Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T13:19:51.535Z Has data issue: false hasContentIssue false
  • English
  • Français

The C1 Closing Lemma, including Hamiltonians

Published online by Cambridge University Press:  19 September 2008

Charles C. Pugh  and
Clark Robinson
Charles C. Pugh
Affiliation:
University of California, Berkeley, California 94720, USA
Clark Robinson
Affiliation:
Northwestern University, Evanston, Illinois 60201, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C1 diffeomorphisms to C1 Hamiltonian vector fields.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 2 , June 1983 , pp. 261 - 313
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Abraham, R. & Marsden, J.. Foundations of Mechanics. Benjamin: New York, 1967.Google Scholar
[2]Abraham, R. & Robbin, J.. Transversal Mappings and Flows. Benjamin: New York, 1967.Google Scholar
[3]Choquet, G.. Lectures on Analysis, Vol. 1. Benjamin: New York, 1969.Google Scholar
[4]Dorroh, J.. Local groups of differentiable transformations. Math. Ann. 192 (1971), 243–249.CrossRefGoogle Scholar
[5]Hart, D.. On the smoothness of generators for flows and foliations. Thesis, University of California: Berkeley, 1980.Google Scholar
[6]Hartman, P.. Ordinary Differential Equations. Wiley: New York, 1964.Google Scholar
[7]Hirsch, M.. Differential Topology: Springer-Verlag: Berlin, 1976.CrossRefGoogle Scholar
[8]Lang, S.. Introduction to Differential Manifolds. Interscience: New York, 1962.Google Scholar
[9]Moser, J.. On the volume element of a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294.Google Scholar
[10]Palis, J.. Vector fields generate few diffeomorphisms. Bull AMS 80 (1973), 503–505.CrossRefGoogle Scholar
[11]Palis, J. & Pugh, C.. Fifty problems in dynamical systems. Lecture Notes in Math. 468, 345–353. Springer-Verlag: Berlin, 1975.Google Scholar
[12]Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1, (1962), 101–120.CrossRefGoogle Scholar
[13]Peixoto, M.. On an approximation theorem of Kupka and Smale. J. Diff. Equat. 3 (1967), 214–227.CrossRefGoogle Scholar
[14]Poincaré, H.. Les Methodes Nouvelles de la Mecanique Célèste, Vol. 1, p. 82. Gauthier-Villars: Paris, 1892.Google Scholar
[15]Pugh, C.. The Closing Lemma. Amer. J. Math. 89 (1967), 956–1009.Google Scholar
[16]Pugh, C.. An improved Closing Lemma and a General Density Theorem. Amer. J. Math. 89 (1967), 1010–1021.Google Scholar
[17]Pugh, C.. On arbitrary sequences of isomorphisms n → n. Trans. Amer. Math. Soc. 184 (1973), 387–400.Google Scholar
[18]Pugh, C.. Against the C 2 Closing Lemma. J. Diff. Equat. 17 (1975), 435–443.CrossRefGoogle Scholar
[19]Robinson, C.. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562–603.CrossRefGoogle Scholar
[20]Robinson, C.. Introduction to the Closing Lemma, structure of attractors in dynamical systems. Lecture Notes in Math. 668, 223–230. Springer-Verlag: Berlin, 1978.Google Scholar
[21]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–819.Google Scholar
[22]Wilson, W.. On the minimal sets of non-singular vector fields. Ann. Math. 84 (1966), 529–536.CrossRefGoogle Scholar