Another look at the Hofer-Zehnder conjecture
type: Item
instance of: scholarly article
title: Another look at the Hofer-Zehnder conjecture
DOI: 10.1007/978-3-031-19111-4_12
author: Erman Çineli
author: Viktor L. Ginzburg
author: Başak Z. Gürel
publication date: 2024-09-11T00:00:00Z
review text: A theorem of \textit{J. Franks} [Invent. Math. 108, No. 2, 403--418 (1992; Zbl 0766.53037); New York J. Math. 2, 1--19 (1996; Zbl 0891.58033)] says that an area-preserving diffeomorphism \(\phi\colon S^2\to S^2\) has either exactly two or infinitely many periodic points. The Hofer-Zehnder conjecture asks for a higher-dimensional analogue of this result. It has been open for a long time, the first substantial result in this direction being the theorem of \textit{E. Shelukhin} [Ann. Math. (2) 195, No. 3, 775--839 (2022; Zbl 1503.53157)] for closed monotone symplectic manifolds \(M\) with semisimple quantum homology saying that if a Hamiltonian diffeomorphism \(\phi\colon M\to M\) has more contractible fixed points, counted homologically, than the total dimension of the homology \(H_*(M)\), then \(\phi\) must have an infinite number of contractible periodic points.\N\NThe paper under review gives another proof of Shelukhin's theorem, which like Shelukhin's proof uses the pair-of-pants product of \textit{P. Seidel} [Geom. Funct. Anal. 25, No. 3, 942--1007 (2015; Zbl 1331.53119)] but highlights different aspects of dynamics and Floer theory. The proof in this paper is a consequence of Theorem 3.1 which relates the Floer graphs for \(\phi\) and \(\phi^2\). (The Floer graph has the capped fixed points as its vertices, they are connected by an arrow if there is an odd number of Floer trajectories.)\N\NFor the entire collection see [Zbl 1515.53004].
reviewed by: Thilo Kuessner
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 37J12
Mathematics Subject Classification ID: 37J39
zbMATH DE Number: 7910414
zbMATH Keywords: periodic orbits
zbMATH Keywords: Hamiltonian diffeomorphisms
zbMATH Keywords: Frank's theorem
zbMATH Keywords: equivariant Floer cohomology
zbMATH Keywords: pseudo-rotations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7910414