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 $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be a smooth potential function. It is known that for any $C>0$, there exists some anisotropic Sobolev space $\mathcal{H}_{C}$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle-Pollicott resonances. In this paper, we show that the density of resonances is bounded by $O\left(\left\langle \omega\right\rangle ^{\frac{n}{1+\beta_{0}}}\right)$ where $\omega=\mathrm{Im}\left(z\right)$, $n=\mathrm{dim}M-1$ and $0