1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

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and

H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS

Vs+ lvt = 0

E(v) < oo

{ v(O,-)*>-=m

Js,

{ v*d>-=O.

lbs'

We leave it as an easy exercise to show that

v(s, t) =(ms+ d, x(tm))

for some orbit x of x = X(x).

Summing up now our construction we see that

lim u(sk, t) = lim vk(O, t) = v(O, t) = x(tm).

k->w k->oo

D

There are some arguments in the previous proof which have to be completed.

Exercise 2.10. 1. For cp E I: with cp' ( s) > 0 for some s E IR we have

[ 1P'(a)da /\ df = oo

(8)

if a+ if is a non-constant holomorphic map.

2. Assume that v : IR x 81 --> IR x M satisfies

Then v has the form

Vs+ lvt = 0

E(v) < oo

f 3 , v(O, ·)* >. = m

fRxS 1 v*d). = 0.

v(s,t) = (ms+d,x(tm))

for some orbit x of x = X(x).

3. Assume we are given a sequence ue: D --> IR x M of smooth maps satisfying

(ue)s + l(ue)t = 0

E(ue)::::; c

IV'ue(ze)I--> oo

ze --> 0,

where the energy is as defined before. Show that the following inequalities

hold.

(9) / :S liminfk_,oo { u'kd>-:S limsupk_, 00 { u'kd>-:S c

}De JD

for every c E (0, 1], where/> 0 is the infimum of all periods of contractible

periodic orbits for X>. in M.