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On the Hofer-Zehnder conjecture on \\(\\mathbb{C}\\text{P}^d\\) via generating functions (with an appendix by Egor Shelukhin)
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On the Hofer-Zehnder conjecture on \\(\\mathbb{C}\\text{P}^d\\) via generating functions (with an appendix by Egor Shelukhin)
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Paper
On the Hofer-Zehnder conjecture on \\(\\mathbb{C}\\text{P}^d\\) via generating functions (with an appendix by Egor Shelukhin)
2022
Overview
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.
Publisher
Cornell University Library, arXiv.org
Subject
