{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T14:22:18Z","timestamp":1772547738876,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,6,3]],"date-time":"2010-06-03T00:00:00Z","timestamp":1275523200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Proc. Camb. Phil. Soc."],"published-print":{"date-parts":[[2010,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:italic>H<\/jats:italic>be a Hamiltonian,<jats:italic>e<\/jats:italic>\u2208<jats:italic>H<\/jats:italic>(<jats:italic>M<\/jats:italic>) \u2282 \u211d and<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>a connected component of<jats:italic>H<\/jats:italic><jats:sup>\u22121<\/jats:sup>({<jats:italic>e<\/jats:italic>}) without singularities. A Hamiltonian system, say a triple (<jats:italic>H<\/jats:italic>,<jats:italic>e<\/jats:italic>,<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>), is Anosov if<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>is uniformly hyperbolic. The Hamiltonian system (<jats:italic>H<\/jats:italic>,<jats:italic>e<\/jats:italic>,<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>) is a<jats:italic>Hamiltonian star system<\/jats:italic>if all the closed orbits of<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>are hyperbolic and the same holds for a connected component of<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004110000253_char1\"\/><\/jats:private-char><jats:sup>\u22121<\/jats:sup>({\u1ebd}), close to<jats:italic>\u0190<jats:sub>H, e<\/jats:sub><\/jats:italic>, for any Hamiltonian<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004110000253_char1\"\/><\/jats:private-char>, in some<jats:italic>C<\/jats:italic><jats:sup>2<\/jats:sup>-neighbourhood of<jats:italic>H<\/jats:italic>, and \u1ebd in some neighbourhood of<jats:italic>e<\/jats:italic>.<\/jats:p><jats:p>In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2<jats:italic>d<\/jats:italic>-dimensional manifold,<jats:italic>d<\/jats:italic>\u2265 2.<\/jats:p>","DOI":"10.1017\/s0305004110000253","type":"journal-article","created":{"date-parts":[[2010,6,3]],"date-time":"2010-06-03T13:15:10Z","timestamp":1275570910000},"page":"373-383","source":"Crossref","is-referenced-by-count":9,"title":["On the stability of the set of hyperbolic closed orbits of a Hamiltonian"],"prefix":"10.1017","volume":"149","author":[{"given":"M\u00c1RIO","family":"BESSA","sequence":"first","affiliation":[]},{"given":"C\u00c9LIA","family":"FERREIRA","sequence":"additional","affiliation":[]},{"given":"JORGE","family":"ROCHA","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2010,6,3]]},"reference":[{"key":"S0305004110000253_ref22","volume-title":"Institut de Math\u00e9matiques de Bourgogne","author":"Vivier","year":"2005"},{"key":"S0305004110000253_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(99)80439-X"},{"key":"S0305004110000253_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700001978"},{"key":"S0305004110000253_ref16","doi-asserted-by":"publisher","DOI":"10.2307\/2007021"},{"key":"S0305004110000253_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809187"},{"key":"S0305004110000253_ref11","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700006726"},{"key":"S0305004110000253_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755316"},{"key":"S0305004110000253_ref2","first-page":"1","article-title":"Geodesic flows on closed Riemannian manifolds of negative curvature","volume":"90","author":"Anosov","year":"1967","journal-title":"Proc. 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