ABSTRACT
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 D 2. Recall that a periodic point p of such an f is a point for which f n (p) = p for some integer n > 0. Assuming n is the least such integer, p is called elliptic if the derivative of f n at p, T p f n , has non-real eigenvalues of norm one. If the eigenvalues of T p f n 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 θ with θ/2π strongly irrational [8, 9]. This result implies that general elliptic orbits are Liapounov stable.
