Symplectic Flows on the Open Ball

In an earlier paper [4] we raised the question: Does every symplectic diffeomorphism of the open unit ball, B2”, in R”’ have a fixed point? We showed that there is an open neighborhood of the identity map in the C’ line topology for which the answer is yes. In this paper it is shown that the answer is no in general. There is a complete, infinitesimally symplectic vectorfield on B4 without any zeros. Thus there are symplectic diffeomorphisms in any C’ weak neighborhood of the identity with no fixed points. This shows that the two-dimensional results are special, for in the open disk every symplectic diffeomorphism has a fixed point, a result Bourgin proved in greater generality for orientation preserving homeomorphisms that preserve a finitely additive measure that is positive on open sets [3]. Furthermore, a complete, infinitesimally symplectic vectorfield on B2 has a zero, as we show in Section 2. The question of volume preserving, orientation preserving diffeomorphisms on the open ball was resolved in 1976 by Asimov [2] with his construction of a divergent free vectorfield on B3, which is complete and has no zeros. This easily extends to all dimensions. In particular, on B4 one had a volume and orientation preserving diffeomorphism with no fixed point, but not a symplectic diffeomorphism.