Locating omega-limit sets using height functions

This work deals with the problem of locating the ω-limit set of a bounded solution of a given autonomous vector field f on a Riemannian manifold. Assuming to know that the ω-limit set Ω is contained in an embedded submanifold S and using an auxiliary function that we call height function W for f and S, we show how to obtain a better estimate of the location of Ω under mild assumptions. Several consequences and an application to a type of polynomial vector fields are presented.