46 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
This follows from the fact that every overtwisted contact form admits a con-
tractible periodic orbit as the asymptotic limit of a finite energy plane.
The application of the pseudoholomorphic curve techniques are not restricted
to finding periodic orbits for Reeb vector fields, but they can be used to study the
structure of 3-manifolds.
This will become more apparent in the next subsection. In fact we only
scratched the surface of the possibilities the holomorphic curve theory offers in
the study of three-manifolds. We will give many more results later.
1.3. Finer aspects of Reeb dynamics and topology
In order to study the dynamics of Reeb vector fields in more detail we need certain
notions.
Assume M is a closed orientable three manifold and A a contact form with
associated contact structure ~ and Reeb vector field X.
We call a complex structure J: ~ ----+ ~ for the vector bundle ~ ----+ M admissible
if d.A(h, Jh) > 0 for h -1- 0. The space of admissible structures is contractible.
Clearly every such J gives~ a natural orientation. Assume next (x, T) is a periodic
solution for X, i.e. x : IR ----+ M
x = X(x), x(O) = x(T).
We call (x, T) contractible if the induced loop
X r : IR/TZ----+ Mis contractible. Pick a smooth map u: D ----+ M satisfying
u(e^2 rrit) = x(tT).
Let Z be a nowhere vanishing section of u*~----+ D. Clearly any two such nowhere
vanishing sections are homotopic through nowhere vanishing sections. Let 'T/ be the
flow associated to X. Then.
d
dt 'T/t(m) = X(TJt(m))
TJo(m) = m, m EM.
Since Lx>-= dix.A + ixd.A = 0 we see that
T'TJt I ~m: ~m ----+ ~ry,(m)
and
d.A(TTJt(m), TTJt(m)) = d.Am·
For the contractible periodic orbit (x, T) we obtain for t E IR the map
TTJt(x(O)) I ~x(O): ~x(O) ----+ ~x(t)·
For v E ~x(O) \ {O} consider TTJt(x(O))v and define f: IR----+ C* using the complex
multiplication induced by J on ~ via
f(t)Z(x(t)) = TTJt(x(O))v.
Denote by .6 ( v, T) the change of argument of f. The winding interval of ( x, T)
with respect to u is the set
1
I(x, T, u, J) = {
2
7r .6.(v, T) Iv E ~x(O) \ {O} }.