Barcode Entropy: Floer Theory & Dynamics

• Barcode entropy is a Floer-theoretic invariant defined via the exponential growth of long bars in persistence modules, capturing dynamical complexity in symplectic systems.
• It utilizes filtered Floer homology and pseudo-chord measures to provide lower bounds on metric and topological entropy across diverse dynamical settings.
• The invariant bridges algebraic-topological methods with classical dynamical complexity while being sensitive to system nuances, though it may miss certain subtleties in higher-dimensional or integrable cases.

Barcode entropy is a Floer-theoretic invariant quantifying the exponential growth rate of long intervals—“bars”—in the persistence modules arising from filtered Floer homology or related persistence-theoretic constructions associated to symplectic and contact dynamics. Serving as a bridge between the qualitative complexity of dynamical systems (measured by metric and topological entropy) and the algebraic structure of Floer-theoretic invariants, barcode entropy provides both detection of underlying dynamical complexity and a method to construct measures realizing large entropy. The concept applies uniformly to Hamiltonian diffeomorphisms, Reeb flows, Lagrangian intersections, geodesic flows, and persistence modules associated to symplectic or relative symplectic (co)homology, with precise definitions refined according to context (Cineli et al., 17 Jul 2025).

1. Formal Definition of Barcode Entropy

Barcode entropy is defined via the exponential growth rate of the number of bars in an appropriate persistence module whose length exceeds a small positive threshold $\varepsilon$. Let $(M,\omega)$ be a closed or convex-at-infinity symplectic manifold, and let $\varphi$ be a compactly supported Hamiltonian diffeomorphism. Fix closed, monotone, Hamiltonian-isotopic Lagrangian submanifolds $L_0,L\subset M$. For each $\varepsilon>0$ consider the filtered Lagrangian Floer homology of $(L_k, L)$, where $L_k = \varphi^k(L_0)$, and let $\beta_\varepsilon(L_k, L)$ be the number of bars in the corresponding barcode of length exceeding $\varepsilon$. The $\varepsilon$-barcode entropy is

$$\hbar_\varepsilon(\varphi;L_0,L) = \limsup_{k\to\infty}\frac{1}{k}\log^+\!\beta_\varepsilon(L_k, L),$$

with $\log^+x = \max\{0,\log x\}$. The barcode entropy is then given by

$$\hbar(\varphi;L_0,L) = \lim_{\varepsilon\to 0^+}\hbar_\varepsilon(\varphi;L_0,L).$$

In the absolute case, with $L_0 = L = \Delta$ (the diagonal in $M\times M$), this definition measures the growth of long bars in the persistence module of (fixed-point) Floer homology of $\varphi^k$.

For Reeb flows on the contact boundary $M=\partial W$ of a Liouville domain $W$, the analogous definition counts bars of extent $>\varepsilon$ with left endpoint below action $s$ in filtered symplectic or wrapped Floer homology; the $\varepsilon$-barcode entropy is

$$\hbar_\varepsilon(\varphi) = \limsup_{s\to\infty}\frac{1}{s}\log^+\!\beta_\varepsilon(s;\varphi),$$

and the barcode entropy is the limit as $\varepsilon\to 0^+$ (Cineli et al., 17 Jul 2025).

2. Pseudo-Chord Measures and Their Dynamical Significance

A major structural advance is the identification and utilization of pseudo-chord measures: a special class of invariant measures for $\varphi$ associated to pairs $(L_0, L)$. These are constructed as weak$^*$ limits of empirical measures along large collections of “approximate Floer chords”—finite orbit segments approximating genuine orbits. More formally, a $U$-approximate chord measure is a weak limit of atomic measures averaging over sequences of points in $L_0$ whose $\varphi^k$-iterates enter an open neighborhood $U\supset L$. Passing to a nested sequence of such neighborhoods yields a pseudo-chord measure. Such measures are always invariant under $\varphi$, and their existence is ensured in settings with sufficient accessibility between $L_0$ and $L$ (notably ergodic or hyperbolic dynamics) (Cineli et al., 17 Jul 2025).

These measures are central to establishing lower bounds on dynamical complexity: for any pseudo-chord measure $\mu$,

$\hbar(\varphi;L_0,L) \leq h_\mu(\varphi),$

where $h_\mu(\varphi)$ is the metric entropy with respect to $\mu$.

3. Comparison with Metric and Topological Entropy

Barcode entropy is sandwiched between topological and metric entropy in a precise sense. The main theorem asserts that for every pseudo-chord measure $\mu$, barcode entropy provides a lower bound for metric entropy: $\hbar(\varphi;L_0,L) \le h_\mu(\varphi)$ (Cineli et al., 17 Jul 2025). By maximizing over all invariant probability measures (via the variational principle),

$$\sup_\mu h_\mu(\varphi) = h_{\mathrm{top}}(\varphi),$$

one recovers the classical upper bound

$$\hbar(\varphi;L_0,L) \le h_{\mathrm{top}}(\varphi).$$

In many two- or three-dimensional cases with hyperbolic dynamics, the lower and upper bounds coincide, yielding

$$\hbar(\varphi;L_0,L) = h_{\mathrm{top}}(\varphi|_K)$$

for locally maximal hyperbolic sets $K$ as well as full equality for Hamiltonian diffeomorphisms of surfaces (Cineli et al., 17 Jul 2025, Cineli et al., 2021, Ginzburg et al., 2022).

