Pointwise convergence of gradient-like systems

S. Łojasiewicz has shown that the ω-limit sets of the trajectories of analytic gradient systems consist of at most one point. We extend this result to the larger class of gradient-like vector fields satisfying an angle condition. In particular, this includes gradient systems, defined by arbitrary C¹ functions from an analytic-geometric category. Corresponding pointwise convergence results are shown for discrete gradient-like algorithms on a Riemannian manifold. This generalizes recent results by Absil, Mahony, and Andrews to the Riemannian geometry setting.