64 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
In order to describe this in more detail assume next we have a finite energy
half cylinder
u: [O, oo) x 81 ---+ JR x Mus+ fut= 0
E(u) < oo.
So that for a sequence sk ---+ oo
u(sk, t)---+ x(tm)for some !ml-periodic orbit x of±= X(x). Assume (x, lml) is nondegenerate.
Then it is possible to introduce suitable coordinates to describe the behaviour
at oo.
We need the following
Lemma 2.14. Let (M, .>..) be a 3-dimensional contact manifold and let x be a T-
periodic solution of the associated Reeb vectorfield ± = X>,(x). Let T be the minimal
period of x such that T = kr for some positive integer k. Then there exists an open
neighbourhood of 81 x {O} in 81 x JR^2 and an open neighbourhood V c M of
P = x(JR) and a diffeomorphism cp: U---+ V mapping 81 x {O} onto P such that
cp* .A= !Aowhere f: U ---+ (0, oo) is a smooth map satisfying
f(iJ, 0, 0) = T
df(iJ, 0, 0) = 0
for all 1J E 81. Here .Ao = diJ + xdy for the coordinates (iJ, x, y) on 81 x JR^2.
Using the above coordinates we can write ii, as follows. View ii, as a map defined
on JR+ x JR which is 1-periodic in the second argument and let JR x JR^2 be the covering
space. Then
u: JR+ x JR ---+ JR x JR x JR^2
u(s, t) = (a(s, t), iJ(s, t),x(s, t), y(s, t)).It satisfies the transformed differential equation, which we will not present here, see
[53]. However it is a good exercise to derive these equations.Exercise 2.15. Using the coordinates provided by the previous lemma write down
the equations for a finite energy half-cylinder.
We have the following
Theorem 2.16. The function ii,: JR+ x JR---+ JR^4 has the following property. Either
(i) There exists c E JR such that(a(s,t),iJ(s,t),x(s,t),y(s,t)) = (ms+c,kt,0,0)
or(ii) There are constant c E JR, d > 0, Ma > 0 for all a E N^2 such that
l8a[a(s, t) - ms - c]I :=::; Mae-ds
l8o.(1J(s, t) - kt]I :=::; Mae-ds