Title: 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.