On the Hofer-Zehnder conjecture on weighted projective spaces

Article Mathematics

On the Hofer-Zehnder conjecture

Egor Shelukhin

Summary: In this paper, we prove a higher-dimensional homological generalization of a well-known result by Franks in 1992, as conjectured by Hofer and Zehnder in 1994. We show that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the manifold's homology, then it must have an infinite number of contractible periodic points.

ANNALS OF MATHEMATICS (2022)

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Article Mathematics, Applied

Another look at the Hofer-Zehnder conjecture

Erman Cineli et al.

Summary: We present a simpler proof of a modified variant of Shelukhin's recent theorem, which extends Franks' two-or-infinitely-many theorem to higher dimensions and establishes a general case of the Hofer-Zehnder conjecture for Hamiltonian diffeomorphisms.

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS (2022)

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Article Mathematics

On the Hofer-Zehnder conjecture on CPd via generating functions

Simon Allais

Summary: In this study, we use generating function techniques to deduce a proof in CPd of the homological generalization of Franks theorem due to Shelukhin. This result proves the Hofer-Zehnder conjecture in the nondegenerate case.

INTERNATIONAL JOURNAL OF MATHEMATICS (2022)

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Article Mathematics, Applied

Quantitative Heegaard Floer cohomology and the Calabi invariant

Daniel Cristofaro-Gardiner et al.

Summary: This article introduces a new family of spectral invariants associated with certain Lagrangian links on compact and connected surfaces. It also resolves open questions in topological surface dynamics and continuous symplectic topology using classical Floer homology. The importance of this work is highlighted by the recovery of the Calabi invariant and the construction of quasi-morphisms.

FORUM OF MATHEMATICS PI (2022)

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Article Mathematics