84 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSwe can make the choices in such a way that the length of uk(ri) is smaller than c
for all k large.We claim that for each singular point zi E 8D, the measure mf of intersected
leaves by uk(Si ) is either small or close to 1. This follows since the endpointsuk(zt) and uk(zi), are close, and uk(8D) is transversal to the leaves, hence winds
all around. So, for given 8 > 0 small, we find segments Si such that for k sufficientlylarge either mf :::; 8 or mf ~ 1 - 8. We shall show that mf ~ 1 - 8. Arguing by
contradiction we assume that for some index i we have mf :::; 8. We consider the
loop o:i =Si U /i c D and denote by A; C D the enclosed set. We shall prove that
(28)
can be made as small as we wish by choosing 8 small. This then contradicts
our bubbling off analysis. Indeed, Zi = lim zf is a singular point. We find as a
consequence of the bubbEng-off analysis small balls Bk C A i such that
(29)
for the fixed constant/> 0. This contradiction shows that mf :::; 8 is not possible.
In order to compute the integral (28) we replace the loop uk(8Ai) = uk(Si) U
uk(ri) C JR x M by a different loop f3f defined as follows: we start from uk(zt) E F,
follow uk(ri) to the point uk(zi) E F, then along the leaf through uk(zi) up to
the boundary 8V of V. Then the short piece along 8V until one reaches the leaf
through uk(zt), then down this leaf to the initial point u k(zt). The integral of>.
over this loop {Jf is small, since uk( /i) is short, mf is small, and since the integrals
: A l·ylinUerover a periodic orhit111crealline
The ··cxplodincM Bishop familyFigure 11. The schematic picture shows how the Bishop family explodes in
the IR-direction by trying to approximate a holomorphic cylinder over a periodic
orbit.