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 ε 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.
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 ε tend vers zéro. Nous supposons que le terme d'ordre 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.
Accepted:
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David Holcman 1; Ivan Kupka 2
@article{CRMATH_2005__341_4_243_0,
author = {David Holcman and Ivan Kupka},
title = {Concentration of the first eigenfunction for a second order elliptic operator},
journal = {Comptes Rendus. Math\'ematique},
pages = {243--246},
publisher = {Elsevier},
volume = {341},
number = {4},
year = {2005},
doi = {10.1016/j.crma.2005.06.035},
language = {en},
}
David Holcman; Ivan Kupka. Concentration of the first eigenfunction for a second order elliptic operator. Comptes Rendus. Mathématique, Volume 341 (2005) no. 4, pp. 243-246. doi : 10.1016/j.crma.2005.06.035. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.035/
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