论文标题
ruelle共鸣的第一乐队接触Anosov流动的Dimension $ 3 $
First band of Ruelle resonances for contact Anosov flows in dimension $3$
论文作者
论文摘要
我们使用半古典措施和不稳定的衍生物表明，平稳的向量字段$ x $生成触点Anosov流量，以$ 3 $的二维流形$ \ MATHCAL {M MATHCAL {M MATHCAL {M MATHCAL {M MATHCAL {M MATHCAL {M MATHCAL {M} $在垂直条$ \ \ in \ Mathbb {C c} c} c} c} c} c} c} c \ c \ re \ re \ re \ c \ c \ c \ cor中只有有限的许多ruelle共振。 [ - ν_ {\ min}+ε， - \ frac {1} {2}ν_ {\ max}-ε] \ cup [ - \ frac {1} {2}才员ν_ {\ max} $是流量的最小和最大扩展率（仅当$ν_ {\ min}>ν_ {\ max}/2 $时，第一条才有意义。我们还以$ s $在$ s $中显示了$ s $（-x-x-s）^{ - 1} $作为$ | {\ rm im}（s）| \ to \ to \ to \ sobolev空间中的\ infty $，并获得具有潜力案例的案例的相似结果。这是faure-tsujii在\ cite {fats1，fats2，fats3}中的结果的简短证明，使用该$ \ dim e_u = \ dim dim e_s = 1 $。
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 $\dim E_u=\dim E_s=1$.
