Deformative magnetic marked length spectrum rigidity

Abstract

Let be a closed surface, be a smooth family of Riemannian metrics on , and let be a smooth family of functions on . We show that if the magnetic curvature of each is negative and the lengths of each periodic orbit remains constant as the parameter varies, then there exists a smooth family of diffeomorphisms such that and . This generalizes a result of Guillemin and Kazhdan (Topology 19 (1980) 301–312) to the setting of magnetic flows.