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

idity of symplectic diffeomorphisms 65
der to generalize this statement, we denote by O the family of open and
sets of R 2n and associate with Ω ∈O
č(Ω) = sup c(U) |U open and U ⊂ Ω
ĉ(Ω) = inf c(U) |U open and U ⊃ Ω .
distinguished family Oc of open sets is defined by the condition
Oc = {Ω ∈O|ĉ (Ω) = č(Ω)} .
ion 7. Assume hj is a sequence of homeomorphisms of R 2n converging
niformly to a homeomorphism h of R 2n .Ifc(hj(U)) = c(U) for all U ∈
ll j, thenc(h(Ω)) = c(Ω) for all Ω ∈Oc.
Ω ∈Oc and ε>0 we find Û,Ǔ ∈Owith Û ⊃ ΩandǓ⊂Ω satisfying
c(Û) ≤ c(Ω) + ε and c(Ǔ) ≥ c(Ω) − ε.
gewehavebytheabovedegreeargumenthj(Û) ⊃ h(Ω) ⊃ hj(Ǔ), and
ing the monotonicity property of the capacity
c( Û)=c(hj(Û)) ≥ c(h(Ω)) ≥ c(hj(Ǔ)) = c(Ǔ),
(Ω) + ε ≥ c(h(Ω)) ≥ c(Ω) − ε. This holds true for every ε>0andthe
is proved.
n. A capacity c is called inner regular, respectively, outer regular if
∈O.
c(U) = č(U) resp. if c(U) = ĉ(U)
ion 8. Assume the capacity c is inner regular or outer regular. Assume ϕj
e homeomorphisms of R 2n and
ϕj → ϕ and ϕ −1
j
→ ϕ−1
niformly. If c(ϕj(U)) = c(U) for all U ∈Oand all j, thenalso
c ϕ(U)
= c(U) for all U ∈O.