Lusternik-Schnirelmann Theory and Closed Reeb Orbits

TL;DR

This paper introduces a new variant of Lusternik-Schnirelmann theory for Floer and symplectic homology, leading to multiplicity results for closed Reeb orbits and insights into spectral invariants and Conley--Zehnder indices.

Contribution

It develops a novel theoretical framework connecting shift operators with spectral invariants, enabling new multiplicity results for Reeb orbits on various contact manifolds.

Findings

Spectral invariants decrease strictly under the shift operator for isolated periodic orbits.

Proved new multiplicity results for simple closed Reeb orbits on the sphere and other contact manifolds.

Revisited and provided a new proof of the jump theorem of Long and Zhu.

Abstract

We develop a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley--Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.