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arxiv: 2606.19071 · v1 · pith:G4JXOY2Jnew · submitted 2026-06-17 · 🧮 math.SG · math.DS

Persistent Entropy of Floer Persistence Barcodes

Wenmin Gong This is my paper

Pith reviewed 2026-06-26 18:03 UTC · model grok-4.3

classification 🧮 math.SG math.DS
keywords persistent entropyFloer persistence barcodesbarcode entropyHamiltonian diffeomorphismssymplectic homologyLiouville domainstopological entropyReeb flows
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The pith

For Hamiltonian diffeomorphisms the persistent entropy of Floer barcodes equals the barcode entropy.

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

The paper defines a new invariant called persistent entropy that records the asymptotic linear growth, under iteration, of the Shannon entropy computed from the lengths of finite bars in action-filtered Floer persistence barcodes. It proves that this quantity coincides exactly with the barcode entropy, which instead tracks the exponential growth rate of the number of sufficiently long bars. The equality holds for all Hamiltonian diffeomorphisms on closed symplectic manifolds. For Liouville domains the paper supplies comparison inequalities between the two entropies together with a subexponential length-growth criterion that forces equality when symplectic homology vanishes. Explicit calculations are given for cotangent disk bundles of negatively curved manifolds, where the persistent entropy recovers the topological entropy of the geodesic flow.

Core claim

Persistent entropy is the asymptotic linear growth rate of the Shannon entropy of the distribution of finite bar lengths in the action-filtered Floer persistence barcode of a Hamiltonian diffeomorphism or Reeb flow. For Hamiltonian diffeomorphisms on closed symplectic manifolds the relative and absolute versions of this quantity equal the corresponding barcode entropies defined by Cineli-Ginzburg-Gürel. On Liouville domains the two entropies satisfy general comparison inequalities, and a subexponential growth condition on bar lengths yields equality beyond the vanishing-symplectic-homology case.

What carries the argument

Persistent entropy, the linear growth rate under iteration of the Shannon entropy determined by the lengths of finite bars in an action-filtered Floer persistence barcode.

If this is right

  • Comparison inequalities hold between persistent entropy and barcode entropy on Liouville domains.
  • Equality follows whenever bar lengths grow subexponentially and symplectic homology vanishes.
  • Persistent entropy of the cotangent disk bundle equals the topological entropy of the geodesic flow on the unit sphere bundle of a negatively curved manifold.
  • Finite-level Shannon entropy satisfies Hofer-stability estimates.
  • New flexibility and rigidity questions arise for barcode and persistent entropies of Reeb flows.

Where Pith is reading between the lines

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

  • The equality suggests that counting long bars and measuring the entropy of their length distribution capture the same dynamical information for Hamiltonian systems.
  • The subexponential growth criterion may extend to other filtered homologies where barcodes are defined.
  • Hofer-stability of the finite-level entropy could be used to bound the difference between persistent and barcode entropies under small perturbations.
  • The relation to geodesic-flow entropy opens the possibility of using persistent entropy to detect positive topological entropy in Reeb flows on contact manifolds.

Load-bearing premise

The distribution of finite bar lengths admits a well-defined Shannon entropy whose asymptotic linear growth rate under iteration exists and does not depend on auxiliary choices made when constructing the barcode.

What would settle it

A Hamiltonian diffeomorphism on a closed symplectic manifold for which the linear growth rate of the Shannon entropy of finite bar lengths differs from the exponential growth rate of the number of not-too-short bars.

read the original abstract

Floer persistence barcodes provide a quantitative way to encode action-filtered Floer homology. Inspired by the Shannon entropy of persistence barcodes in topological data analysis, we introduce a Floer-theoretic entropy invariant, called \textit{persistent entropy}, which measures the asymptotic linear growth rate, under iteration, of the Shannon entropy determined by the distribution of finite bar lengths. This is complementary to the barcode entropy of \c{C}ineli--Ginzburg--G\"{u}rel, which records the exponential growth rate of the number of not-too-short bars. We prove that, for Hamiltonian diffeomorphisms, the relative and absolute persistent entropies coincide with the corresponding barcode entropies. For Liouville domains, we prove general comparison inequalities and a subexponential length-growth criterion which gives equality beyond the case of vanishing symplectic homology. We also compute the persistent entropy of cotangent disk bundles of negatively curved manifolds and relate it to the topological entropy of the geodesic flow. In addition, we prove Hofer-stability estimates for finite-level Shannon entropy and derive flexibility and rigidity-type questions for barcode and persistent entropies of Reeb flows.

