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arxiv: 2606.09164 · v1 · pith:YFBHLCQ7new · submitted 2026-06-08 · 🧮 math.SG

A lower bound for relative symplectic cohomology barcode entropy

Jonghyeon Ahn This is my paper

Pith reviewed 2026-06-27 14:20 UTC · model grok-4.3

classification 🧮 math.SG
keywords barcode entropyrelative symplectic cohomologyReeb flowtopological entropyhyperbolic invariant setLiouville domainpersistence moduleFloer theory
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The pith

The barcode entropy of relative symplectic cohomology SH_M(K) is bounded below by the topological entropy of the Reeb flow on any hyperbolic invariant set it contains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a lower bound relating a Floer-theoretic growth rate to boundary dynamics. Barcode entropy is defined as the exponential growth rate of the number of sufficiently long bars in the persistence module coming from the relative symplectic cohomology of a Liouville domain K inside M. The proof shows this quantity is at least as large as the topological entropy of the Reeb flow on the boundary of K, restricted to any hyperbolic invariant set that the flow admits. A reader would care because the result gives an explicit dynamical lower bound for an algebraic invariant that arises in symplectic geometry.

Core claim

We show that the barcode entropy of SH_M(K) is bounded below by the topological entropy of the Reeb flow restricted to a hyperbolic invariant set when the Reeb flow on ∂K possesses such a set.

What carries the argument

Barcode entropy of the persistence module SH_M(K), which extracts the exponential growth rate of not-too-short bars and is compared directly to the topological entropy of the Reeb flow on a hyperbolic invariant set.

If this is right

  • Positive topological entropy of the Reeb flow on a hyperbolic set forces the barcode entropy of SH_M(K) to be positive.
  • The growth rate of bars in the persistence module is controlled from below by a concrete dynamical quantity on the boundary.
  • The bound supplies a way to read off lower estimates for the algebraic complexity of relative symplectic cohomology from the existence of chaotic Reeb orbits.
  • The result applies whenever the boundary Reeb flow contains a hyperbolic set, regardless of the ambient manifold M.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same comparison technique might extend to other Floer-theoretic invariants that admit persistence-module structures.
  • Examples with explicitly computable Reeb flows and hyperbolic sets, such as certain ellipsoids, could be used to test sharpness of the bound.
  • If the hyperbolic-set hypothesis can be relaxed, the result would apply to a wider class of contact boundaries.

Load-bearing premise

The Reeb flow on the boundary of K must possess a hyperbolic invariant set.

What would settle it

Construct or compute an explicit Liouville domain K where the Reeb flow on ∂K has a hyperbolic invariant set of positive topological entropy yet the barcode entropy of SH_M(K) is strictly smaller than that entropy value.

read the original abstract

In this paper, we continue to study the barcode entropy of relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. This barcode entropy measures the exponential growth rate of the number of not-too-short bars in the persistence module $SH_M(K)$. We prove that this Floer-theoretic invariant admits a nontrivial lower bound in terms of the topological entropy of the Reeb flow on $\partial K$ when the Reeb flow possesses a hyperbolic invariant set. More precisely, we show that the barcode entropy of $SH_M(K)$ is bounded below by the topological entropy of the Reeb flow restricted to a hyperbolic invariant set.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves a conditional lower bound: under the hypothesis that the Reeb flow on the boundary of a Liouville domain K ⊂ M possesses a hyperbolic invariant set Λ, the barcode entropy of the relative symplectic cohomology SH_M(K) is at least the topological entropy of the Reeb flow restricted to Λ. The argument constructs a persistence module from Floer data on K and compares its bar growth rate to the orbit growth on Λ via standard techniques in symplectic cohomology and persistence modules.

Significance. If the result holds, it supplies an explicit dynamical lower bound for a Floer-theoretic entropy invariant, connecting barcode entropy to Reeb dynamics on hyperbolic sets. This is potentially useful for producing nontrivial estimates in contact and symplectic geometry when hyperbolic sets are known to exist. The manuscript uses standard Floer-theoretic and persistence-module machinery whose hypotheses align with the stated assumptions.

minor comments (2)
  1. The abstract and introduction state the main theorem but do not indicate the precise section where the comparison between the persistence module of SH_M(K) and the orbit growth on Λ is carried out; adding an explicit reference to the relevant proposition or theorem number would improve readability.
  2. Notation for the barcode entropy (e.g., how 'not-too-short bars' are quantified) and the precise definition of the hyperbolic invariant set Λ should be recalled or cross-referenced at the statement of the main theorem for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The report recommends minor revision but lists no specific major comments or requested changes. We therefore have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central result is a conditional lower bound: barcode entropy of SH_M(K) is at least the topological entropy of the Reeb flow restricted to an explicitly hypothesized hyperbolic invariant set Λ ⊂ ∂K. The argument proceeds via standard Floer data on the Liouville domain, construction of the persistence module for relative symplectic cohomology, and comparison of bar growth rates to orbit growth on Λ. No equation or step reduces by definition to a fitted quantity, self-citation load-bearing premise, or ansatz smuggled from prior work by the same author. The hypothesis on the existence of Λ is stated openly and is not derived from the paper's own constructions. This is the normal case of an independent dynamical lower bound.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all technical setup (filtered Floer complexes, persistence modules, Reeb dynamics) is treated as background.

pith-pipeline@v0.9.1-grok · 5630 in / 1089 out tokens · 15637 ms · 2026-06-27T14:20:03.425664+00:00 · methodology

discussion (0)

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Reference graph

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23 extracted references · 8 canonical work pages

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