NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 39
The binding orbit has period 2πγ1(0), since
and it is nondegenerate and elliptic.
′′ γ 1
γ1(0)β(0) =− (0)
γ ′′
2
(0) /∈ Z,
Example 2.2
For the contact form Tdθ+(1/k)(xdy−ydx) = Tdθ+(r 2 /k)dφ the central orbit
S 1 ×{0} is degenerate, but
is a local model near the binding if
In this case,
and we note that
If
λ = (1 − r 2 )(T dθ+ r2
k dφ)
k, T > 0, kT /∈ Z, and kT ≥ 1
2 .
µ(r) = 2rT
k (1 − r2 ) 2 > 0 and
A(r) = 1
µ 2 (r)
γ ′′
2
(r)γ ′
1
α(r)
β(r) =−γ ′ 2 (r)
γ ′ 1
′′ ′
(r) − γ 1 (r)γ 2 (r) =
1 − 2r2
=
(r) kT
γ ′′
1 (0)
γ ′′
2
m
=
n
(0) =−kT,
4kr
T (1 − r2 .
) 4
for integers n, m, then the invariant torus Tr is foliated with periodic orbits. The
case m = 0 is only possible if r = 1/ √ 2.Ifr is sufficiently small, then |m| ≥2.
Indeed, we would otherwise be able to find sequences rl ↘ 0 and {nl} ⊂Z such that
kT/(1 − 2r 2 l ) = nl, which is impossible. The binding orbit has period 2πT while
the periodic orbits close to the binding orbit have much larger periods equal to
τ = 2πTm (1 − r2 ) 2
.
1 − 2r2 Example 2.3
Consider the contact form λ = T (1 − r 2 )dθ + (r 2 /k)dφ on S 1 × D. Itisalsoa
local model near the binding if k, T > 0, kT ≥ 1/2, andkT is not an integer. We