48 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
We will not prove this result in this paper. Nevertheless it is useful to under-
stand it in a special case.
Consider the standard contact form Ao and recall the biholomorphic map :
C^2 \ {O} ---> IR x 83. On C^2 \ {O} we define a holomorphic foliation F' of C^2 \ {O}
consisting of the sets C x { c }, where c E C \ {O} and (C \ {O}) x {O}. Taking the
images under we obtain a finite energy foliation of mutually disjoint holomorphic
surfaces F. We denote it by F. Observe that with a E IR and F E Falso a+ F
belongs to F (Observe that we have a natural IR-action on IR x 8^3 ). The only fixedpoint for the IR-action is the cylinder over P = 81 x {O}, i.e. IR x P. Consider the
projection IR x 83 ---> 83. Then the fixed point projects onto the periodic orbit P for
the Hopf fibration and the other surfaces to open disks bounded by P. Moreover
any IR-orbit projects to the same embedded disk. What we obtain is called in low-
dimensional topology an open book decomposition of 83 with disk-like page and
binding P. We observe that all orbits different from Pare transversal to the pages.
So fixing one page we obtain a return map. Theorem 1.34 may be viewed as ageneralisation of this fact for a contact form f Ao, for suitable f.
Let us also fix ideas about the proof that there are 2 or infinitely many periodic
orbits. Since we have a (global) disk-like surface of section we can define a return
map
'!/; : iJ---> iJ.
Clearly dA induces an area form on iJ of finite area. Moreover
'l/;*dAIV = dAIV.We need the following theorem due to Brouwer (The Brouwer translation theorem).
Theorem 1.35. Leth: IR^2 ---> IR^2 be a homeomorphism. If h does not have a fixed
point there exists a non-empty open subset U such that
hj (U) n U = 0 for all j = 1, 2, 3, ....
An immediate corollary is the following result:Corollary 1.36. Let B be the open unit disk in C and cp : B ---> B an area pre-
serving homeomorphism. Then cp has a fixed point.Proof. Clearly B is homeomorphic to IR^2. If cp does not have a fixed point the
Brouwer translation theorem gives as a nonempty open set U C B satisfying
<pj (U) n U = 0 for j = 1, 2 ....
This implies that cpi(U) n <pj (U) = 0 for i =/. j. Consequently
N7r = area(B);:::: L~>J(U) = Narea(U)---> oo,
j=lwhich is impossible. D
The disk (iJ, dA) is homeomorphic by an area preserving map to some open disk Br.
Hence '!/; has a fixed point. Removing the fixed point we obtain an area preserving
map
'!/;:A-> A,
where A is an open annulus. Now we apply a striking recent result by J. Franks,
[31]: