Barcode entropy and relative symplectic cohomology

TL;DR

This paper links the growth rate of a Floer-theoretic invariant called barcode entropy in relative symplectic cohomology to the dynamical complexity of Reeb flows on boundary manifolds, providing a quantitative relationship.

Contribution

It establishes a bound connecting barcode entropy of relative symplectic cohomology to the topological entropy of Reeb flows, revealing a new quantitative link between symplectic invariants and dynamical systems.

Findings

Barcode entropy is bounded above by Reeb flow topological entropy.

The bound depends on the embedding of the Liouville domain.

Provides a quantitative measure linking Floer theory and dynamics.

Abstract

In this paper, we study the barcode entropy--the exponential growth rate of the number of not-too-short bars--of the persistence module associated with the relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. Our main result establishes a quantitative link between this Floer-theoretic invariant and the dynamics of the Reeb flow on $\partial K$. More precisely, we show that the barcode entropy of the relative symplectic cohomology $SH_M(K)$ is bounded above by a constant multiple of the topological entropy of the Reeb flow on the boundary of the domain, where the constant depends on the embedding of $K$ into $M$.