Definition of "symplectomorphism"

In prequantizing symplectomorphisms of the type T*g, the only special property which we used was the fact that they preserve canonical 1-forms.

1977, Alan Weinstein, Lectures on Symplectic Manifolds, American Mathematical Society, page 34

Poincare's argument is based on the fact that the fixed points of a symplectomorphism of the annulus are precisely the critical points of the function F ( x , y ) = ∫ ( f d v − g d u ) {\displaystyle \textstyle F(x,y)=\int {(fdv-gdu)}} , where u = ( X + x ) / 2 {\displaystyle u=(X+x)/2} , v = ( Y + y ) / 2 {\displaystyle v=(Y+y)/2} , true under the assumption that the Jacobian ∂ ( u , v ) / ∂ ( x , y ) {\displaystyle \partial (u,v)/\partial (x,y)} is different from zero.

2001, A. Dzhamay, G. Wassermann (translators), V. I. Arnol'd, A. B. Givental' Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 39

The symplectomorphisms of a symplectic manifold ( M , ω ) {\displaystyle (M,\omega )} form the group / Sympl ( M , ω ) = { f : M ⟶ ≃ M | f ∗ ω = ω } {\displaystyle {\text{Sympl}}(M,\omega )=\lbrace f:M{\overset {\simeq }{\longrightarrow }}M\ \vert \ f^{*}\omega =\omega \rbrace } .

2008, Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2nd printing with corrections, page 63