Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan
Address at time of publication: Department of Mathematics, Utsunomiya University, Mine-machi 321-8505, Japan
Email: kazsaka@cc.kanagawa-u.ac.jp, sakaik01@kanagawa-u.ac.jp

Naoya Sumi
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Email: sumi@comp.metro-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-01-02748-9
Keywords: Topologically stable, structurally stable, Axiom A, strong transversality condition, vector fields, flows
Received by editor(s): October 12, 1999
Received by editor(s) in revised form: June 28, 2000
Published electronically: April 9, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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