Barcode entropy and relative symplectic cohomology

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 $S{H}_M(K)$ 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 $\mathrm{\partial }K$. More precisely, we show that the barcode entropy of the relative symplectic cohomology $S{H}_M(K)$ 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 $M$.