# Holomorphic differentials, thermostats and Anosov flows

## TL;DR

This paper introduces a new family of thermostat flows on the unit tangent bundle of Riemannian 2-manifolds, connecting holomorphic differentials with dynamical properties like Anosov behavior and invariant measures.

## Contribution

It establishes a parametrization of these flows via weighted holomorphic differentials and analyzes their dynamical properties, including dominated splitting and conditions for Anosov flows.

## Key findings

- Flows include geodesic flows of negatively curved metrics and Hilbert metric flows.
- Identifies conditions under which flows are Anosov.
- Studies invariant measures and regularity of foliations in special cases.

## 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.