A Lipschitz version of the $\lambda$-Lemma and a characterization of homoclinic and heteroclinic orbits

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney’s Extension Theorem on compact manifolds to obtain a version of the well-known Lambda Lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the context of finite dimension with a norm weaker than $C^1$-norm and stronger than $C^0$-norm.