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

idity of symplectic diffeomorphisms 59
3. Let ϕj : B(1) → (R 2n ,ω0) be a sequence of symplectic embeddings
ng locally uniformly to a map ϕ : B(1) → R 2n .Ifϕ is differentiable at
enϕ ′ (0) = A is a symplectic map, i.e., A ∗ ω0 = ω0.
ee that, in general, a volume preserving diffeomorphism cannot be apted
by symplectic diffeomorphisms in the C 0 -topology. By using, locally,
charts we deduce immediately from Theorem 3
4. (Eliashberg, Gromov) The group of symplectic diffeomorphisms of a
symplectic manifold (M,ω) isC 0 -closed in the group of all diffeomorf
M.
e early seventies M. Gromov proved the alternative that the group of symiffeomorphisms
either is C 0 -closed in the group of all diffeomorphisms or
osure is the group of volume preserving diffeomorphisms. That symplecmorphisms
can be distinguished from volume preserving diffeomorphisms
l properties which are stable under C 0 -limits was announced in the early
y Y. Eliashberg in his preprint “Rigidity of symplectic and contact struc-
981) [78], which in full form has not been published. Proofs are partially
d in Eliashberg [71], 1987. Gromov gave a proof of Theorem 4 in [107]
e techniques of pseudoholomorphic curves. Both Eliashberg und Gromov
the C 0 -stability from non embedding results. In his book [108] Gromov
alled Nash-Moser techniques of hard implicit function theorems, while
rg [71], 1987, uses an analogue of Theorem 3. Following the strategy of
d and H. Hofer in [68], we shall show next, that Theorem 3 is an easy
nce of the existence of any capacity function c.
convenient in the following to extend the capacity to all subsets of R 2n .
we take a capacity function c given on the open subsets U ⊂ R 2n and
r an arbitrary subset A ⊂ R 2n :
c(A) =inf c(U) A ⊂ U and U open .
e monotonicity property
A ⊂ B =⇒ c(A) ≤ c(B)
e for all subsets of R 2n . From the symplectic invariance of the capacity
sets, one deduces the invariance
c ϕ(A) = c(A)
ery symplectic embedding ϕ defined on an open neighborhood of A.
Theorem 3. Without loss of generality we shall assume in the following
) = 0. We first claim that the linear map ϕ ′ (0) = A is an isomorphism.
because ϕ is differentiable at 0, we have ϕ(x) =Ax + O(|x|), so that for
balls Bε of radius ε>0 and centered at 0,
m ϕ(Bε)