LECTURE 2. ANALYTICAL TOOLS 63In the previous theorem we showed that every sequence sk ---> oo has a subse-
quence so that limk_, 00 u(sk, t) = x(mt). Under certain circumstances we can even
deduce that
lim u(s, t) = x(mt).
S-+ 00If m = 0, viewing u as defined on a punctured disk, one can extend u in a smooth
way over the puncture, see [l]. So assume that m =/= 0 and that
u(sk, t)---> x(mt)
where x(m·) is isolated as an !ml-periodic orbit in C^00 (1R/Z, M). Then assume
arguing indirectly that there exists another sequence Sk' ---> oo with
u(s~, t)---> y(mt).
Take an open C^00 -neighbourhood Ux of x(m·) and an open neigbourhood U of the
(compact) set of all z (m·), where z is a !ml-periodic orbit for X other than x. Byassumption we can take these neighbourhoods in such a way that Ux n U = 0.
Without loss of generality passing to subsequences we may assume
f
Sk < Sk < Sk+l·
Pick s'k E (sk, s~) such that (fork large)
u(s'k,·) f/.UxUUy.
By the previous discussion we find that a suitable subsequence u(s'k, ·) will also
converge to a I ml-periodic orbit of X not in Ux U U which is impossible.
A condition which guarantees that periodic orbits are isolated is the following
Definition 2.11. A periodic orbit (x, T) for the Reeb vectorfield X is called non-
degenerate ifT77r(x(O)) I ~x(O): ~x(O)---> ~x(O) does not have 1 in the spectrum. Here
7) is the flow associated to X.
The following exercise relates non-degeneracy to the property of being an iso-
lated periodic orbit.
Exercise. 2.12. Let (x , T), T =/= 0 , be a T-periodic orbit for the Reeb vector field
X. Then there exists an c > 0 and an open neighbourhood U of x(T·) in C^00 (S^1 , M)
such that the following holds. If (y, T), T =/= 0, is a periodic orbit such that IT-Tl :Sc
and y(T·) EU, then (y , T) = (x, T).
Assuming non-degeneracy of the asymptotic limit more can be said. For proofs
we refer to [53, 54]. The next theorem follows basically from the previous discus-
sion.
Theorem 2.13. Assume u: [O, oo) x 81 ---> IR x M is a finite energy half-cylinder
us+ ](u)ut = o
E(u) < oo.
Assume there exists a sequence Sk ---> oo such that
u(sk, t)---> x(mt)
to some !ml-periodic orbit, m =/= 0, so that (x , lml) is nondegenerate. Then in factlim u(s, t) = x(mt) in C^00 (S1, M).
S-+00
The latter limit is in fact of exponential nature {this is not obvious).