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Comptes Rendus Mathématique
Volume 341, n° 4
pages 243-246 (août 2005)
Doi : 10.1016/j.crma.2005.06.035
Received : 9 June 2005 ;  accepted : 25 June 2005
Concentration of the first eigenfunction for a second order elliptic operator
Concentration de la première fonction propre pour un opérateur elliptique du second ordre
 

David Holcman a , Ivan Kupka b
a Department of Mathematics, Weizmann Institute of Science 76100 Rehovot, Israel 
b Département de mathématiques, 175, rue du Chevaleret, 75013 Paris, France 

Abstract

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant goes to zero. We assume that the first order term is given by a vector field b , whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure [Y. Kifer, Principal eigenvalues, topological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1990) (1) 1-47] is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis. To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

Nous étudions sur une variété Riemannienne compacte, les limites semiclassiques de la première fonction propre associée à un opérateur positif du second ordre positif divers quand la constante de diffusion tend vers zéro. Nous supposons que le terme dʼordre un est un champ de vecteur b , dont les ensembles récurrents sont des points hyperboliques ou des cycles ou des tores à deux dimensions. Les limites de la fonction propre normalisée sont concentrées sur les ensembles récurrents de dimension maximale où la pression topologique est atteinte. Sur le cycles et les tores, les mesures limites sont absolument continues par rapport à la mesure de probabilitè invariante par b . Nous avons déterminé ces limites en utilisant une analyse de type blow-up. Pour citer cet article : D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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