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

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idity of symplectic diffeomorphisms 63 ave demonstrated that A∗ω0 = λω0. By conformality a linear antisymplecpreserves the capacities. Composing the maps ψj and ψ with the symplec- −1 −1 1 B = √λ 1 A if λ>0 and with the antisymplectic map B = √−λA we are therefore reduced to the case A = α 1, with α>0 ave to show that α =1.Ifα
Chapter 2 Symplectic capacities y. A linear map A in R 2n preserving the capacities of ellipsoids, c(A(E)) = ither symplectic or antisymplectic. A ∗ ω0 = ω0 or A ∗ ω0 = −ω0. be any capacity function and consider a homeomorphism h of R 2n satis- c(h(E)) = c(E) for all (small) ellipsoids E. h is in addition differentiable, we conclude from Theorem 5 that h is orphism which is either symplectic or antisymplectic, h ′ (x) ∗ ω0 = ±ω0. nalogous to a measure preserving homeomorphism, i.e., a homeomorphism g (2.3). Here we conclude that det h ′ (x) =±1 incaseh is differentiable. hat every capacity function c singles out the distinguished group of homesms preserving the capacity of all open sets. The elements of this group of orphisms have the additional property that they are symplectic or antiic in case they are differentiable. This group can, therefore, be viewed as gical version of the group of symplectic diffeomorphisms. It is not known this group is closed under locally uniform limits. But the following weaker old true. 6. Assume hj : R 2n → R 2n is a sequence of homeomorphisms satisfying c(hj(E)) = c(E) esp. all small) ellipsoids E. Assume hj converges locally uniformly to a orphism h of R 2n .Then esp. all small) ellipsoids E. c(h(E)) = c(E) nce h −1 ◦ hj → id locally uniformly, we conclude for every ellipsoid E and ε
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