MULTIDIMENSIONAL SINGULAR λ-LEMMA

The well known lambda-Lemma [3] states the following: Let f be a C-1-diffeomorphism of R-n with a hyperbolic fixed point at 0 and m- and p-dimensional stable and unstable manifolds W-S and W-U, respectively (m+p = n). Let D be a p-disk in W-U and w be another p-disk in W-U meeting W-S at some point A transversely. Then U-n >= 0 f(n)(w) contains p-disks arbitrarily C-1-close to D. In this paper we will show that the same assertion still holds outside of an arbitrarily small neighborhood of 0, even in the case of non-transverse homoclinic intersections with finite order of contact, if we assume that 0 is a low order non-resonant point.