A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics

Area-preserving maps and flows play an essential role in the study of motions of mechanical systems, especially in celestial mechanics (see [1, 14]). Since one is often interested in the behavior of an area-preserving map around a fixed point and in the number and type of critical points and periodic orbits of an area-preserving flow, the ./i'xed-point index of a map and the index oj" a singularity or a closed orbit of a flow can yield much information about the map or flow. For example, the fact that the index of an isolated singularity of an area-preserving flow can never be greater than +1 has aided in setting a lower bound for the number of stationary points of certain area-preserving flows. It has also been a useful necessary condition for a flow to be area-preserving. For these reasons, the conjecture that the fixed-point index of an area-preserving homeomorphism of a 2manifold is always less than or equal to + 1 has drawn attention. In this paper, we answer this conjecture in the affirmative for smooth maps and then put this bound to work to show that certain maps must have at least two fixed points and certain flows at least two periodic orbits. An important application is the following generalization of a famous theorem of Liapunov: a Hamiltonian vectorfield on M 4 must have two distinct one-parameter families of periodic orbits around a non-degenerate minimum (or maximum) of the Hamiltonian, even when the pure imaginary characteristic exponents are in resonance.