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 M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document} has only finitely many Ruelle resonances in the vertical strips {s∈C|Re(s)∈[-νmin+ε,-12νmax-ε]∪[-12νmin+ε,0]}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ s\in \mathbb {C}\ |\ \mathrm{Re}(s)\in [-\nu _{\min }+\varepsilon ,-\frac{1}{2}\nu _{\max }-\varepsilon ]\cup [-\frac{1}{2}\nu _{\min }+\varepsilon ,0]\}$$\end{document} for all ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}, where 0<νmin≤νmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\nu _{\min }\le \nu _{\max }$$\end{document} are the minimal and maximal expansion rates of the flow (the first strip only makes sense if νmin>νmax/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{\min }>\nu _{\max }/2$$\end{document}). We also show polynomial bounds in s for the resolvent (-X-s)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-X-s)^{-1}$$\end{document} as |Im(s)|→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathrm{Im}(s)|\rightarrow \infty $$\end{document} 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, using that dimEu=dimEs=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim E_u=\dim E_s=1$$\end{document}.