恒温器弱叶理的规律性
Regularity of weak foliations for thermostats
摘要：
设$M$是具有黎曼度量$g$的闭定向曲面。我们考虑由磁场$Ω$和具有外场${\bf e}$的恒温器的影响下的粒子运动所确定的流动$ϕ$。我们证明了如果$ϕ$是Anosov，那么它具有弱稳定和不稳定的$C^{1,1}$类叶理当且仅当外场${\bf e}$具有全局势$U$，$g_{1}:=e^{-2U}g$具有常曲率，并且$e^{-U}Ω$是$g_1$的面积形式的常倍数.我们还给出了一个弱叶理是$C^{1,1}$类的充分必要条件，我们表明，{ \it组合}的恒温器和磁场的效果可以产生一个Anosov流与弱稳定叶理类$C^{\infty}$和弱不稳定叶理这是{ \it不} $C^{1,1}$。最后，我们研究Anosov恒温器依赖于二次的速度，我们的特点是光滑弱叶理。特别是，我们表明，准fuchsian流定义的Ghys在\cite{Ghy1}可以出现在这种方式。
英文摘要：
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $ϕ$ determined by the motion of a particle under the influence of a magnetic field $Ω$ and a thermostat with external field ${\bf e}$. We show that if $ϕ$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and only if the external field ${\bf e}$ has a global potential $U$, $g_{1}:=e^{-2U}g$ has constant curvature and $e^{-U}Ω$ is a constant multiple of the area form of $g_1$. We also give necessary and sufficient conditions for just one of the weak foliations to be of class $C^{1,1}$ and we show that the {\it combined} effect of a thermostat and a magnetic field can produce an Anosov flow with a weak stable foliation of class $C^{\infty}$ and a weak unstable foliation which is {\it not} $C^{1,1}$. Finally we study Anosov thermostats depending quadratically on the velocity and we characterize those with smooth weak foliations. In particular, we show that quasi-fuchsian flows as defined by Ghys in \cite{Ghy1} can arise in this fashion.
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