Regularity of weak foliations for thermostats

Abstract

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $\varphi$ determined by the motion of a particle under the influence of a magnetic field $\Omega$ and a thermostat with external field $\mathbf{e}$. We show that if $\varphi$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and only if the external field $\mathbf{e}$ has a global potential $U$, $g_1:=e^{-2U}g$ has constant curvature and $e^{-U}\Omega$ 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.

Cited by in corpus (1)

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