Stability of degenerated fixed points of analytic area preserving mappings

Abstract

It js well known that hyperbolic points of an analytic area preserving mapping (APM) T are unstable. As a Corollary of Moser's twist thcorem the elliptic ones are stable provided the eigenvalues l. of DT at the fixed point are nota k-th root of t.he unity, k~ lf2p+2 p ~l. and any of the first p coefficients of the Birkhoff normal form is non-zero. To end the study of the stability of fixed μoints we study the parabolic ar degenerated case. Elliptic points far which stability can not be decided using directly Moser' s theorem (specially if " is a third or fourth root of the uni ty) can be reduced to the parabolic case taking a suitable power of T. The main result is that a degenerated fixed point of an analytic APM is stable if and only if the generating function of T, with the part which generates the identity suppressed, has a strict extremum at the fixed point. Sorne examples and comment are included.