A Piecewise Smooth $\lambda$-Lemma

In this paper we provide extensions of the $\lambda$-Lemma (also known as Inclination Lemma) for piecewise smooth vector fields and maps. In order to achieve our main result, we investigate the regularity of time-T-maps of piecewise smooth vector fields defined at crossing orbits. We prove that these maps are homeomorphisms and also piecewise smooth diffeomorphisms.