Genericity of weak-mixing measures on geometrically finite manifolds

Let $M$ be a manifold with pinched negative sectional curvature. We show that when $M$ is geometrically finite and the geodesic flow on $T^{1}M$ is topologically mixing then the set of mixing invariant measures is dense in the set ${\mathcal{M}}^1(T^{1}M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_{\delta }$ subset of ${\mathcal{M}}^1(T^{1}M)$. We also show how to extend these results to manifolds with cusps or with constant negative curvature.