Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-27T08:14:37.473Z Has data issue: false hasContentIssue false

On the generic existence of homoclinic points

Published online by Cambridge University Press:  19 September 2008

Fernando Oliveira
Affiliation:
Departamento de Matemática, ICEX-UFMG, Caixa Postal 702, 30161-Belo Horizonte-MG, Brazil
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.

This work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Birkhoff, G. D.. Collected Mathematical Papers Vol. II, p. 350, American Mathematical Society, 1950.Google Scholar
[2]Mather, J. N.. Invariant subsets for area-preserving homeomorphisms of surfaces, mathematical analysis and applications, Part B. Advances in Mathematics Supplementary Studies, Volume 7B.Google Scholar
[3]Newhouse, S. E.. Quasi-elliptic periodic points in conservative dynamical systems. Amer. J. Math. Vol. 99 (1977) No. 5, 10611087.Google Scholar
[4]Pixton, D.. Planar homoclinic points. J. Differential Equations 44 (1982), 365382.Google Scholar
[5]Poincaré, H.. Les Methodes Nouvelles de la Mécanique Celeste, Tome II. 1899.Google Scholar
[6]Robinson, C.. Generic Properties of conservative systems, I, II, Amer. J. of Math. 92 (1970), 562603, 897906.Google Scholar
[7]Robinson, C.. Closing stable and unstable manifolds on the two-sphere. Proc. Amer. Math. Soc. 41 (1973), 299303.Google Scholar
[8]Smale, S.. Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology, Princeton University Press, Princeton, New Jersey, 1965.Google Scholar
[9]Stillwell, J.. Classical Topology and Combinatorial Group Theory. Graduate Texts in Mathematics, Volume 72, Springer-Verlag.Google Scholar
[10]Takens, F.. Homoclinic points in conservative systems. Invent. Math. 18 (1972), 267292.Google Scholar
[11]Zehnder, E.. Homoclinic points near elliptic fixed points. Comm. Pure Appl. Math. XXVI (1973), 131182.Google Scholar