4. Conceptual and Technical Origins

At its core, barcode entropy refines the connection between dynamical complexity and algebraic-topological invariants arising from Floer theory and persistent homology. The barcode associated to the action-filtered Floer (co)homology persistence module encodes the appearance, persistence, and disappearance of homology classes as one varies the action threshold. Long bars correspond, heuristically, to orbits or intersection points that remain significant under perturbation or iteration. The growth rates of long bars, via action-filtration, capture the proliferation of dynamically distinct phenomena such as periodic points, Floer chords, or closed geodesics.

Key technical tools include Lagrangian tomographs (families of Hamiltonian-isotopic Lagrangians sampling volume growth), Crofton-type integral-geometric bounds relating intersection counts to volume, and Yomdin’s theorem on volume growth and entropy. Lower bounds are ultimately enabled by energy quantization or crossing-energy theorems, guaranteeing that non-trivial Floer strips or Morse trajectories crossing certain regions incur an action drop bounded below, ensuring the isolation and persistence of associated bars (Cineli et al., 17 Jul 2025, Ginzburg et al., 2022).

5. Applications and Consequences

Barcode entropy provides deep connections and applications in symplectic topology and Hamiltonian dynamics, including the following notable cases:

• Hyperbolic Sets: For locally maximal transitive hyperbolic subsets intersecting the relevant Lagrangians in an appropriate fashion, barcode entropy detects and tends to equal the entropy of the hyperbolic set (Cineli et al., 17 Jul 2025).
• Hyperbolic Toral Symplectomorphisms: For linear Anosov maps (e.g., on $\mathbb{T}^2$) and monotone Lagrangian circles tangent to (un)stable manifolds, barcode entropy equals the measure-theoretic/topological entropy, realized on the Bowen–Margulis measure (Cineli et al., 17 Jul 2025).
• Geodesic Flows: On negatively curved manifolds, the Morse-theoretic barcode entropy (associated to the energy functional on loop space) is equal to the topological entropy of the geodesic flow—the bound is sharp and realized by the Bowen–Margulis measure (Ginzburg et al., 2022, Cineli et al., 17 Jul 2025).

The technique also targets the robustness of volume growth and entropy under perturbations, with implications for rigidity/stability of volume–growth rates and for Hofer geometry (Meiwes, 2024).

6. Counterexamples and Limitations

Barcode entropy does not always fully capture topological entropy in higher dimensions. There exist Hamiltonian diffeomorphisms on symplectic manifolds with positive topological entropy but zero barcode entropy, explicitly constructed using pseudo-rotations whose Floer complexes have only infinite bars for all iterates (Cineli, 2023). Thus, barcode entropy is strictly a lower bound and, in dimensions greater than two, may fail to detect dynamical complexity concentrated in dynamically subtle ways not seen by the persistence structure.

Barcode entropy is also invariant under action-filtration preserving symplectic isomorphisms and, in many cases, under deformations of Liouville fillings (Fender et al., 2023, Fernandes, 2024, Barut et al., 11 Mar 2025). In integrable (toric) systems, barcode entropy vanishes, and the barcode growth is merely polynomial, distinguishing these from systems with exponential bar growth (Barut et al., 11 Mar 2025).

7. Synthesis and Broader Context

Barcode entropy consolidates several strands at the interface of Floer theory, persistent homology, and dynamics. It acts as a quantitative, computable proxy for dynamical entropy, with algebraic and analytic underpinnings:

• In low dimensions and hyperbolic settings, it recovers classical invariants, coinciding with topological entropy (Cineli et al., 2021, Ginzburg et al., 2022, Cineli et al., 17 Jul 2025).
• It establishes refined lower bounds for the metric entropy via pseudo-chord measures, going beyond earlier topological bounds (Cineli et al., 17 Jul 2025).
• The algebraic formalism—persistent modules and their barcodes—enables invariance results, structural classification, and computational strategies using filtered Floer theory and energy quantization.
• Applications span geodesic flows, contact and symplectic manifolds, and the study of the growth of periodic orbits and intersection points.
• Limitations highlight the need for further Floer-theoretic or persistence-theoretic refinements to fully capture all dynamical complexity, especially in higher dimensions or settings lacking significant hyperbolicity.

Barcode entropy thus occupies a central role in quantifying and relating persistent homological complexity to the classical measure-theoretic and topological complexity of symplectic and contact dynamical systems (Cineli et al., 17 Jul 2025, Ginzburg et al., 2022, Cineli et al., 2021).