Semi-classical limits of the first eigenfunction and concentration on the recurrent sets of a dynamical system

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. If the drift of the diffusion is given by a Morse-Smale vector field b, the limits of the eigenfunctions concentrate on the recurrent set of b. A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set. We consider generalized Morse-Smale vector fields, the recurrent set of which is composed of hyperbolic critical points, limit cycles and two dimensional torii. Under some compatibility conditions between the flow of b and the Riemannian metric g along each of these components, we prove that the support of a limit lies on those recurrent components of maximal dimension, where the topological pressure is achieved. Also, the restriction of the limit measure to either a cycle or a torus is absolutely continuous with respect to the unique invariant probability measure of the restriction of b to the cycle or the torus. When the torii are not charged, the restriction of the limit measure is absolutely continuous with respect to the arclength on the cycle and we have determined the corresponding density. Finally, the support of the limit measures and the support of the measures selected by the variational formulation of the topological pressure (TP) are identical.