On the Hofer–Zehnder conjecture on ℂPd via generating functions

Abstract

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in [Formula: see text] of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of [Formula: see text] that has at least [Formula: see text] nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of [Formula: see text]-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.