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On the Barcode Entropy of Reeb Flows

Erman Cineli, Viktor L. Ginzburg, Basak Z. Gurel, Marco Mazzucchelli

TL;DR

The paper develops a precise link between barcode entropy, a Floer-theoretic invariant, and topological entropy for Reeb flows. Central to the approach is the Crossing Energy Theorem, which provides a uniform positive energy lower bound for Floer cylinders crossing hyperbolic invariant sets, enabling a lower bound on barcode growth by the topological entropy of those sets. Together with existing upper bounds, this yields equality of barcode entropy and topological entropy in dimension three, thereby connecting localized hyperbolic dynamics with global symbolic growth captured by persistence modules. The results extend prior Hamiltonian and geodesic-flow insights to Reeb dynamics, highlighting barcode entropy as a robust detector of dynamical complexity in contact-type settings.

Abstract

In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows.

On the Barcode Entropy of Reeb Flows

TL;DR

The paper develops a precise link between barcode entropy, a Floer-theoretic invariant, and topological entropy for Reeb flows. Central to the approach is the Crossing Energy Theorem, which provides a uniform positive energy lower bound for Floer cylinders crossing hyperbolic invariant sets, enabling a lower bound on barcode growth by the topological entropy of those sets. Together with existing upper bounds, this yields equality of barcode entropy and topological entropy in dimension three, thereby connecting localized hyperbolic dynamics with global symbolic growth captured by persistence modules. The results extend prior Hamiltonian and geodesic-flow insights to Reeb dynamics, highlighting barcode entropy as a robust detector of dynamical complexity in contact-type settings.

Abstract

In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows.
Paper Structure (19 sections, 13 theorems, 112 equations)

This paper contains 19 sections, 13 theorems, 112 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main definitions and results
  4. Preliminaries, conventions, and notation
  5. Persistence modules
  6. Semi-admissible Hamiltonians, periodic orbits and the action functional
  7. Filtered Floer and symplectic homology
  8. Floer equation
  9. Floer homology and continuation maps
  10. Symplectic homology
  11. Barcode entropy revisited
  12. Barcode entropy of a persistence module
  13. Barcode entropy via Floer homology
  14. Crossing energy
  15. Crossing Energy Theorem
  16. ...and 4 more sections

Key Result

Theorem 1

For any Liouville domain $(W,d\alpha)$, we have

Theorems & Definitions (33)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1: FLS
  • Theorem 2
  • Corollary 3
  • Remark 1.3: Other types of barcode entropy
  • Remark 1.4: Barcode entropy for geodesic flows
  • Example 2.1: Homology of sublevels
  • Remark 2.2: Locally finite approximations
  • Remark 3.1
  • ...and 23 more