On the relation between action and linking

TL;DR

This paper introduces numerical invariants for contact forms in three dimensions, using asymptotic cycles to relate action and linking, and extends results on mean actions of periodic points to Anosov Reeb flows.

Contribution

It develops a new framework connecting action and linking via asymptotic cycles and proves a generalized result for Anosov Reeb flows.

Findings

Introduces numerical invariants of contact forms.

Establishes estimates using asymptotic cycles.

Extends results to Anosov Reeb flows.

Abstract

We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.