  Inspired by the techniques of Givental and Th\'eret, we provide a proof with
generating functions of a recent result of Ginzburg-G\"urel 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.
