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An Arnold-type principle for non-smooth objects

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Abstract

In this article, we study the Arnold conjecture in settings where objects under consideration are no longer smooth but only continuous. The example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point shows that the conjecture does not admit a direct generalization to continuous settings. However, it appears that the following Arnold-type principle continues to hold in \(C^0\) settings: suppose that X is a non-smooth object for which one can define spectral invariants. If the number of spectral invariants associated to X is smaller than the number predicted by the (homological) Arnold conjecture, then the set of fixed/intersection points of X is homologically non-trivial, hence it is infinite. We recently proved that the above principle holds for Hamiltonian homeomorphisms of closed and aspherical symplectic manifolds. In this article, we verify this principle in two new settings: \(C^0\) Lagrangians in cotangent bundles and Hausdorff limits of Legendrians in 1-jet bundles which are isotopic to 0-section. An unexpected consequence of the result on Legendrians is that the classical Arnold conjecture does hold for Hausdorff limits of Legendrians in 1-jet bundles.

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Notes

  1. Here, \(\cap \) refers to the intersection product in homology. The cup length can be equivalently defined in terms of the cup product in cohomology.

  2. Note that we do not make any assumptions regarding non-degeneracy of Hamiltonian diffeomorphisms here.

  3. Sandon has recently presented a reformulation of the Arnold conjecture for contactomorphisms; see [34, 35].

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Acknowledgements

We dedicate this article to Claude Viterbo whose works in mathematics have deeply influenced ours. Not only that, Claude’s constant support and interest in our research, since the very beginnings of our careers, has been a great source of encouragement to each one of us. Lemma 4.5 was proven jointly with Rémi Leclercq. We are grateful to him for generously sharing his ideas with us. We also thank Alberto Abbondandolo for pointing out to us the paper of Pierre Pageault [31] used in the proof of Proposition 1.2. Our proofs of Theorems 1.1 and 1.5 were inspired by the paper of Wyatt Howard [15].

The first author was partially supported by ERC Starting Grant 757585 and ISF Grant 2026/17. The second author was partially supported by the ANR project “Microlocal” ANR-15-CE40-0007. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the third author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester. The third author greatly benefited from the lively research atmosphere of the MSRI and would like to thank the members and staff of the MSRI for their warm hospitality. The third author was partially supported by ERC Starting Grant 851701.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Lev Buhovsky

  2. Sorbonne Université and Université Paris Cité, CNRS, IMJ–PRG F-75005, Paris, France

    Vincent Humilière & Sobhan Seyfaddini

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  1. Lev Buhovsky
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  2. Vincent Humilière
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  3. Sobhan Seyfaddini
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Correspondence to Lev Buhovsky.

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Dedicated to Claude Viterbo on the occasion of his 60th birthday.

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This article is part of the topical collection “Symplectic geometry—A Festschrift in honour of Claude Viterbo’s 60th birthday” edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk.

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Buhovsky, L., Humilière, V. & Seyfaddini, S. An Arnold-type principle for non-smooth objects. J. Fixed Point Theory Appl. 24, 24 (2022). https://doi.org/10.1007/s11784-022-00934-z

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