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Wrapped Floer homology and hyperbolic sets

Rafael A. Fernandes

TL;DR

This work establishes a direct link between a Floer-theoretic invariant, the wrapped Floer homology barcode entropy, and deterministic chaos measures of the Reeb flow on the boundary of a Liouville domain. By defining and leveraging a relative framework, the authors prove that, in the presence of a locally maximal hyperbolic set $K$ for the Reeb flow, the barcode entropy $\hbar^{HW}$ satisfies $\hbar^{HW}(M,L_0 \rightarrow L_1) \ge h_{top}(K)$. The core methodology combines the filtered wrapped Floer homology construction with a pair of dynamical-analytic results, the Crossing Energy Theorem (with boundary conditions) and the Location Constraint, to translate low-energy Floer data into exponential growth dictated by hyperbolic dynamics. This demonstrates that barcode entropy captures local dynamical complexity independent of global topology or filling, linking Floer-theoretic invariants to classical dynamical entropy and providing a robust tool for probing chaotic Reeb dynamics through symplectic topology.

Abstract

In this paper, we continue the quest to understand the interplay between wrapped Floer homology barcode and topological entropy. Wrapped Floer homology barcode entropy is defined as the exponential growth, with respect to the left endpoints, of the number of not-too-short bars in its barcode. We prove that, in the presence of a topologically transitive, locally maximal hyperbolic set for the Reeb flow on the boundary of a Liouville domain, the barcode entropy is bounded from below by the topological entropy restricted to the hyperbolic set.

Wrapped Floer homology and hyperbolic sets

TL;DR

This work establishes a direct link between a Floer-theoretic invariant, the wrapped Floer homology barcode entropy, and deterministic chaos measures of the Reeb flow on the boundary of a Liouville domain. By defining and leveraging a relative framework, the authors prove that, in the presence of a locally maximal hyperbolic set for the Reeb flow, the barcode entropy satisfies . The core methodology combines the filtered wrapped Floer homology construction with a pair of dynamical-analytic results, the Crossing Energy Theorem (with boundary conditions) and the Location Constraint, to translate low-energy Floer data into exponential growth dictated by hyperbolic dynamics. This demonstrates that barcode entropy captures local dynamical complexity independent of global topology or filling, linking Floer-theoretic invariants to classical dynamical entropy and providing a robust tool for probing chaotic Reeb dynamics through symplectic topology.

Abstract

In this paper, we continue the quest to understand the interplay between wrapped Floer homology barcode and topological entropy. Wrapped Floer homology barcode entropy is defined as the exponential growth, with respect to the left endpoints, of the number of not-too-short bars in its barcode. We prove that, in the presence of a topologically transitive, locally maximal hyperbolic set for the Reeb flow on the boundary of a Liouville domain, the barcode entropy is bounded from below by the topological entropy restricted to the hyperbolic set.
Paper Structure (13 sections, 8 theorems, 95 equations)

This paper contains 13 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Wrapped Floer homology
  5. Exact conical Lagrangians
  6. Hamiltonians and action filtration
  7. Floer homology and continuation maps
  8. Filtered wrapped Floer homology
  9. Wrapped Floer homology barcode entropy
  10. Proof of Theorem B
  11. Exponential growth of chords and topological entropy
  12. Crossing Energy and proof of Theorem B
  13. Location Constraint and proof of Crossing Energy Theorem

Key Result

Lemma 1

For a convex semi-admissible Hamiltonian $H$, with $H(r,x)= h(r)$ in $[1,r_{max}(H))\times \Sigma$ and $H(r,x) = rT - B$, in $r \geq r_{max}$, the reparametrization function $\widetilde{A_H} =\widetilde{A_h}$ that transforms lengths in actions given by $\widetilde{A_h}(t)=rh^{'}(r) - h(r)$, where $h

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1: Corolary 1, fernandes2024barcode
  • Definition 2
  • Definition 3
  • Proposition 2: Proposition 4, fernandes2024barcode
  • Proposition 3
  • proof
  • ...and 14 more