## CEMAPRE Seminar

### On the stability of the set of hyperbolic closed orbits and equilibria for Hamiltonians

Célia Ferreira
(FCUP-CMUP)

Abstract: A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e$, is said to be Anosov if there exists a connected component $E(H,e)$ 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 $E^*(H,e)$ of the energy level $H^{-1}(e)$ such that all the closed orbits and all the critical points of $E^*(H,e)$ are hyperbolic, and the same holds for a connected component of the energy level $H'^{-1}(e)$, close to $E^*(H,e)$, for any Hamiltonian $H'$, in some $C^2$-neighbourhood of $H$, and $e'$ in some neighbourhood of $e$. In this seminar we prove that for any four-dimensional Hamiltonian star level $(H,e)$ if the surface $E^*(H,e)$ does not contain critical points, then $X_H^t|_{E^*(H,e)}$ is Anosov; if $E^*(H,e)$ has critical points, then there exists $e'$, arbitrarily close to $e$, such that $X_H^t|_{E^*(H',e')}$ is Anosov.

Wednesday, February 24, 2010
Time: 14h30
Room: Sala 306, Edificio Quelhas, ISEG
http://cemapre.iseg.ulisboa.pt/seminars/cemapre/