Properties of pseudoholomorphic curves in symplectisations .1. Asymptotics

Given an oriented, compact, 3-dimenional contact manifold (M, lambda) we study maps (u) over tilde = (a, u) : C --> R x M satisfying the Cauch-Riemann type equation (u) over tilde(s) + (J) over tilde() (J) over tilde(t) = 0, with a very special almost complex structure (J) over tilde related to the contact form lambda on M. If the energy is positive and bounded, 0 < E((u) < infinity, then the asymptotic behavior of u : C --> M as \z\ --> infinity is intimately related to the dynamics of the Reeb vector field X(lambda) on M. Assuming the periodic solutions of X(lambda) to be non degenerate, we shall show that lim(R-->infinity) u(Re-2 pi it) = x(Tt) for a T-periodic solution x with E((u) over tilde) = T. The main result is an asymptotic formula which demonstrates the exponential nature of this limit. Some consequences for the geometry of the maps u : C --> M are deduced.