Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Hamiltonian elliptic dynamics on symplectic -manifolds

Author(s): Mário Bessa; João Lopes Dias
Journal: Proc. Amer. Math. Soc. 137 (2009), 585-592.
MSC (2000): Primary 37J25, 37D30; Secondary 37C27
Posted: August 20, 2008
MathSciNet review: 2448579
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We consider -Hamiltonian functions on compact -dimensional symplectic manifolds to study the elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through . Moreover, this implies that, for far from Anosov regular energy surfaces of a -generic Hamiltonian, the elliptic closed orbits are generic.

References:

1.: A. Arbieto and C. Matheus,
A pasting lemma and some applications for conservative systems,
Ergod. Th. & Dynam. Sys. 27, 5 (2007), 1399-1417. MR 2358971
2.: M.-C. Arnaud,
The generic symplectic -diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point,
Ergod. Th. & Dynam. Sys. 22, 6 (2002), 1621-1639. MR 1944396 (2003k:37080)
3.: M. Bessa,
The Lyapunov exponents of generic zero divergence three-dimensional vector fields,
Ergod. Th. & Dynam. Sys. 27, 5 (2007), 1445-1472. MR 2358973
4.: M. Bessa and J. Lopes Dias,
Generic dynamics of -dimensional Hamiltonian systems,
Commun. Math. Phys 281, 3 (2008), 597-619.
5.: M. Bessa and P. Duarte,
Abundance of elliptical dynamics on conservative -flows,
Dynamical Systems-An International Journal (to appear).
6.: J. Bochi,
Genericity of zero Lyapunov exponents,
Ergod. Th. & Dynam. Sys. 22, 6 (2002), 1667-1696. MR 1944399 (2003m:37035)
7.: J. Bochi and B. Fayad,
Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for cocycles,
Bull. Braz. Math. Soc. 37 (2006), 307-349. MR 2267781 (2007m:37121)
8.: C. Bonatti, L. Díaz and M. Viana,
Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective,
Encycl. of Math. Sci. 102. Math. Phys. 3. Springer-Verlag, Berlin, 2005. MR 2105774 (2005g:37001)
9.: V. Horita and A. Tahzibi.
Partial hyperbolicity for symplectic diffeomorphisms.
Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 5 (2006), 641-661. MR 2259610 (2007j:37047)
10.: S. Newhouse,
Quasi-elliptic periodic points in conservative dynamical systems,
Amer. J. Math. 99 (1977), 1061-1087. MR 0455049 (56:13290)
11.: C. Pugh and C. Robinson,
The $C\sp 1$ closing lemma, including Hamiltonians,
Ergod. Th. & Dynam. Sys. 3 (1983), 261-313. MR 742228 (85m:58106)
12.: R. C. Robinson,
Generic properties of conservative systems, I and II,
Amer. J. Math. 92 (1970), 562-603 and 897-906. MR 0273640 (42:8517), MR 0279403 (43:5125)
13.: R. Saghin and Z. Xia,
Partial hyperbolicity or dense elliptical periodic points for -generic symplectic diffeomorphisms,
Trans. Amer. Math. Soc. 358, 11 (2006), 5119-5138. MR 2231887 (2007d:37044)
14.: T. Vivier,
Robustly transitive -dimensional regular energy surfaces are Anosov,
Institut de Mathématiques de Bourgogne, Dijon, Preprint 412 (2005). http://math.u- bourgogne.fr/topo/prepub/pre05.html

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|