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On the stability of the set of hyperbolic closed orbits of a hamiltonian
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Abstract(s)
A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e
\in \mathbb{R}$, is said to be Anosov if there exists a connected component
$\mathcal{E}_{H,e}$ of $H^{-1}({e})$ which is uniformly hyperbolic for the
Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star
system if there exists a connected component $\mathcal{E}^\star_{H,e}$ of the
energy level $H^{-1}({{e}})$ such that all the closed orbits and all the
critical points of $\mathcal{E}^\star_{H,e}$ are hyperbolic, and the same holds
for a connected component of the energy level $\tilde{H}^{-1}({\tilde{e}})$,
close to $\mathcal{E}^\star_{H,e}$, for any Hamiltonian $\tilde{H}$, in some
$C^2$-neighbourhood of $H$, and $\tilde{e}$ in some neighbourhood of $e$.
In this article we prove that for any four-dimensional Hamiltonian star level
$(H,e)$ if the surface $\mathcal{E}^\star_{H,e}$ does not contain critical
points, then $X_H^t|_{\mathcal{E}^\star_{H,e}}$ is Anosov; if
$\mathcal{E}^\star_{H,e}$ has critical points, then there exists $\tilde{e}$,
arbitrarily close to $e$, such that $X_H^t|_{\mathcal{E}^\star_{H,\tilde{e}}}$
is Anosov.
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Citation
M. Bessa, C. Ferreira, J. Rocha, On the Stability of the Set of Hyperbolic Closed Orbits of a Hamiltonian, Mathematical Proceedings of the Cambridge Philosophical Society, 149, 2, 373-383, 2009
Publisher
Cambridge University Press