An elementary alternative to PFH spectral invariants

TL;DR

This paper introduces a simpler alternative to PFH spectral invariants, enabling more elementary proofs of key results in symplectic geometry and dynamical systems, including closing lemmas and the simplicity conjecture.

Contribution

It presents a new, elementary spectral invariant approach that simplifies proofs of complex results previously reliant on PFH spectral invariants.

Findings

Provided elementary proofs for quantitative closing lemmas

Confirmed the simplicity conjecture using new invariants

Demonstrated the effectiveness of elementary invariants in symplectic topology

Abstract

Inspired by Hutchings' elementary alternative to ECH capacities, we introduce an elementary alternative to spectral invariants defined via periodic Floer homology (PFH). We use these spectral invariants to provide more elementary proofs of a number of results which have recently been obtained using PFH spectral invariants. Among these results are quantitative closing lemmas for area-preserving surface diffeomorphisms and the simplicity conjecture.