On an approximation theorem of Kupka and Smale

Peixoto, M. M.

doi:10.1016/0022-0396(67)90026-5

Cited by 119 publications

(82 citation statements)

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“…The proof for diffeomorphisms follows that given for Proof. This lemma is an equivariant version of Lemma 3 in [20]. The proof depends on a compactness argument similar to that given by Peixoto and we omit details.…”

Section: Proposition D a Is An Elementary Periodic Orbit Of X If Andmentioning

confidence: 86%

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Equivariant dynamical systems

Field¹

1980

Trans. Amer. Math. Soc.

151

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Abstract.The basic properties of vector fields and diffeomorphisms invariant under the action of a compact Lie group are presented. A Kupka-Smale density theorem for equivariant dynamical systems and an existence theorem for equivariant Morse-Smale systems on an arbitrary compact G-manifold are proved.Introduction. In this work we develop the basic properties of vector fields and diffeomorphisms invariant under the action of a compact Lie group. Earlier versions of this theory were announced in [7], [8] though these results are weaker in certain significant respects than those presented here.We prove two fundamental theorems: A Kupka-Smale density theorem for equivariant dynamical systems (Theorem B of §9) and an existence theorem for equivariant Morse-Smale vector fields on a compact G-manifold (Theorem D of §10). The latter theorem has interesting implications for the theories of equivariant dynamical systems and equivariant differential topology, notably for a generalisation of the C° isotopy and approximation theorems of Shub and Smale [23], [27] to equivariant maps and we intend to pursue these matters elsewhere. We hope also to make applications of part of our theory to bifurcation problems involving a loss of symmetry (cf. S. Schecter, Bifurcations with symmetry [17, pp. 224-249]). The definition of Í2G in §10 is particularly relevant here as is the theory of equivariant general position or G-transversality for the invariant manifolds of elementary critical elements (see §9 and also below). In § §1, 2 we outline the basic properties of smooth G-manifolds and equivariant flows and diffeomorphisms that we need. In §3 we define elementary invariant (7-orbits in terms of the normal hyperbolicity conditions of Hirsh, Pugh and Shub [15] and then give a spectral characterisation of such orbits. In §4 we make a similar discussion of periodic orbits followed in § §5, 6 by the basic local theory of equivariant dynamical systems. In §7, we prove that Cr 1-generic (resp. 2-generic) equivariant vector fields and diffeomorphisms form an open dense set (resp. residual set), r > 1. Apart from the complications introduced by the presence of the G-action, the proofs follow Peixoto [20]. Stable manifold theory is discussed in §8. In §9, we prove a Kupka-Smale density theorem for C°° equivariant dynamical systems. As there is yet no Cr-theory of equivariant general position, r < oo, we have no C density theorem. It should be pointed out that the theory of equivariant

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Section: Proposition D a Is An Elementary Periodic Orbit Of X If Andmentioning

confidence: 86%

“…Our approach is similar to Peixoto's proof of the Kupka-Smale theorem [20]. The proof for diffeomorphisms follows that given for Proof.…”

Section: Proposition D a Is An Elementary Periodic Orbit Of X If Andmentioning

confidence: 89%

Equivariant dynamical systems

Field¹

1980

Trans. Amer. Math. Soc.

151

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“…a countable intersection of open and dense sets). Peixoto [23] extended this result to non-compact manifolds without boundary. Since for dissipative vector fields on R n + the non-compact proof applies to the interior of any face of R n + , it follows that Kupka-Smale vector fields as defined above are residual in P n .…”

Section: Can We Approximate F By a Vector Field With Robust Communities?mentioning

confidence: 87%

Successional stability of vector fields in dimension three

Schreiber

1999

Proc. Amer. Math. Soc.

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Abstract. A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the C 1 Whitney topology by a successionally stable vector field.

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“…According to the theorem of Kupka-Smale, the weak transversality condition is a generic property (see [3], [22], [35], and [50]). In contrast, Smale [49] gives an example of an open set N ç Diff (M) such that every ƒ G N satisfies axiom A and no feN satisfies the strong transversality condition.…”

Section: Theorem Let F E Diff(m) Where M Is Compact Arid Suppose F Imentioning

confidence: 99%

Topological conjugacy and structural stability for discrete dynamical systems

Robbin¹

1972

Bull. Amer. Math. Soc.

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ABSTRACT. Topological conjugacy and various concepts of structural stability are defined, motivated, and criticized. Two basic problems emerge: characterization of structural stability and classification up to topological conjugacy. Solutions to these problems are outlined for linear automorphisms and the general characterization problem is discussed.This paper is an expanded version of an invited address given at the summer meeting of the American Mathematical Society in September, 1971. Much of the material here was also covered in a series of lectures given at the Institut des Hautes Etudes Scientifiques in the fall of 1971.Our notation is that of AMS 1970 subject classifications. Primary 58F00; Secondary 58F10, 58F15. Key words and phrases. Cascade, flow, topological conjugacy, differentiable conjugacy, structural stability, north pole-south pole map, hyperbolic toral automorphism, strong structural stability, absolute structural stability, relative structural stability, symplectic manifold, twist stability, topological stability, semistability, Yin-Yang problem, hyperbolic linear automorphism, in-set, out-set, on-set, Anosov diffeomorphism, selector, conjugacy selector, adjoint representation, ergodic, infinitesimally ergodic, Sobolev space, hyperbolic invariant set, nonwandering set, Q-hyperbolic, P-hyperbolic, axiom A, weak axiom A, essential spectrum, strong transversahty condition, weak transversality condition, in-set, stable manifold, out-set, unstable manifold.

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On an approximation theorem of Kupka and Smale

Cited by 119 publications

References 3 publications

Equivariant dynamical systems

Equivariant dynamical systems

Successional stability of vector fields in dimension three

Topological conjugacy and structural stability for discrete dynamical systems

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