Quasi-Elliptic Periodic Points in Conservative Dynamical Systems.

Abstract

1. One of the most important and beautiful subjects in the theory of dynamical systems concerns the orbit structure near an elliptic periodic point of an area preserving diffeomorphism f of the two dimensional disk D2. Recall that a periodic point p of such an f is a point for which f ( p) = p for some integer n >0. Assuming n is the least such integer, p is called elliptic if the derivative of fn at p, Tpfn, has non-real eigenvalues of norm one. If the eigenvalues of Tpfn have norm different from one, p is called hyperbolic. It has been known for a long time that elliptic periodic orbits occur in many problems in mechanics, in particular, the restricted three body problems [3, 8]. When f is real analytic, Birkhoff established a normal form for f near an elliptic fixed point provided the eigenvalues of Tf are not roots of unity. If this normal form is not linear, he showed that the fixed point is a limit of infinitely many periodic points, and that among these accumulating periodic points both elliptic and hyperbolic types appear [28]. A theorem due to Kolmogorov, Arnold, and Moser asserts that many f-invariant circles enclose a general elliptic fixed point p, and that on each of these circles f behaves like a rotation through an angle 9 with 0/27r strongly irrational [8, 9]. This result implies that general elliptic orbits are Liapounov stable. By contrast, according to a theorem of Hartman and Grobman [12, 16], the local structure near a hyperbolic fixed point p is not complicated. The diffeomorphism f behaves topologically like its derivative. The points asymptotic to p under forward and backward iterates form smooth curves (the stable and unstable manifolds of p) meeting transversely at p, and the other nearby orbits lie on continuous curves which are easily described. Recently, E. Zehnder has shown that, generically, many hyperbolic periodic orbits near an elliptic periodic orbit have homoclinic points (non-periodic intersections of the stable and unstable manifolds of the same hyperbolic periodic orbit) [28]. Thus the rather intricate picture in Figure 1 (taken from [3]) generically occurs near any elliptic periodic point p. Each circle is invariant under a power