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On the barcode entropy of Lagrangian submanifolds

Matthias Meiwes

TL;DR

The article develops and analyzes relative barcode entropy for Hamiltonian diffeomorphisms relative to pairs of Lagrangian submanifolds, connecting Floer-theoretic barcode growth to dynamical entropy. It establishes lower bounds linking $\hbar(\psi;L_0,L_1)$ to the topological entropy of invariant hyperbolic sets, with a sharp 2D analogue: in dimension two the entropy bounds extend to closed curves in the complement of horseshoe orbits. It also proves stability results under Hofer and $\gamma$-distances, and introduces a strong version $\mathsf{H}^R$, for which positive values are exhibited in twisted product examples, indicating robust entropy features. Collectively, these results illuminate how symplectic topology (via Lagrangian Floer data) encodes and stabilizes dynamical complexity, with applications to volume growth and topological entropy robustness in Hamiltonian systems.

Abstract

This article deals with relative barcode entropy, a notion that was recently introduced by Cineli, Ginzburg, and Gurel. We exhibit some settings in closed symplectic manifolds for which the relative barcode entropy of a Hamiltonian diffeomorphism and a pair of Lagrangian submanifolds is positive. In analogy to a result in the absolute case by the above authors, we obtain that the topological entropy of any horseshoe K is a lower bound if the two Lagrangians contain a local unstable resp. stable manifold in K. In dimension 2, we also estimate the relative barcode entropy of a pair of closed curves that lie in special homotopy classes in the complement of certain periodic orbits in K. Furthermore, we define a variant of relative barcode entropy and exhibit first examples for which it is positive. As applications, certain robustness features of the volume growth and the topological entropy are discussed.

On the barcode entropy of Lagrangian submanifolds

TL;DR

The article develops and analyzes relative barcode entropy for Hamiltonian diffeomorphisms relative to pairs of Lagrangian submanifolds, connecting Floer-theoretic barcode growth to dynamical entropy. It establishes lower bounds linking to the topological entropy of invariant hyperbolic sets, with a sharp 2D analogue: in dimension two the entropy bounds extend to closed curves in the complement of horseshoe orbits. It also proves stability results under Hofer and -distances, and introduces a strong version , for which positive values are exhibited in twisted product examples, indicating robust entropy features. Collectively, these results illuminate how symplectic topology (via Lagrangian Floer data) encodes and stabilizes dynamical complexity, with applications to volume growth and topological entropy robustness in Hamiltonian systems.

Abstract

This article deals with relative barcode entropy, a notion that was recently introduced by Cineli, Ginzburg, and Gurel. We exhibit some settings in closed symplectic manifolds for which the relative barcode entropy of a Hamiltonian diffeomorphism and a pair of Lagrangian submanifolds is positive. In analogy to a result in the absolute case by the above authors, we obtain that the topological entropy of any horseshoe K is a lower bound if the two Lagrangians contain a local unstable resp. stable manifold in K. In dimension 2, we also estimate the relative barcode entropy of a pair of closed curves that lie in special homotopy classes in the complement of certain periodic orbits in K. Furthermore, we define a variant of relative barcode entropy and exhibit first examples for which it is positive. As applications, certain robustness features of the volume growth and the topological entropy are discussed.
Paper Structure (18 sections, 8 theorems, 85 equations)

This paper contains 18 sections, 8 theorems, 85 equations.

Table of Contents

  1. Introduction and main results
  2. Barcode entropy for Lagrangian submanifolds that contain local stable and unstable disks
  3. Barcode entropy of closed curves in the complement of some horseshoe orbits
  4. Stability with respect to the Hofer- and the $\gamma$-distance
  5. Strong barcode entropy and some further examples
  6. Preliminaries on relative barcode entropy
  7. Hamiltonian diffeomorphisms and mapping cylinder
  8. The chain complex
  9. Rescaling and lifts of holomorphic strips
  10. Relative barcode entropy
  11. Pseudo-orbits property for small energies and the proof of Theorem \ref{['mainthm:higherdim']}
  12. Holomorphic curves with small energy and pseudo-orbits
  13. Local stable/unstable manifolds and growth of chords
  14. Proof of Theorem \ref{['thm:barcode_lagr']}: Links in Horseshoes and crossing energy
  15. Restricting the Floer chain complex to link complements.
  16. ...and 3 more sections

Key Result

Proposition 1.5

Let $\psi \in \mathrm{Ham}(M,\omega)$, and $(L_0,L_1)$ be an admissible pair. Then the following holds:

Theorems & Definitions (23)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5
  • proof : Proof of Proposition \ref{['prop:strong_prop']}
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • ...and 13 more