1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

60 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS

The first observation is the following

Proposition 2.8. Let u be as in (6). Then

(7) m= lim r u(s,·)*>.
S-+ 00 1 51
exists.

If m > 0 we call the puncture positive, if m < 0 negative, and m = 0 removable.

Proof. By Stokes' theorem

{ u(s,-)*>.= { u(O,-)*>.+ { u*d>.

l 51 l 51 110 ,s] x 81

Hence the map

=:Co+ r ~ [lnuslJ + lnutlJ]dsdt.

110 ,s]x8^1

s ~ { u(s,-)* >.
1 51
is monotone. On the other hand

{ u*d>.:::; { u*d>.:::; E(u) < oo.

110 ,s]x8^1 110 ,oo)x81

Hence the limit exists. D

Theorem 2.9. Let u be as in (7) and (sk) a sequence converging to oo. Then (sk)
has a subsequence still denoted by ( s k) such that

lim u(sk, t) = x(tm) in C^00

k-+oo

for some orbit x of x = X(x). Herem is th e number introduced in (7). We note
that in case m f=. 0 th e map x has to be a necessarily periodic orbit for X with
period lml.

Proof. Let us first show that the gradient 'Vu is uniformly bounded:

l'Vu( s, t) I :::; c V(s,t) E [O,oo) x S^1.

As norm we take

l(h, k)lta,u) = h^2 + (>.(u)(k))^2 + d>.(k, J(u)k).

Let us view u as a map

[O, oo) x JR ~ JR x M.

Arguing indirectly we find a sequence (sk, tk) with tk E [O, 1], Sk ~ +oo such that

l'Vu(sk, tk)I ~ +oo.

Define with uk = (ak, uk) by

iik(z) = (a(zk + z ) - a(zk), u(zk + z))

with zk = sk + itk. Then