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.

References in corpus (6)

• Dynamical Ensembles in Nonequilibrium Statistical Mechanics
• Dynamical ensembles in stationary states
• Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics
• SRB states and nonequilibrium statistical mechanics close to equilibrium
• Reversible Anosov diffeomorphisms and large deviations
• The longitudinal KAM-cocycle of a magnetic flow

Cited by in corpus (7)

• Holomorphic differentials, thermostats and Anosov flows
• Hopf type rigidity for thermostats
• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• Resonant forms at zero for dissipative Anosov flows
• The attenuated ray transforms on Gaussian thermostats with negative curvature
• On the injectivity of the X-ray transform for Anosov thermostats
• Thermostats without conjugate points