论文标题
以零的响应性Anosov流动为零的谐振形式
Resonant forms at zero for dissipative Anosov flows
论文作者
论文摘要
我们以$ 3 $ - manifolds的流动为零地研究了零以零的共振差分形式。我们特别注意耗散案例，即，Anosov流动并不能保留绝对连续的措施。这样的流有两个明显的西奈 - 荷利 - $ 3 $ - 形式，$ω_ {\ text {srb}}}^{\ pm} $，以及同胞类$ [i _ {x} {x}ω_ {\ text {\文本{srb}}}}}}^{在确定共振$ 1 $ - 形式的空间中。当这两个类都消失时，我们将流量与$ \ textit {helicity} $相关联，该{helicity} $自然扩展了与零同源卷保持流量相关的经典概念。我们提供了一种一般理论，其中包括共鸣$ 1 $ - 形式和srb测量以及地图$ x \ mapsto的本地几何形状[i _ {x}ω_ {\ text {x}ω_{接下来，我们研究几个相关的示例类。其中包括与Holomormormormorgic二次差异相关的恒温器，从而引起了Ghys引入的准芬斯流动。对于这些流量，我们在零时明确计算所有共振$ 1 $ - forms，我们表明$ [i_ {x}ω_ {\ text {srb}}}}}^{\ pm}] = 0 $，并给出了螺旋的明确配方。此外，我们表明，准芬太西流的通用时间更改为半神经，因此在零时消失的ruelle zeta函数的顺序为$-χ（m）$，与地理流动案例相同。相比之下，我们表明，如果$（m，g）$是负弯曲的闭合表面，则由（小）谐波$ 1 $ - form驱动的高斯恒温器具有ruelle zeta函数，其在零下消失的顺序为$χ（m）-1 $。
We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $Ω_{\text{SRB}}^{\pm}$, and the cohomology classes $[ι_{X}Ω_{\text{SRB}}^{\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant $1$-forms. When both classes vanish we associate to the flow a $\textit{helicity}$ that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$-forms and SRB-measures as well as the local geometry of the maps $X\mapsto [ι_{X}Ω_{\text{SRB}}^{\pm}]$ near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant $1$-forms at zero, we show that $[ι_{X}Ω_{\text{SRB}}^{\pm}]=0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $-χ(M)$, the same as in the geodesic flow case. In contrast, we show that if $(M,g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$-form has a Ruelle zeta function whose order of vanishing at zero is $-χ(M)-1$.
