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A counter-example to a C2 closing lemma

Published online by Cambridge University Press:  19 September 2008

C. Gutierrez
C. Gutierrez
Affiliation:
IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22 460, Brazil
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Abstract

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Let M be a compact manifold that contains a two-dimensional punctured torus. Given p ∈ M and an integer r ≥ 2, there exists X ∈ ∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood of X|(M−{p}) in r(M−{p}), no Y ∈ has closed orbits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 4 , December 1987 , pp. 509 - 530
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[Cx]Coexeter, H. S. M.. Introduction to Geometry. University of Toronto, New York, London, John Wiley and Sons, Inc. 1961.Google Scholar
[De]Denjoy, A.. Sur les courbes définies par les équations differentielles à la surface du tore. J. Mathematique 9(11) (1932), 333–375.Google Scholar
[Gu.1]Gutierrez, C.. Smoothing continuous flows on two-manifolds and recurrences. Ergod. Th. & Dynam. Sys. 6 (1986), 17–44.CrossRefGoogle Scholar
[Gu.2]Gutierrez, C.. On the C r-closing lemma for flows on the torus T 2. Ergod. Th. & Dynam. Sys. 6 (1986), 45–56.CrossRefGoogle Scholar
[Her]Herman, M.. Sur la conjugaison differentiable des diféomorphismes du cercle a des rotations. Pub. Math. #49, pp. 5–234.Google Scholar
[H-P-S]Hirsh, M., Pugh, C. & Shub, M.. Invariant Manifolds. Lecture Notes in Mathematics. Edited by Dold, A. and Eckmann, B.. Springer-Verlag. 1977.CrossRefGoogle Scholar
[La]Lang, S.. Introduction to Diophantine Approximations. Addison-Wesley Publishing Company. 1966.Google Scholar
[Ma]Mañé, R.. An ergodic closing lemma. Ann. of Math. 116 (1982), 503–541.CrossRefGoogle Scholar
[Me-Pa]de Melo, W. & Palis, J.. Geometric Theory of Dynamical Systems. Springer-Verlag. New York Inc. (1982).Google Scholar
[N-P-T]Newhouse, S., Palis, J. & Takens, F.. Bifurcations and stability of families of diffeomorphisms. Publ. IHES. #57 (1983), 5–71.CrossRefGoogle Scholar
[Pa-Sm]Palis, J. & Smale, S.. Structural stability theorems. In Global Analysis, Proc. Symp. Pure Math., 14 (1970). AMS, 223–231.CrossRefGoogle Scholar
[Pe]Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1 (1962) 101–120.CrossRefGoogle Scholar
[Px]Pixton, D.. Planar homoclinic points. J. Diff. Eq. 44 (1982), #3, 365–382.CrossRefGoogle Scholar
[Pg.1]Pugh, C.. Against the C 2-closing lemma. Jour. Diff. Eq. 17 (1975), 435–443.CrossRefGoogle Scholar
[Pg.2]Pugh, C.. An improved closing lemma and general density theorem. Amer. J. Math. 89 (1967) 1010–1021.CrossRefGoogle Scholar
[Pg.3]Pugh, C.. The C 1 connecting lemma: A counter-example. Preprint. U. C. Berkeley, 1984.Google Scholar
[Sl]Slater, N. B.. Gaps and steps for the sequence nθ mod 1. Proc. Camb. Phil. Soc. 63 (1967) 1115–1123.CrossRefGoogle Scholar
[So]Sotomayor, J.. Generic one-parameter families of vector fields on two-dimensional manifolds. Publ. Math. IHES. 44 1974.Google Scholar
[Ta]Takens, F.. Homoclinics points in conservative systems. Invent. Math. 18 (1972) 267–292.CrossRefGoogle Scholar