First band of Ruelle resonances for contact Anosov flows in dimension $3$
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 $\{ s\in \mathbb{C}\ |\ {\rm Re}(s)\in [-ν_{\min}+ε,-\frac{1}{2}ν_{\max}-ε]\cup [-\frac{1}{2}ν_{\min}+ε,0]\}$ for all $ε>0$, where $0<ν_{\min}\leq ν_{\max}$ are the minimal and maximal expansion rates of the flow (the first strip only makes sense if $ν_{\min}>ν_{\max}/2$). 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 $\dim E_u=\dim E_s=1$.