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LION, JEAN-MARIE MOUSSU, ROBERT Sanz Sánchez, Fernando 2024-06-22T10:18:41Z 2024-06-22T10:18:41Z 2002 Ergod. Th. & Dynam. Sys. (2002), 22, 525–534 0143-3857 https://uvadoc.uva.es/handle/10324/68194 10.1017/S0143385702000251 525 02 534 Ergodic Theory and Dynamical Systems 22 1469-4417 A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thom’s Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times) ; this can also be applied to some gradient vector fields. Travail financ´e par le CNRS et le r´eseau europ´een TMR Sing.Ec.Diff. et Feuilletages application/pdf fra Cambridge University Press info:eu-repo/semantics/restrictedAccess Cambridge University Press Champs de vecteurs analytiques et champs de gradients info:eu-repo/semantics/article info:eu-repo/semantics/submittedVersion SI