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

idity of symplectic diffeomorphisms 67
rom the definition of γ we infer
µ Q(r) = r 2n n = γ Q(r) .
ery symplectic embedding is volume preserving we find for the capacity
the open and bounded set Ω ⊂ R2n that
µ(Ω) ≥ sup µ Q(r) = γ(Ω)
r
n ,
e supremum is taken over those r, for which there is a symplectic embed-
) → Ω. If Q is any open cube having its edges parallel to the coordinate
conclude that
s that
µ(Q) = γ(Q) n = γ
n h(Q)
µ(Ω) ≤ µ h(Ω) ,
≤ µ h(Q) .
Ω ∈O. Indeed assume Ω ∈O;thengivenε>0 we find, in view of the
y of the Lebesgue measure, finitely many disjoint open cubes Qj contained
h that µ(Ω) − ε ≤ µ(Qj). Hence by the estimate above
µ(Ω) − ε ≤
µ h(Qj)
=
µ h(
Qj)
j
≤ µ h(Ω) .
ds true for every ε>0 and the claim follows. By the same argument,
µ(h −1 (Ω)) and hence µ(Ω) ≤ µ(h(Ω)) ≤ µ(h −1 ◦ h(Ω)) = µ(Ω), proving
osition.
y. If a homeomorphism of R 2n preserves all the capacities of open sets in
n it also preserves the Lebesgue measure.
ur considerations so far are based on the existence of a capacity not yet
ed. In the next chapter we shall construct a very special capacity function
ynamically by means of Hamiltonian systems.