Band structure of the Ruelle spectrum of contact Anosov flows

Abstract

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.