Multidimensional singular λ-lemma
Southwest Texas State University, Department of Mathematics
The well-known λ-Lemma  states the following: Let ƒ be a C1-diffeomorphism of ℝn with a hyperbolic fixed point at 0 and m- and p-dimensional stable and unstable manifolds Ws and Wu, respectively (m + p = n). Let D be a p-disk in Wu and w be another p-disk in Wu meeting Ws at some point A transversely. Then ⋃n≥0 ƒn(w) contains p-disks arbitrarily C1-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.
Homoclinic tangency, Invariant manifolds, Lambda-Lemma, Order of contact, Resonance
Rayskin, V. (2003). Multidimensional singular λ-lemma. <i>Electronic Journal of Differential Equations, 2003</i>(38), pp. 1-9.