1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

170 D. SALAMON, FLOER HOMOLOGY

as the subset of all those Hamiltonians for which all the periodic solutions x E P(H)
are nondegenerate and for which the operator Du is surjective for every pair x± E

P(H) and every connecting orbit u E M(x-, x+; H , J). Then the implicit function

theorem shows that the moduli spaces M(x-, x+; H, J) of connecting orbits are

finite dimensional smooth manifolds of the right dimension whenever H E Hreg·
Hence, to complete the proof of Theorem 1.24, it remains to show that the set
'Hreg is indeed of the second category in the sense of Baire, i.e. can be expressed
as a countable intersection of open and dense sets (and hence, in particular, is
nonempty). This follows, via standard transversality arguments, from an infinite
dimensional version of Sard's theorem which is due to Smale [52]. The details are

carefully carried out in [13] and [47] and will not be reproduced here.

2. 
   
   
   
   7. Exponential convergence
   
   
   
   

The purpose of this final section of the Fredholm chapter is to prove that all fi-
nite energy solutions of (7) converge exponentially to periodic orbits, and thus to
complete the proof of Proposition 1.21.

Lemma 2.11. Suppose that the operator D =Os+ Jo8t + S satisfies the require-

ments of Theorem 2.2 and, in addition,

lim sup ll8sS(s, t)ll = 0,

s->±oo O:S:t:S:l

sup 11 8tS(s, t)ll < oo.

s,t

Then there exists a constant 8 > 0 such that th e following holds. For every C^2 -

function ~ : JR x JR/Z ......, JR^2 n which satisfies D~ = 0 and does not diverge to oo as

s ......, ±oo there exists a constant c > 0 such that, for alls, t E JR,

l~(s, t)I :<:::: ce-6lsl.

Proof. Consider the function f : JR ......, JR defined by

1 f l

f(s) =

2

Jo l~(s, t)l^2 dt.

By assumption, this function is twice continuously differentiable and its second
derivative is given by

J"(s) fo

1

(las~(s, t) l^2 + (~(s, t), OsOs~(s, t))) dt

2 fo

1

l8s~(s, t)l^2 dt - fo

1

(~(s, t), 08 S(s, t)~(s, t)) dt

> 2 fo

1

1Jo8t~(s, t) + S(s, t)~(s, t)l2 dt - € fo

1

l~(s, t)l^2 dt

> 8

2

fo

1

l~(s, t) l

2

dt

82 f ( s).

Here the penultimate inequality only holds for sufficiently large s and it uses the

fact that the operator A(s) = Jo8t +Sis invertible for larges. Now the inequality

J" 2: 82 f

in plies that e-^6 s (!' ( s) + 8 f ( s)) is nondecreasing. Hence J' ( s) + 8 f ( s) :<:::: 0 for s 2: so

since otherwise f(s) would diverge exponentially as s ......, oo. Hence the function