An Arnold-type principle for non-smooth objects

Lev Buhovsky*, Vincent Humilière, Sobhan Seyfaddini

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 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 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.

Original languageEnglish
Article number24
JournalJournal of Fixed Point Theory and Applications
Issue number2
StatePublished - Jun 2022


FundersFunder number
National Science Foundation851701, DMS-1440140
European Research Council757585
Israel Science FoundationANR-15-CE40-0007, 2026/17


    • Arnold conjecture
    • C Lagrangian
    • C rigidity
    • Hamiltonian homeomorphisms
    • Legendrian
    • spectral invariants
    • symplectic


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