We use generating function techniques developed by Givental, Th\'eret and ourselves to deduce a proof in $\mathbb{ C}\text{ P}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer-Zehnder conjecture in the non-degenerated case: every Hamiltonian diffeomorphism of $\mathbb{ C}\text{ P}^d$ that has at least $d+2$ non-degenerated periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.
Categories: math.SG math.DS