Proof of the transverse intersection of the stable and unstable manifolds in the two-dimensional area preserving map

Abstract

Abstract In this paper, we study the area preserving map T included in the Hénon family of maps. The map T defined on the plane is characterized by a parameter a (≥ 0). There exist the saddle fixed point P and the elliptic point Q at a > 0. Let u be the primary homoclinic point at which the unstable manifold Wu starting from the saddle fixed point P intersects the symmetry axis denoted by the dominant axis of Q. The stable manifold Ws also intersects Wu at u. At u, let ξu(u) be the inclination of Wu and ξs(u) be that of Ws. Using a new geometric method, we prove the transversality of the intersection for any value of a (> 0). Here, the transversality means that ξu(u) < ξs(u) holds.