Regularity of weak foliations for thermostats

Abstract

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 e. We show that if ϕ is Anosov, then it has weak stable and unstable foliations of class C1,1 if and only if the external field e has a global potential U, g1 ≔ e−2Ug has constant curvature and e−UΩ is a constant multiple of the area form of g1. We also give necessary and sufficient conditions for just one of the weak foliations to be of class C1,1 and we show that the combined effect of a thermostat and a magnetic field can produce an Anosov flow with a weak stable foliation of class C∞ and a weak unstable foliation which is not C1,1. 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 (1992 Ann. Inst. Fourier 42 209–47) can arise in this fashion.