Structurally stable flows

Pyo, Dong; Choi, Sung Kyu; Park, Jong Suh

doi:10.1017/s0004972700037035

Cited by 2 publications

(8 citation statements)

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“…In fact, Chi, Choi and Park [l] have recently given a proof for Theorem A. Unfortunately, the proof given in [1] contains some serious gaps as we point out below, and the aim of the present note is to provide a way of making it up.…”

Section: Introductionmentioning

confidence: 90%

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Combined two stabilities imply Axiom A for vector fields

Wen

1993

Bull. Austral. Math. Soc.

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Section: Introductionmentioning

confidence: 90%

“…(This does not affect the proof of Hurley [5] because he used Theorem M.) In the present note we shall use Theorem L to fill the gap. Another point which is missed in [1] is the problem of dealing with singularities. This will be made up by Lemma 4 of Section 3 below.…”

Section: Introductionmentioning

confidence: 99%

Combined two stabilities imply Axiom A for vector fields

Wen

1993

Bull. Austral. Math. Soc.

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“…To say that a vector field is Axiom A means that its nonwandering set carries a hyperbolic structure and is the closure of its closed orbits (a closed orbit is either a fixed point or a nontrivial periodic orbit). See [2,11,12] for a more detailed description of Axiom A. If Z is a vector field, let Aj(Z) denote the closure of the set of (non-fixed) hyperbolic…”

mentioning

confidence: 99%

“…Hurley [2] periodic orbits of Z whose unstable manifolds have dimension i, and let E(£) be the set of hyperbolic fixed points of Z. The chain recurrent set of a vector field Z will be denoted R(Z); its definition can be found in [3,4,6,12].…”

mentioning

confidence: 99%

“…R{Z) is a compact, nonempty, flow-invariant subset of M that contains all closed orbits. The connected components of R(Z) are called chain components (a different but equivalent definition was used in [2,12]; a proof of the equivalence of the two definitions can be found in [6] PROOF: Conclusions 1 and 2 are contained in Theorem C of [4]. Conclusions 3, 4 and 5 are, respectively, Corollary 7, Proposition 4, and Proposition 5 of [6].…”

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Bistable vector fields are axiom A

Hurley

1995

Bull. Austral. Math. Soc.

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BISTABLE VECTOR FIELDS ARE AXIOM A MIKE HURLEYRecently L. Wen showed that if a C 1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.In [5] it was shown that if a C 1 diffeomorphism on a compact manifold is bistable (that is, both topologically stable and structurally stable), then the diffeomorphism is an Axiom A diffeomorphism. The proof goes by showing that such a diffeomorphism satisfies conditions that Mane had shown [8] would imply Axiom A. (This result from [5] was later superseded by Mane's solution of the C 1 stability conjecture that structural stability alone implied Axiom A for diffeomorphisms [9].) Later, Chi, Choi and Park [2] addressed the question of whether bistability implies Axiom A for vector fields, and announced that it does (the stability conjecture for vector fields in dimensions greater than 3 is still open). There were some problems with their approach, as described by Wen in [12]. In addition to pointing out the difficulties with the argument in [2], Wen shows that bistability implies Axiom A for vector fields. The proof goes by checking that a bistable vector field satisfies conditions that Liao [7] has shown to imply Axiom A. For the most part, Wen's argument follows the same general lines as in the diffeomorphism case, using results from [4,5]; however it introduces a special argument (Lemma 4) to deal with the singular (fixed) points of the vector field. The point of this paper is to show that the special argument dealing with the fixed points can be replaced by a simpler argument from [6] that is more in the spirit of the rest of Wen's proof. (The results of [6] are given in terms of flows rather than vector fields, but this is not important as the same proofs work in either context.) For the most part we shall refer the reader to [12] for detailed definitions and background.The theorem of Liao that is the key to establishing Axiom A is given below. To say that a vector field is Axiom A means that its nonwandering set carries a hyperbolic structure and is the closure of its closed orbits (a closed orbit is either a fixed point or a nontrivial periodic orbit). See [2,11,12] for a more detailed description of Axiom A. If Z is a vector field, let Aj(Z) denote the closure of the set of (non-fixed) hyperbolic Received 21st March, 1994Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/95 SA2.00+0.00.

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Structurally stable flows

Abstract: We show that a C 1 -flow on a compact Riemannian manifold is structurally stable and topologically stable if and only if it satisfies Axiom A and the strong transversality condition. This improves Smale's conjecture for flows.

Cited by 2 publications

References 12 publications

Combined two stabilities imply Axiom A for vector fields

Combined two stabilities imply Axiom A for vector fields

Bistable vector fields are axiom A

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