Band structure of the Ruelle spectrum of contact Anosov flows

Faure, Frédéric; Tsujii, Masato

Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 385-391.

Si X est un champ de vecteur dʼAnosov de contact sur une variété compacte lisse M et si V∈C∞(M), il est connu que lʼopérateur différentiel A=−X+V a un spectre discret appelé résonances de Ruelle–Pollicott dans des espaces de Sobolev spécifiques. On montre que, pour |Imz|→∞, les valeurs propres de A sont incluses dans des bandes verticales et que, dans les gaps entre ces bandes, la résolvante de A est bornée uniformément par rapport à |Im(z)|. Dans chaque bande isolée, la densité des valeurs propres est donnée par une loi de Weyl. Dans la première bande, la plupart des valeurs propres se concentrent sur la ligne verticale Re(z)=〈D〉M, qui est la moyenne spatiale de la fonction D(x)=V(x)−12divX|Eu(x), où Eu est la distribution instable. Ce spectre en bande permet dʼexprimer le comportement asymptotique des fonctions de corrélations dynamiques.

If X is a contact Anosov vector field on a smooth compact manifold M and V∈C∞(M), it is known that the differential operator A=−X+V has some discrete spectrum called Ruelle–Pollicott resonances in specific Sobolev spaces. We show that for |Imz|→∞ the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band, the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate to the vertical line Re(z)=〈D〉M, the space average of the function D(x)=V(x)−12divX|Eu(x) where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.

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