Fractal Weyl law for the Ruelle spectrum of Anosov flows

Abstract

On a closed manifold M, we consider a smooth vector field X that generates an Anosov flow. Let V ∈ C∞(M; R) be a smooth function called potential. It is known that for any C>0, there exists some anisotropic Sobolev space HC such that the operator A=−X +V has intrinsic discrete spectrum on Re(z)>−C called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by O(⟨ω⟩ n 1+β0) where ω=Im(z), n=dim M−1 and 0