On the normal exponential map in singular conformal metrics

Brake orbits and homoclinics of autonomous dynamical systems correspond, via Maupertuis principle, to geodesics in Riemannian manifolds endowed with a metric which is singular on the boundary (Jacobi metric). Motivated by the classical, yet still intriguing in many aspects, problem of establishing multiplicity results for brake orbits and homoclinics, as done in Giambò et al. (2005, 2010, 2011), and by the development of a Morse theory in Giambò et al. (2014) for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.