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Barcode entropy for Reeb flows on contact manifolds with Liouville fillings

Elijah Fender, Sangjin Lee, Beomjun Sohn

TL;DR

We define the SH-barcode entropy by applying persistence-modules to filtered symplectic homology of Liouville fillings and extract a barcode, whose exponential growth rate of long bars yields the entropy. We prove that this entropy is an invariant of the boundary contact manifold, independent of the filling, and that it provides a general lower bound for the topological entropy of the Reeb flow. The work develops two compatible notions of barcode entropy, proves their equivalence, and relates SH-barcodes to RFH-barcodes, using exact triangles and a Lagrangian-tomograph Crofton framework. A central result shows that the SH-barcode entropy is bounded above by the topological entropy of the Reeb flow, linking symplectic-growth phenomena to dynamical complexity with potential implications for entropy detection in contact dynamics.

Abstract

We study the topological entropy of Reeb flows on contact manifolds with Liouville fillings. With the theory of persistence modules, we define SH-barcode entropy from the symplectic homology of a filling. We prove that the SH-barcode entropy is independent of the choice of the filling and that the barcode entropy provides a lower bound for the topological entropy of the Reeb flow.

Barcode entropy for Reeb flows on contact manifolds with Liouville fillings

TL;DR

We define the SH-barcode entropy by applying persistence-modules to filtered symplectic homology of Liouville fillings and extract a barcode, whose exponential growth rate of long bars yields the entropy. We prove that this entropy is an invariant of the boundary contact manifold, independent of the filling, and that it provides a general lower bound for the topological entropy of the Reeb flow. The work develops two compatible notions of barcode entropy, proves their equivalence, and relates SH-barcodes to RFH-barcodes, using exact triangles and a Lagrangian-tomograph Crofton framework. A central result shows that the SH-barcode entropy is bounded above by the topological entropy of the Reeb flow, linking symplectic-growth phenomena to dynamical complexity with potential implications for entropy detection in contact dynamics.

Abstract

We study the topological entropy of Reeb flows on contact manifolds with Liouville fillings. With the theory of persistence modules, we define SH-barcode entropy from the symplectic homology of a filling. We prove that the SH-barcode entropy is independent of the choice of the filling and that the barcode entropy provides a lower bound for the topological entropy of the Reeb flow.
Paper Structure (27 sections, 33 theorems, 164 equations)

This paper contains 27 sections, 33 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Further directions
  4. Acknowledgment
  5. Preliminary: Persistence module
  6. Non-Archimedean normed vector space
  7. Persistence module
  8. Interleaving distance
  9. Barcodes and the bottleneck distance
  10. Two barcode entropy
  11. Setting and notations
  12. Hamiltonian Floer homology and the corresponding persistence module
  13. A barcode entropy from the symplectic homology
  14. Hamiltonian Floer homologies of degenerate Hamiltonian functions
  15. Barcode entropy of a convex radial Hamiltonian
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $(Y,\alpha)$ be a closed non-degenerate contact manifold with Liouville fillings $(W_1, \lambda_1)$ and $(W_2, \lambda_2)$ whose first Chern classes vanish. Then, the $\operatorname{SH}$-barcode entropy of $(Y,\alpha)$ defined by the symplectic homology of $(W_1, \lambda_1)$ and that of $(W_2, \

Theorems & Definitions (105)

  • Theorem 1.1: = Theorem \ref{['invariance: main thm']}
  • Theorem 1.2: =Theorem \ref{['thm vs topological entropy formal']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4: Theorem 3.4 of Usher-Zhang
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 95 more