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Barcode entropy and relative symplectic cohomology

Jonghyeon Ahn

TL;DR

The paper defines and studies barcode entropy for the relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a closed symplectic manifold $M$, linking the growth of not-too-short bars to the Reeb flow on $\partial K$. It develops the necessary persistence-module and Floer-theoretic framework, introduces action-filtered relative cohomology, and defines corresponding barcode entropies. The main result is an explicit upper bound $\hbar(SH_M(K)) \le C(M,K)\, h_{\mathrm{top}}(\varphi_α^1)$, where $C(M,K)$ reflects the embedding via a collar, and $φ_α^t$ is the Reeb flow on the boundary. This establishes a quantitative bridge between ambient boundary dynamics and the persistence structure of a relative Floer-theoretic invariant, with the bound guaranteeing finiteness and exponential growth control of the barcode entropy.

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 $M$.

Barcode entropy and relative symplectic cohomology

TL;DR

The paper defines and studies barcode entropy for the relative symplectic cohomology of a Liouville domain embedded in a closed symplectic manifold , linking the growth of not-too-short bars to the Reeb flow on . It develops the necessary persistence-module and Floer-theoretic framework, introduces action-filtered relative cohomology, and defines corresponding barcode entropies. The main result is an explicit upper bound , where reflects the embedding via a collar, and is the Reeb flow on the boundary. This establishes a quantitative bridge between ambient boundary dynamics and the persistence structure of a relative Floer-theoretic invariant, with the bound guaranteeing finiteness and exponential growth control of the barcode entropy.

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 of a Liouville domain embedded in a symplectic manifold . Our main result establishes a quantitative link between this Floer-theoretic invariant and the dynamics of the Reeb flow on . More precisely, we show that the barcode entropy of the relative symplectic cohomology 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 into .
Paper Structure (18 sections, 12 theorems, 185 equations)

This paper contains 18 sections, 12 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Preliminaries
  5. From persistence module theory
  6. From Floer theory
  7. Hamiltonian Floer cohomology
  8. Relative symplectic cohomology
  9. From dynamical systems theory
  10. Relative symplectic cohomology of a Liouville domain
  11. Reformulation of the definition
  12. Filtered relative symplectic cohomology
  13. Barcode entropies
  14. Definition
  15. Comparison of barcode entropies
  16. ...and 3 more sections

Key Result

Theorem A

Let $(M,\omega)$ be a closed symplectic manifold and $K \subset M$ be a Liouville domain. Assume that $(M,\omega)$ is symplectically aspherical and $(\partial K, \alpha)$ is index-bounded. Then there exists a constant $C=C(M,K) >0$, depending on the pair $(M,K)$, such that where $\varphi_\alpha^t$ is the Reeb flow on $(\partial K, \alpha)$.

Theorems & Definitions (26)

  • Theorem A: Theorem \ref{['thmare']}
  • Definition 2.1
  • Theorem 2.2: Normal form theorem
  • Theorem 2.3: Yomdin
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 16 more