On the Hofer–Zehnder conjecture for semipositive symplectic manifolds

Marcelo S. Atallah; Han Lou

doi:10.3934/jmd.2025018

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2025, Volume 21: 755-804. Doi: 10.3934/jmd.2025018

This volume Previous Article Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus Next Article Mapping class group orbit closures for Deroin–Tholozan representations

On the Hofer–Zehnder conjecture for semipositive symplectic manifolds

Marcelo S. Atallah^1, and
Han Lou^2,

1.
Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada
2.
Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Received: January 11, 2024

Revised: July 21, 2025

Published online: December 23, 2025

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Abstract

We show that, on a closed semipositive symplectic manifold with semisimple quantum homology, any Hamiltonian diffeomorphism possessing more contractible fixed points, counted homologically, than the total Betti number of the manifold, must have infinitely many periodic points. This generalizes to the semipositive setting the beautiful result of Shelukhin on the Hofer–Zehnder conjecture. The key component of the proof is a new study of the effect of reduction modulo a prime on the bounds on filtered Floer homology that arise from semisimplicity. This relies on the theory of algebraic extensions of non-Archimedean normed fields.

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Mathematics Subject Classification: Primary: 53D22, 53D40, 37J12, 37J06; Secondary: 57R17, 37J06.

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