On the Hofer-Zehnder conjecture on $\mathbb{C}\text{P}^d$ via generating functions (with an appendix by Egor Shelukhin)

TL;DR

This paper proves the Hofer-Zehnder conjecture for $ ext{CP}^d$ using generating functions, establishing that Hamiltonian diffeomorphisms with sufficiently many non-degenerate periodic points have infinitely many, without relying on Floer homology.

Contribution

It provides a new proof of the Hofer-Zehnder conjecture in $ ext{CP}^d$ via generating functions, avoiding Floer homology and $J$-holomorphic curves.

Findings

Proves the Hofer-Zehnder conjecture in $ ext{CP}^d$ for non-degenerate cases.

Establishes a Smith-type inequality for Hamiltonian diffeomorphisms using generating functions.

Offers a new approach that bypasses Floer homology techniques.

Abstract

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.