There are several example of fixed point theorems to symplectic geometry. As per my knowledge ,it has several application on compact symplectic manifold.
An example is the theorem that any area-preserving mapping (symplectic map) $\psi$ of the two-dimensional sphere into itself exists at least two distinct fixed points although an arbitrary orientation-preserving mapping may have only one single fixed points.
You may want to learn about Floer Lagrangian Homology. It aims to study the minimal number of intersections of two Lagrangian submanifolds in a symplectic manifold, in particular, it gives you information about the minimal number of fixed points of Hamiltonian diffeomorphisms (and hence, periodic orbits in geometric dynamical systems). Also, there is an old (but extremely interesting) paper by Moser where the topic of fixed points is explored in some detail:
http://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6241-11511_2006_Article_BF02545741.pdf
Along with Moser's work.
Article Fixed points of symplectic periodic flows
Article Symplectic rigidity, symplectic fixed points and global pert...
http://matwbn.icm.edu.pl/ksiazki/bcp/bcp45/bcp4511.pdf
And references therein.
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