52 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICStogether with t he cylinder
(C \ {O}) x {O}
under iI> give a finite energy foliation of ( S^3 , .A, J).
This particular finite energy foliation, which projected to our previous open
book decomposition, looks very special. Nevertheless it exhibits already certain
interesting features, which are special cases of some abstract phenomenon. For
example, if F = Fa for some a f- 0, then F has the form JR x P, where P is a
closed integral curve for X , i.e. a periodic orbit. If F n Fa = 0 for some a f-0,
then with T : JR x S^3 ---+ S^3 being the the projection the set F = T(F) is an
embedded submanifold of M transversal to X. Moreover if for F and G in :F we
have FnG f-0, then there exists a E JR with Fa = G. In our case the collection {F}
consists of the set P = 81 x {O} and planes asymptotic to P. If we compactify all
these planes at infinity by adding the circle P , then the compactified planes become
disks defining an open book decomposion of S^3 with disklike page transversal in
the interior to X.
There is also another finite energy foliation for (S^3 , .A, J). Namely consider the
collection of all JR x P, where P runs over all Hopf circles.
By a small perturbation of .A, replacing it by f .A, where f is close to 1, it is
possible to destroy most period orbits. So the generic picture is usually one having
only a finite number of fixed points.
What about the existence of finite energy foliations?
Definition 1.48. We call a contact form on M non-degenerate if the periodic
orbits and their iterates are non-degenerate.
The following proposition follows from well-known results in Hamiltonian dy-
namics, [78].
Proposition 1.49. Fix a contact form T on the closed three-manifold M. Con-
sider the subset 31 of C^00 (M, (0, oo)) consisting of all f such that .A= fr is non-
degenerate. Let 32 be the subset of 31 consisting of all f such that in addition thestable and unstable manifolds of hyperbolic orbits intersect transversally. Then (^31)
and32 are Baire subsets ofC^00 (M,(O,oo)).
Now consider the standard contact form .Ao on S^3.Definition 1.50. A finite energy foliation :F for (S^3 , J, f >- 0 ) is called stable if the
following holds
• The function f belongs to 31.
- Every leaf of the foliation is the image of a finite energy sphere.
- For every F E :F the asymptotic limits are simply covered (i.e. have minimal
period) and their Conley-Zehnder indices are contained in {1, 2, 3}. - Every leaf F has precisely one positive puncture, but an arbitary number of
negative punctures. The negative asymptotic limits are mutually different
and have self-linking number -1.
• Every leaf F which is not a fixed point for the JR-action satisfies μ(F) - 2 +
Ur E {1, 2}. Here μ(F) is defined by
μ(F) = μ(zo) - L μ(z),
zEr-
where zo is the positive puncture and r-denotes the set of negative punc-
tures.