NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 53
where the map r satisfies for every (i, j) ∈ N 2 a decay estimate of the form
with Mij and d positive constants.
|∇ i
s ∇j
t r(s, t)| ≤Mij e −ds
Our situation is less general than in [31], so we will explain the notation in the context
of this paper. The setup is a manifold M with contact form λ and contact structure
ξ = ker λ. Consider a periodic orbit ¯P of the Reeb vector field Xλ with period T ,and
we may assume here that T is its minimal period. We introduce P (t) := ¯P (Tt/2π)
such that P (0) = P (2π). IfJ : ξ → ξ is a dλ-compatible complex structure, then
the set of all ˜J -holomorphic half-cylinders
ũ = (a,u) :[R,∞) × S 1 → R × M, S 1 = R/2πZ
for which |a(s, t) − Ts/2π| and |u(s, t) − P (t)| decay at some exponential rate (in
local coordinates near the orbit P (S 1 )) is denoted by M(P,J). Note that it is assumed
here that the domain [R,∞) × S 1 is endowed with the standard complex structure. A
smooth map U :[R,∞)×S 1 → P ∗ ξ for which U(s, t) ∈ ξP (t) is called an asymptotic
representative of ũ if there is a proper embedding ψ :[R,∞) × S 1 → R × S 1
asymptotic to the identity so that
ũ ψ(s, t) = Ts/2π, exp P (t) U(s, t) , ∀ (s, t) ∈ [R,∞) × S 1
(exp is the exponential map corresponding to some metric on M, e.g., the one induced
by λ and J ). Every ũ ∈ M(P,J) has an asymptotic representative (see [31]). The
asymptotic operator AP,J is defined as follows:
(AP,Jh)(t) :=− T
2π J P (t) d
ds
s=0
Dφ−s(φs(P (t)))h(φs(P (t))) ,
where φs is the flow of the Reeb vector field and where h is a section in P ∗ ξ →
S 1 . Because the Reeb flow preserves the splitting TM = R Xλ ⊕ ξ we have also
(AP,Jh)(t) ∈ ξP (t).
We compute the asymptotic operator AP,J for the binding orbit
¯P (t) =
t
, 0, 0 ∈ S
γ1(0) 1 × R2 .
Recall that the above periodic orbit has minimal period T = 2πγ1(0). Using
φs P (t) = φs(t,0, 0) = t + s
, 0, 0 ,
γ1(0)