×

The \(C^1\) closing lemma. (Le “closing lemma” en topologie \(C^1\).) (French) Zbl 0920.58039

The \(C^1\) closing lemma asserts that, given a dynamical system \(f\) in a compact manifold \(M\) and a recurrent point \(p\) of \(f\), there exists a dynamical system \(g\) in any neighborhood of \(f\) in the \(C^1\) topology such that \(p\) is a periodic point of \(g\). The lemma was first proved by C. C. Pugh [Am. J. Math. 89, 956-1009 (1967; 167.21803)]. The proof was very complicated and had a minor error. An improved proof was given by C. C. Pugh and C. Robinson [Ergodic Theory Dyn. Syst. 3, 261-313 (1983; 548.58012)]. But their proof is still hard. Mai gave a simpler proof of the lemma based on Liao’s techniques and his own ones of movements under a sequence of linear transformations [J.-H. Mai, Sci. Sin., Ser. A 29, 1020-1031 (1986; Zbl 0616.58024); Chin. Sci. Bull. 34, 179-184 (1989; Zbl 0686.58030)]. Mañé studied the stability properties of discrete dynamical systems, i.e., \(C^1\)-diffeomorphisms of closed manifolds [Ann. Math. (2) 116, 503-540 (1982; Zbl 0511.58029)].
In the paper under review, the author uses Mai’s algebraic result to simplify the proof of the \(C^1\) closing lemma and solves a new case of symplectic fields. Moreover, the author generalizes the \(C^1\) ergodic closing lemma of Mañé (the ergodic version of the orbit closing lemma) to the Borelian positive measures, finite on every compact, defined on noncompact manifolds.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37A99 Ergodic theory
37C10 Dynamics induced by flows and semiflows
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] R. ABRAHAM et J. MARSDEN - Foundations of mechanics , Benjamin N.Y., 1967 . Zbl 0158.42901 · Zbl 0158.42901
[2] V. ARNOLD - Méthodes mathématiques de la mécanique classique , MIR, 1976 . MR 57 #14033a | Zbl 0385.70001 · Zbl 0385.70001
[3] V. ARNOLD et A. AVEZ - Problèmes ergodiques de la mécanique classique , Gauthier-Villars, 1967 . MR 35 #334 | Zbl 0149.21704 · Zbl 0149.21704
[4] C. GUTIERREZ - ”A counter-exemple to a C2 closing lemma” , Erg. Th. and Dyn. Syst. 7 ( 1987 ), p. 509-530. MR 89k:58240 | Zbl 0642.58036 · Zbl 0642.58036
[5] M. HERMAN - ”Exemple de flots hamiltoniens dont aucune perturbation en topologie C\infty n’a d’orbites périodiques sur un ouvert de surfaces d’énergie” , C.R.A.S. ( 1991 ), no. 313, p. 49-51. MR 92m:58046 | Zbl 0759.58016 · Zbl 0759.58016
[6] MAI JIEHUA - ”A simpler proof of C1 closing lemma” , Scientia Sinica 10 ( 1986 ), no. XXIV, p. 1020-1031. Zbl 0616.58024 · Zbl 0616.58024
[7] , ”A simpler proof of the extended C1 closing lemma” , Chinese Science Bull. 34-3 ( 1989 ), p. 180-184. · Zbl 0686.58030
[8] R. MAÑÉ - ”An ergodic closing lemma” , Annals of Mathematics 116 ( 1982 ), p. 503-540. MR 84f:58070 | Zbl 0511.58029 · Zbl 0511.58029
[9] J. MOSER - ”On the volume element on a manifold” , Trans. Amer. Math. Soc. 120 ( 1965 ), p. 286-294. MR 32 #409 | Zbl 0141.19407 · Zbl 0141.19407
[10] , ”Proof of a generalized form of a fixed point theorem due to G.D. Birkhoff” , Springer Lect. Notes in Math. 597 ( 1977 ), p. 464-494. MR 58 #13205 | Zbl 0358.58009 · Zbl 0358.58009
[11] J. PALIS et W. DE MELO - Geometric theory of dynamical systems , Springer-Verlag, 1982 . MR 84a:58004 | Zbl 0491.58001 · Zbl 0491.58001
[12] J. PALIS et C. PUGH - ”Fifty problems in dynamical systems” , L.N. in Math. 468 ( 1974 ), p. 345-353. MR 58 #31134 | Zbl 0304.58011 · Zbl 0304.58011
[13] C. PUGH - ”The closing lemma” , Amer. J. Math. 89 ( 1967 ), p. 956-1009. MR 37 #2256 | Zbl 0167.21803 · Zbl 0167.21803
[14] , ”An improved closing lemma and a general density theorem” , Amer. J. Math. 89 ( 1967 ), p. 1010-1021. MR 37 #2257 | Zbl 0167.21804 · Zbl 0167.21804
[15] C. PUGH et C. ROBINSON - ”The C1 closing lemma, including hamiltonians” , Erg. Th. & Dyn. Syst. 3 ( 1983 ), p. 261-314. MR 85m:58106 | Zbl 0548.58012 · Zbl 0548.58012
[16] C. ROBINSON - ”Introduction to the closing lemma” , Springer Lect. Notes in Math. 668 ( 1978 ), p. 223-230. MR 80d:58061 | Zbl 0403.58020 · Zbl 0403.58020
[17] M. SHUB - ”Stabilité globale des systèmes dynamiques” , Asterisque 56 ( 1978 ). MR 80c:58015 | Zbl 0396.58014 · Zbl 0396.58014
[18] L. WEN - ”On the C1-stability conjecture of flows” , J. Diff. Equations 129 ( 1996 ), p. 334-357. MR 97j:58082 | Zbl 0866.58050 · Zbl 0866.58050
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.