24 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGYHowever , below we will use the strong form (iii).
The interesting question is: do symplectic capacities exist? A moment's reflec-
t ion shows t hat the fact t hat t hey do is essentially equivalent to the nonsqueezing
theorem. Let us define the Gromov width we by
we ( U, w) = sup{ 7rr^2 : B^2 n ( r) embeds symplectically in U}.
Then we clearly satisfies the conditions (i), (ii), and also we(B^2 n(r)) = 7rr^2. The
only difficult thing to check is t hat we(Z(l )) < oo, but in fact
we(Z(r)) = 7rr^2
by the nonsqueezing theorem. Thus we is a capacity. There are now several
other known capacities, (cf work by Ekeland-Hofer [EH], Hofer-Zehnder [HZ],
Viterbo [V]) mostly defined by looking at properties of the periodic flows of certain
Hamiltonian functions H t hat are associated to U.
The main result is
Theore m 4.2 (Ekeland-Hofer). A (local) orientation-preserving diffeomorphism
¢ of (R^2 n, wo) is symplectic iff it preserves the capacity of all open subsets of R^2 n,
ie iff there is a capacity c such that c( ¢( U)) = c( U) for all open U.
The proof is based on the corresponding result at t he linear level.Pro posit io n 4.3. A linear map L that preserves the capacity of ellipsoids is either
symplectic or antisymplectic, ie L * ( w 0 ) = ±wo.
Proo f. If L is neither symplectic nor antisymplectic the same can be said of its
transpose LT. Therefore there are vectors v , w so that
wo(v, w) i= ±wo(LT v, L T w).By perturbing v, w and using the openness of the above condit ion we can suppose
t hat both wo(v,w) and w 0 (LTv,LTw) are nonzero. Then, replacing LT by its
inverse if necessary, we can arrange that
0 < A^2 = lwo(LT v, L T w)I < wa(v, w) = l.
Now construct two standard bases of R^2 n, the first starting as
and t he second starting as
I L T VU1=T ,
LTw
v'1 = ±--A '
U2, I ....Let A, resp. A', be the symplectic linear map t hat takes t he standard /basis
e1,e2,e3, ... ofR^2 n to u 1 ,v 1 ,u2,... , resp. u~,v~,u~, .... Then, taking C to be
(A')-^1 LT A, we have
C(e1) = Ae1, C(e2) = Ae2.Thus the matrices for C and er h ave the form
A (^0) *
*
A 0 0 0
0 A 0 A (^0 0)
C=^0 0 C T =
(^0 0) *