An Arnold-type principle for non-smooth objects

TL;DR

This paper explores an Arnold-type principle for non-smooth objects in symplectic topology, demonstrating its validity for certain continuous objects and deriving implications for the classical Arnold conjecture.

Contribution

It extends the Arnold-type principle to $C^0$ Lagrangians and Legendrian limits, showing its applicability beyond smooth settings and confirming the classical conjecture in new cases.

Findings

The Arnold-type principle holds for $C^0$ Lagrangians in cotangent bundles.

The principle is valid for Hausdorff limits of Legendrians in 1-jet bundles.

The classical Arnold conjecture holds for Hausdorff limits of Legendrians.

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