On the Hofer-Zehnder conjecture on CPd via generating functions

We use generating function techniques developed by Givental, Theret and ourselves to deduce a proof in CPd 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 CPd that has at least d + 2 nondegenerate 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 Hoer theory, which lends itself to adaptation to the setting of generating functions.

Automated summary

In this study, we use generating function techniques to deduce a proof in CPd of the homological generalization of Franks theorem due to Shelukhin. This result proves the Hofer-Zehnder conjecture in the nondegenerate case.