On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C} \mathrm{P}^d$ via generating functions

Inspired by the techniques of Givental and Theret, we provide a proof with generating functions of a recent result of Ginzburg-Gurel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.