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 $\{s\in \mathbb{C}|\mathrm{Re}(s)\in [-{\nu }_{\min }+\epsilon ,-\frac{1}{2}{\nu }_{\max }-\epsilon ]\cup [-\frac{1}{2}{\nu }_{\min }+\epsilon ,0]\}$ for all $\epsilon >0$, where $0<{\nu }_{\min }\le {\nu }_{\max }$ 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 $|\mathrm{Im}(s)|\to \infty$ 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 $\dim E_u=\dim E_s=1$.

2 figures; revised version, to appear in Comm. Math. Phys

References in corpus (1)

• Smooth mixing Anosov flows in dimension three are exponential mixing