On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M, g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p, q is an element of M that are non-conjugate in a suitable sense, under the assumption that (M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies