On the normal exponential map in singular conformal metrics

TL;DR

This paper investigates the properties of the normal exponential map in singular conformal metrics, motivated by applications to brake orbits and homoclinics in dynamical systems, and develops a global perspective on this geometric structure.

Contribution

It provides a new global analysis of the normal exponential map in singular conformal metrics, extending Morse theory and geometric methods to this setting.

Findings

Analysis of the normal exponential map in singular metrics

Extension of Morse theory to singular conformal metrics

Insights into geodesic multiplicity in dynamical systems

Abstract

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 [6, 7, 10], and by the development of a Morse theory in [8] for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.