Symplectic Invariants and Hamiltonian Dynamics - Hofer / Zehnder ...

finition and application to embeddings 55
e may assume that Ω contains the origin. Observe that dim W ⊥ =2.
e, in the second case, W ⊥ is a symplectic subspace and R 2n = W ⊥ ⊕ W .
a symplectic basis (e1,f1) inW ⊥ we can, therefore, assume by a linear
ic change of coordinates that
W = {(x, y) | x1 = y1 =0} .
is bounded, we have for z ∈ Ω+W ,thatx 2 1 + y2 1 0, so that the point (x, y) ∈ Ω+W if x2 1 + x22 0
ntly small. Consequently, by the monotonicity of a capacity, c(Ω × W ) ≥
= πR2 . This holds true for every R>0sothatc(Ω + W )=+∞as
w of the monotonicity property, the symplectic invariants c(M,ω) repreparticular,
obstructions of symplectic embeddings. An immediate consef
the axioms is the celebrated squeezing theorem of Gromov [107] which
to the concept of a capacity.
1. (Gromov’s squeezing theorem) There is a symplectic embedding ϕ :
Z(R) if and only if R ≥ r.
ϕ is a symplectic embedding, then using the monotonicity property of
city, together with (2.1) and (2.2), we have
theorem follows.
πr 2 = c B(r) ≤ c Z(R) = πR 2 ,
next result also illustrates the difference between volume preserving and
ic diffeomorphisms. We consider in (R 4 ,ω0) with symplectic coordinates
2,y2) the product of symplectic open 2-balls B(r1) × B(r2). By a linear
ic map we can assume that r1 ≤ r2.
ion 4. There is a symplectic diffeomorphism ϕ : B(r1) × B(r2) → B(s1) ×
and only if r1 = s1 and r2 = s2.
that, in contrast, there is a linear volume preserving diffeomorphism ψ :
(1) → B(r) × B( 1)
for every r>0. As r → 0, we evidently have
r
⎧
⎪⎨
1
c B(r) × B r → 0
⎪⎩
1
vol B(r) × B = const.
r