{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T12:12:25Z","timestamp":1773403945326,"version":"3.50.1"},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2007,10,1]],"date-time":"2007-10-01T00:00:00Z","timestamp":1191196800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ergod. Th. Dynam. Sys."],"published-print":{"date-parts":[[2007,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We prove that for a <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup>-generic (dense <jats:italic>G<\/jats:italic><jats:sub><jats:italic>\u03b4<\/jats:italic><\/jats:sub>) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point <jats:italic>p<\/jats:italic>\u2208<jats:italic>M<\/jats:italic> that either the Lyapunov exponents at <jats:italic>p<\/jats:italic> are zero or <jats:italic>X<\/jats:italic> is an Anosov vector field. Then we prove that for a <jats:italic>C<\/jats:italic><jats:sup>1<\/jats:sup>-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. <jats:italic>p<\/jats:italic>\u2208<jats:italic>M<\/jats:italic> that either the Lyapunov exponents at <jats:italic>p<\/jats:italic> are zero or <jats:italic>p<\/jats:italic> belongs to a compact invariant set with dominated splitting for the linear Poincar\u00e9 flow.<\/jats:p>","DOI":"10.1017\/s0143385707000107","type":"journal-article","created":{"date-parts":[[2007,8,23]],"date-time":"2007-08-23T08:42:24Z","timestamp":1187858544000},"page":"1445-1472","source":"Crossref","is-referenced-by-count":25,"title":["The Lyapunov exponents of generic zero divergence three-dimensional vector fields"],"prefix":"10.1017","volume":"27","author":[{"given":"M\u00c1RIO","family":"BESSA","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2007,10,1]]},"reference":[{"key":"S0143385707000107_ref6","volume-title":"Advances in Dynamical Systems","author":"Bochi","year":"2004"},{"key":"S0143385707000107_ref12","first-page":"197","article-title":"A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems","volume":"19","author":"Oseledets","year":"1968","journal-title":"Trans. Moscow Math. Soc."},{"key":"S0143385707000107_ref8","first-page":"59","article-title":"Persistently transitive vector fields on three-dimensional manifolds","volume":"160","author":"Doering","year":"1987","journal-title":"Proc. Dynam. Sys. Bifur. Theory"},{"key":"S0143385707000107_ref13","doi-asserted-by":"publisher","DOI":"10.2307\/2373361"},{"key":"S0143385707000107_ref2","unstructured":"[2] Bessa M. . The Lyapunov exponents of conservative continuous-time dynamical systems. Thesis, IMPA (C048\/2006), 2005."},{"key":"S0143385707000107_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02584629"},{"key":"S0143385707000107_ref5","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2005.161.1423"},{"key":"S0143385707000107_ref9","first-page":"1269","volume-title":"Proc. Int. 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