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

2 major / 3 minor

Summary. The manuscript introduces persistent entropy, defined as the asymptotic linear growth rate (under iteration) of the Shannon entropy computed from the distribution of finite bar lengths in action-filtered Floer persistence barcodes. It proves that, for Hamiltonian diffeomorphisms, the relative and absolute versions of this invariant coincide with the corresponding barcode entropies of Cineli–Ginzburg–Gürel; it establishes comparison inequalities between the two entropies together with a subexponential length-growth criterion that yields equality for Liouville domains beyond the vanishing symplectic-homology case; it computes the invariant explicitly for cotangent disk bundles of negatively curved manifolds and relates the value to the topological entropy of the geodesic flow; and it derives Hofer-stability estimates for the finite-level Shannon entropy while posing flexibility/rigidity questions for Reeb flows.

Significance. If the stated equalities and comparisons hold, the work supplies a new, complementary entropy invariant in symplectic geometry that quantifies the spread of bar lengths rather than their exponential proliferation. The explicit computations linking persistent entropy to geodesic-flow entropy and the Hofer-stability results strengthen its potential utility for distinguishing Hamiltonian and Reeb dynamics.

major comments (2)
  1. [§3] §3 (definition of persistent entropy): the Shannon entropy is taken over the normalized distribution of finite bar lengths, but the manuscript does not explicitly address whether this distribution remains a probability measure when the total number of bars tends to infinity under iteration; this normalization step is load-bearing for the claimed linear growth rate to be well-defined and independent of auxiliary choices.
  2. [Theorem 4.3] Theorem 4.3 (subexponential length-growth criterion for Liouville domains): the criterion is used to obtain equality between persistent and barcode entropies, yet the proof sketch does not verify that the length-growth hypothesis is preserved under the iteration maps employed in the Floer construction; a counter-example or explicit verification would be needed to confirm the criterion applies in the stated generality.
minor comments (3)
  1. [§2, §4] Notation for relative versus absolute versions is introduced in §2 but used interchangeably in some statements of §4; a uniform convention would improve readability.
  2. [§5] The computation for cotangent disk bundles in §5 would benefit from an explicit table listing the persistent entropy values alongside the corresponding topological entropies for the model manifolds considered.
  3. [References] A few references to the barcode-entropy literature appear without page numbers; adding them would facilitate verification of the comparison statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We respond to each major comment below and will make the indicated revisions.

read point-by-point responses

  1. Referee: [§3] §3 (definition of persistent entropy): the Shannon entropy is taken over the normalized distribution of finite bar lengths, but the manuscript does not explicitly address whether this distribution remains a probability measure when the total number of bars tends to infinity under iteration; this normalization step is load-bearing for the claimed linear growth rate to be well-defined and independent of auxiliary choices.

    Authors: We appreciate the referee's observation. For each fixed iterate the action-filtered Floer complex is finite-dimensional, so the barcode consists of finitely many bars and the normalized distribution is always a probability measure. The linear growth rate is then taken in the limit of the number of iterations. We will insert a short clarifying paragraph in §3 making this finiteness explicit and confirming independence of auxiliary choices. revision: yes

  2. Referee: [Theorem 4.3] Theorem 4.3 (subexponential length-growth criterion for Liouville domains): the criterion is used to obtain equality between persistent and barcode entropies, yet the proof sketch does not verify that the length-growth hypothesis is preserved under the iteration maps employed in the Floer construction; a counter-example or explicit verification would be needed to confirm the criterion applies in the stated generality.

    Authors: The referee correctly notes that the proof sketch is brief on this point. The subexponential length-growth hypothesis is stated directly on the bar lengths of the persistence module; the iteration map on the free loop space preserves the relevant action bounds by the standard properties of the Hamiltonian action functional, so the hypothesis carries over. To make the argument fully rigorous we will expand the proof of Theorem 4.3 with an explicit verification of preservation under iteration. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces persistent entropy as an independent invariant: the asymptotic linear growth rate of Shannon entropy computed from the distribution of finite bar lengths in action-filtered Floer persistence barcodes. Barcode entropy is separately defined as the exponential growth rate of the count of not-too-short bars. The claimed results are explicit proofs that the two coincide for Hamiltonian diffeomorphisms, plus comparison inequalities for Liouville domains; these are presented as theorems rather than identities by construction. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the stated definitions or results. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard existence and functoriality properties of Floer homology and persistence modules in symplectic geometry, plus the new definition of persistent entropy; no free parameters or invented entities with independent evidence are indicated.

axioms (1)
  • domain assumption Standard properties of action-filtered Floer homology and the associated persistence barcodes for Hamiltonian diffeomorphisms and Liouville domains.
    Invoked as the foundation for defining both barcode entropy and the new persistent entropy.
invented entities (1)
  • Persistent entropy no independent evidence
    purpose: To quantify the asymptotic linear growth rate of Shannon entropy determined by the distribution of finite bar lengths in Floer persistence barcodes.
    Newly introduced invariant whose properties are proved in the paper.

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