We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in [Formula: see text]Pd 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]Pd 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.

We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus g; it implies that a symplectic diffeomorphism isotopic to the identity with more than 2g − 2 fixed points, counted homologically, has infinitely many periodic points. Furthermore, we present examples of symplectic diffeomorphisms with a prescribed number of periodic points. In particular, we construct symplectic flows on surfaces possessing only one fixed point and no other periodic orbits.

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