Persistent Entropy of Floer Persistence Barcodes

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 Çineli--Ginzburg--Gürel, 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.

• The paper introduces persistent entropy, a new Floer-theoretic invariant measuring the Shannon entropy growth of bar-length distributions in persistence barcodes.
• It demonstrates that persistent entropy equals barcode entropy for Hamiltonian diffeomorphisms, showing that length distributions add no extra asymptotic information on the exponential scale.
• The study establishes stability estimates in the Hofer metric and provides comparisons in Liouville domains, highlighting rigidity and flexibility in symplectic dynamics.

Persistent Entropy of Floer Persistence Barcodes: Formal Summary

Motivation and Context

The article "Persistent Entropy of Floer Persistence Barcodes" (2606.19071) introduces a new Floer-theoretic invariant termed persistent entropy, based on the iteration-induced linear growth rate of Shannon entropy calculated from distributions of finite bar lengths in Floer persistence barcodes. This construction is explicitly inspired by analogs in topological data analysis and serves as a counterpart to the barcode entropy defined by Çineli-Ginzburg-Gürel, which counts the exponential growth of the number of nontrivial bars in persistence barcodes.

Barcode entropy has established connections with topological entropy for Hamiltonian systems, providing both upper and lower bounds in dynamical systems, especially on closed surfaces. However, barcode entropy operates solely on the count of bars, ignoring the distribution of their lengths—a facet with substantial relevance in symplectic topology. The paper seeks to clarify whether persistent entropy, which takes bar-length distribution seriously, yields strictly stronger invariants, and systematically investigates its theoretical and structural properties.

Definitions and Constructions

The main technical groundwork is laid out via precise definitions for:

• Relative and Absolute Persistent Entropy: Both are given as asymptotic growth rates of Shannon entropy of finite bar-length distributions under iteration of Hamiltonian diffeomorphisms. Relative variants pertain to pairs of Lagrangian submanifolds, while absolute variants arise from Floer complexes associated directly to Hamiltonian dynamics.
• Slow Versions: Polynomial (logarithmic) growth rates are considered for zero-entropy systems, paralleling slow entropy in ergodic theory.
• Barcodes and Persistence Modules: The framework is closely aligned with standard definitions in Floer theory and persistence modules, including the bottleneck distance and structural theorems ensuring well-defined invariance and stability.

The Shannon entropy is computed for normalized bar-length vectors, encoding the effective diversity of bar lengths. Notably, these definitions extend naturally to exact Lagrangians and Liouville domains, with the latter facilitating contact-geometric analogues via symplectic homology barcodes.

Main Results

Hamiltonian Diffeomorphisms

The central claims are:

• Equality of Persistent and Barcode Entropies: For any Hamiltonian diffeomorphism on a closed monotone symplectic manifold, the persistent entropy (linear growth rate of Shannon entropy) equals the barcode entropy (exponential growth rate of bar counts), both in the relative and absolute sense:

$$h_{\mathrm{per}}(\varphi) = h_{\mathrm{bar}}(\varphi)$$

Similarly, for slow versions:

$$h_{\mathrm{sper}}(\varphi) \leq h_{\mathrm{sbar}}(\varphi)$$

with equality holding under subexponential boundary-depth growth conditions.

• Structural Implications: This result shows that on the exponential scale, the bar-length distribution and the bar count encode strictly the same asymptotic information for Hamiltonian diffeomorphisms. The bar distribution does not detect more than the number of sufficiently long bars in this regime.
• Stability Estimates: The paper provides quantitative estimates for finite-level Shannon entropy with respect to Hofer distance, demonstrating that persistent entropy is a stable numerical statistic at the finite barcode level. Continuity in Hofer metric is established, underlining compatibility with Floer-theoretic geometry.

Liouville Domains and Reeb Flows

• Comparison Theorems: For Liouville domains, persistent entropy is bounded above by barcode entropy, i.e.,

$$h_{\mathrm{per}}(X, \lambda) \leq h_{\mathrm{bar}}(X, \lambda)$$

Equality holds when the symplectic homology vanishes or when bar lengths grow subexponentially in the action window.

• Flexibility and Rigidity: The article constructs explicit examples for cotangent disk bundles of negatively curved manifolds, establishing the equality of persistent entropy, barcode entropy, and topological entropy for geodesic flows. This realizes the entropy rigidity threshold from Katok's theorem within the context of persistent entropy.
• Positivity Results: Lower bounds are derived for persistent entropy in the presence of hyperbolic invariant sets, tying persistent entropy to topological entropy and ensuring positivity in cases where dynamical complexity is present.
• Flexibility Statements: The construction of contact forms with arbitrarily small entropy and the realization of all values strictly greater than Katok's constant for surfaces of genus $k \geq 2$ demonstrates the flexibility of persistent entropy above critical thresholds.

Implications and Theoretical Significance

The structural equality between persistent and barcode entropy for Hamiltonian diffeomorphisms is a nontrivial Floer-theoretic result, establishing that the additional length-distribution information encoded in Shannon entropy cannot asymptotically distinguish systems beyond what barcode entropy already detects. In contrast, differences may arise in contact settings (Liouville domains) unless bar-length growth is controlled, with the gap potentially detectable via infinite bars or highly unbalanced length distributions.

Persistent entropy inherits dynamical lower bounds from barcode entropy and, through stability results, offers robust numerical statistics for applications in Hamiltonian dynamics. Its compatibility with Hofer geometry further recommends it as a metric-sensitive quantitative invariant.

The explicit link between persistent entropy, topological entropy, and geodesic flow entropy in negatively curved manifolds unifies symplectic, dynamical, and topological perspectives. The flexibility results suggest further applications in entropy rigidity and collapse phenomena for Reeb and Hamiltonian flows.

Speculation on Future Developments

Open problems remain regarding strict gaps between persistent and barcode entropy in the contact case and the realizability of such gaps in symplectic geometry. Further research may explore the detection mechanisms for such phenomena, especially via infinite bars or the distribution of long finite bars. Extensions to higher-dimensional settings, metric entropy, and operator-theoretic perspectives on persistence modules are natural directions. The flexibility of persistent entropy above structural thresholds invites investigation into entropy collapses and rigidity phenomena across broader classes of symplectic and contact dynamics.

Conclusion

The paper rigorously establishes the persistent entropy as a well-founded, stable Floer-theoretic invariant, equating it asymptotically with barcode entropy in standard Hamiltonian settings and identifying conditions for equality and gaps in contact-geometric scenarios. Its numerical, metric, and dynamical stability, along with structural rigidity results and applications to entropy theory, mark a significant development in symplectic topology, Floer homology, and persistence theory (2606.19071).