Vector fields with topological stability

Moriyasu, Kazumine; Sakai, Kazuhiro; Sumi, Naoya

doi:10.1090/S0002-9947-01-02748-9

Vector fields with topological stability
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by Kazumine Moriyasu, Kazuhiro Sakai and Naoya Sumi PDF

Trans. Amer. Math. Soc. 353 (2001), 3391-3408 Request permission

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.

References

Philip Fleming and Mike Hurley, A converse topological stability theorem for flows on surfaces, J. Differential Equations 53 (1984), no. 2, 172–191. MR 748238, DOI 10.1016/0022-0396(84)90038-X
John Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301–308. MR 283812, DOI 10.1090/S0002-9947-1971-0283812-3
Shaobo Gan, Another proof for $C^1$ stability conjecture for flows, Sci. China Ser. A 41 (1998), no. 10, 1076–1082. MR 1667545, DOI 10.1007/BF02871842
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Mike Hurley, Consequences of topological stability, J. Differential Equations 54 (1984), no. 1, 60–72. MR 756545, DOI 10.1016/0022-0396(84)90142-6
Mike Hurley, Bistable vector fields are Axiom A, Bull. Austral. Math. Soc. 51 (1995), no. 1, 83–86. MR 1313115, DOI 10.1017/S0004972700013915
Shantao Liao, Qualitative theory of differentiable dynamical systems, Science Press Beijing, Beijing; distributed by American Mathematical Society, Providence, RI, 1996. Translated from the Chinese; With a preface by Min-de Cheng. MR 1449640
Kazumine Moriyasu, The topological stability of diffeomorphisms, Nagoya Math. J. 123 (1991), 91–102. MR 1126184, DOI 10.1017/S0027763000003664
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
Charles C. Pugh and Clark Robinson, The $C^{1}$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261–313. MR 742228, DOI 10.1017/S0143385700001978
R. Clark Robinson, Structural stability of $C^{1}$ flows, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 262–277. MR 0650640
Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425–437. MR 494300, DOI 10.1216/RMJ-1977-7-3-425
Kazuhiro Sakai, Diffeomorphisms with persistency, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2249–2254. MR 1317049, DOI 10.1090/S0002-9939-96-03275-3
K. Sakai, Topologically stable flows on surfaces, Far East J. Appl. Math. 1 (1997), 133-143.
Lan Wen, Combined two stabilities imply Axiom A for vector fields, Bull. Austral. Math. Soc. 48 (1993), no. 1, 23–30. MR 1227431, DOI 10.1017/S0004972700015410
Lan Wen, On the $C^1$ stability conjecture for flows, J. Differential Equations 129 (1996), no. 2, 334–357. MR 1404387, DOI 10.1006/jdeq.1996.0121
Lan Wen and Zhihong Xia, $C^1$ connecting lemmas, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5213–5230. MR 1694382, DOI 10.1090/S0002-9947-00-02553-8