Bi-Lipschitz -triviality of map germs and Newton filtrations

This article is devoted to criteria of Lipschitz equisingularity for families of real analytic map germs from ( R n , 0 ) to ( R p , 0 ) with n ⩾ p like g λ ( x ) = g ( x ) + λ h ( x ) , where λ is a small real number. The main result of this article, Theorem 4.2 states a condition on the order of the terms of h ( x ) , in such a way that the family g λ is bi-Lipschitz A -trivial. Theorem 4.2 gives the conditions in terms of Newton polyhedron associated to the germ g. The tools used here are based in the construction of convenient controlled vector fields.