Band structure of the Ruelle spectrum of contact Anosov flows

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) − 1 2divX|Eu(x)where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.