Holomorphic differentials, thermostats and Anosov flows

Abstract

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian $2$-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

arXiv commentFinal version, exposition improved, references added, 23 pages

References in corpus (5)

• Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics
• SRB states and nonequilibrium statistical mechanics close to equilibrium
• Regularity of weak foliations for thermostats
• Entropy production in Gaussian thermostats
• Hopf type rigidity for thermostats

Cited by in corpus (8)

• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• Geometric Theory of Weyl Structures
• Minimal Lagrangian connections on compact surfaces
• Vortices over Riemann surfaces and dominated splittings
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
• Smooth orbit equivalence rigidity for dissipative geodesic flows
• Thermostats without conjugate points