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On a closed manifold $M$, we consider a smooth vector field $X$ thatgenerates an Anosov flow. Let $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be asmooth function called potential. It is known that for any $C>0$, there existssome 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 resonances. In this paper, we show a fractal Weyl law: thedensity 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 $0<\beta_{0}\leq1$ is the H\"older exponent of thedistribution $E_{u}\oplus E_{s}$ (strong stable and unstable). We also obtainsome more precise results concerning the wave front set of the resonances andthe invertibility of the transfer operator. Since the dynamical distributions$E_{u}$,$E_{s}$ are non smooth, we use some semi-classical analysis based onwave packet transform associated to an adapted metric $g$ on $T^{*}M$ andconstruct some specific anisotropic Sobolev spaces.

Fractal Weyl law for the Ruelle spectrum of Anosov flows