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

Abstract

We use generating function techniques developed by Givental, Théret 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.

49 pages, 2 figures

References in corpus (1)

• On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}{\text{P}}^d$ via generating functions

Cited by in corpus (3)

• On the Hofer-Zehnder conjecture on weighted projective spaces
• Barcode entropy for Reeb flows on contact manifolds with Liouville fillings
• Invariant Sets and Hyperbolic Closed Reeb Orbits