A thermostat is a dynamical system modelling the motion of a particle on a surface under the influence of a force that is always orthogonal to its velocity. Since this force is allowed to depend on the particle's velocity, the system can be dissipative. As a generalization of geodesic and magnetic flows, thermostats provide a unified point of view to study many non-conservative dynamical systems, where the classical Hamiltonian structure is absent. We present rigidity results for thermostats that are Anosov, projectively Anosov, or have no conjugate points. For instance, we investigate rigidity under smooth orbit equivalences, and we explain how Eberlein's characterization of Anosov geodesic flows translates to thermostats. Our goal is to understand which results from the geodesic case extend to thermostats, despite the lack of volume preservation, while also highlighting the nuances that appear from greater dynamical complexity. To this end, we present examples and counterexamples illustrating the richer dynamics of thermostats.
mathematics, chaotic dynamical systems, geometry
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