First band of Ruelle resonances for contact Anosov flows in dimension $3$

Abstract: We show, using semiclassical measures and unstable derivatives, that a smooth vector field $X$ generating a contact Anosov flow on a $3$-dimensional manifold $\mathcal{M}$ has only finitely many Ruelle resonances in the vertical strips ${ s\in \mathbb{C}\ |\ {\rm Re}(s)\in [-\nu_{\min}+\epsilon,-\frac{1}{2}\nu_{\max}-\epsilon]\cup [-\frac{1}{2}\nu_{\min}+\epsilon,0]}$ for all $\epsilon>0$, where $0<\nu_{\min}\leq \nu_{\max}$ are the minimal and maximal expansion rates of the flow (the first strip only makes sense if $\nu_{\min}>\nu_{\max}/2$). We also show polynomial bounds in $s$ for the resolvent $(-X-s)^{-1}$ as $|{\rm Im}(s)|\to \infty$ in Sobolev spaces, and obtain similar results for cases with a potential. This is a short proof of a particular case of the results by Faure-Tsujii in \cite{FaTs1,FaTs2,FaTs3}, using that $\dim E_u=\dim E_s=1$.