On the creation of conjugate points for thermostats

Abstract

Let $(M,g)$ be a closed oriented Riemannian surface, and let $SM$ be its unit tangent bundle. We show that the interior in the ${\mathcal{C}}^2$ topology of the set of smooth functions $\lambda :SM\to \mathbb{R}$ for which the thermostat $(M,g,\lambda )$ has no conjugate points is a subset of those functions for which the thermostat is projectively Anosov. Moreover, we prove that if a reversible thermostat is projectively Anosov, then its non-wandering set contains no conjugate points.

arXiv commentRevision after referee process. 18 pages, 4 figures