Abstract: Dynamically defined Lagrange spectra is a subset of the real line which describes the asymptotic behaviour of orbits of the system with respect to a reference potential. This spectrum is motivated by classical problems in number theory but can be used to understand the long range behaviour of orbits of generic dynamics. The purpose of this talk is to present some ideas on how to guarantee richness (positivity of the Hausdorff dimension) of dynamically defined Lagrange spectra for generic dynamics admitting transverse homoclinic intersection in a compact manifold of any dimension large or equal than 2.