×
Mañé, Ricardo

A proof of the \(C^ 1\) stability conjecture. (English) Zbl 0678.58022

Publ. Math., Inst. Hautes Étud. Sci. 66, 161-210 (1988).
The fact that strong transversality plus Axiom A is a necessary and sufficient condition for a \(C^ r\) diffeomorphism on a closed manifold to be \(C^ r\) structurally stable is known as the Palais-Smale conjecture.
Sufficiency was proved in the 70’s by J. Robin and C. Robinson and necessity was reduced to proving that \(C^ r\) structural stability implies Axiom A. This problem became known as the stability conjecture. The main result of this paper is the following:
Theorem: Every \(C^ 1\) structurally stable diffeomorphism of a closed manifold satisfies Axiom A.
Reviewer: M.Teixeira

Cited in 11 Reviews
Cited in 108 Documents

MSC:

37C75 Stability theory for smooth dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Keywords:

diffeomorphism; structural stability
PDF BibTeX XML Cite
\textit{R. Mañé}, Publ. Math., Inst. Hautes Étud. Sci. 66, 161--210 (1988; Zbl 0678.58022)
Full Text: DOI Numdam
  • logo of hbz
  • logo of WorldCat

References:

[1] A. Andronov, L. Pontrjagin, Systèmes grossiers,Dokl. Akad. Nauk. SSSR,14 (1937), 247–251. · JFM 63.0728.01
[2] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature,Proc. Steklov Inst. Math.,90 (1967), 1–235. · Zbl 0176.19101
[3] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms,Springer Lecture Notes in Math.,470 (1975). · Zbl 0308.28010
[4] J. Franks, Necessary conditions for stability of diffeomorphisms,Trans. A.M.S.,158 (1971), 301–308. · Zbl 0219.58005
[5] J. Guckenheimer, A strange, strange attractor, inThe Hopf bifurcation and its applications, Applied Mathematics Series,19 (1976), 165–178, Springer Verlag.
[6] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds,Springer Lecture Notes in Math.,583 (1977). · Zbl 0355.58009
[7] R. Labarca, M. J. Pacifico, Stability of singular horseshoes,Topology,25 (1986), 337–352. · Zbl 0611.58033
[8] S. T. Liao, On the stability conjecture,Chinese Ann. Math.,1 (1980), 9–30. · Zbl 0449.58013
[9] R. Mañé, Persistent manifolds are normally hyperbolic,Trans. A.M.S.,246 (1978), 261–283. · Zbl 0362.58014
[10] R. Mañé, Expansive diffeomorphisms, inDynamical Systems-Warwick 1974,Springer Lecture Notes in Math.,468 (1975), 162–174.
[11] R. Mañé, Characterization of AS diffeomorphisms,Proc. ELAM III, Springer Lecture Notes in Math.,597 (1977), 389–394.
[12] R. Mañé, An ergodic closing lemma,Ann. of Math.,116 (1982), 503–540. · Zbl 0511.58029
[13] R. Mañé, On the creation of homoclinic points,Publ. Math. I.H.E.S.,66 (1987), 139–159. · Zbl 0669.58024
[14] S. Newhouse, Lectures on dynamical systems,Progr. in Math.,8 (1980), 1–114. · Zbl 0444.58001
[15] J. Palis, A note on {\(\Omega\)}-stability, inGlobal Analysis, Proc. Sympos. Pure Math., A.M.S.,14 (1970), 221–222.
[16] J. Palis, S. Smale, Structural stability theorems, inGlobal Analysis, Proc. Sympos. Pure Math., A.M.S.,14 (1970), 223–231. · Zbl 0214.50702
[17] V. A. Pliss, Analysis of the necessity of the conditions of Smale and Robbin for structural stability of periodic systems of differential equations,Diff. Uravnenija,8 (1972), 972–983. · Zbl 0286.34088
[18] V. A. Pliss, On a conjecture due to Smale,Diff. Uravnenija,8 (1972), 268–282. · Zbl 0243.34077
[19] C. Pugh, The closing lemma,Amer. J. Math.,89 (1967), 956–1009. · Zbl 0167.21803
[20] J. Robbin, A structural stability theorem,Ann. of Math.,94 (1971), 447–493. · Zbl 0224.58005
[21] C. Robinson, Cr structural stability implies Kupka-Smale, inDynamical Systems, Salvador, 1971, Academic Press 1973, 443–449.
[22] C. Robinson, Structural stability of C1 diffeomorphisms,J. Diff. Eq.,22 (1976), 28–73. · Zbl 0343.58009
[23] M. Shub, Stabilité globale des systèmes dynamiques,Astérisque, 56 (1978). · Zbl 0396.58014
[24] S. Smale, Diffeomorphisms with many periodic points, inDifferential and Combinatorial Topology, Princeton Univ. Press, 1964, 63–80.
[25] S. Smale, Differentiable dynamical systems,Bull. A.M.S.,73 (1967), 747–817. · Zbl 0202.55202
[26] S. Smale, The {\(\Omega\)}-stability theorem, inGlobal Analysis, Proc. Sympos. Pure Math., A.M.S.,14 (1970), 289–297.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.