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Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

Pazit Haim-Kislev, Ofir Karin

TL;DR

The paper addresses the challenge of computing invariants for Hamiltonian diffeomorphisms by formulating a generating function barcode (GF-barcode) built from Morse-theoretic sublevel sets of a generating function quadratic at infinity. It develops a finite-time, composition-friendly algorithm that converts a product of small diffeomorphisms into a GFQI, samples and filters a compact domain, and extracts a barcode with a provable bottleneck-distance bound to the true GF-barcode. The main technical contributions include a composition formula for GF-generating functions, a Viterbo-style method to obtain a GFQI, a reduction to compact-domain GF-homology, and a complete pipeline (cell complex construction, filtration, and matrix reduction) to compute the GF-barcode with quantified error. The implementation for 2D radial Hamiltonians demonstrates practical applicability, yielding barcodes that closely approximate conjectured structures and validating the approach for concrete dynamical systems.

Abstract

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of $ \mathbb{R}^{2n} $ by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it.

Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

TL;DR

The paper addresses the challenge of computing invariants for Hamiltonian diffeomorphisms by formulating a generating function barcode (GF-barcode) built from Morse-theoretic sublevel sets of a generating function quadratic at infinity. It develops a finite-time, composition-friendly algorithm that converts a product of small diffeomorphisms into a GFQI, samples and filters a compact domain, and extracts a barcode with a provable bottleneck-distance bound to the true GF-barcode. The main technical contributions include a composition formula for GF-generating functions, a Viterbo-style method to obtain a GFQI, a reduction to compact-domain GF-homology, and a complete pipeline (cell complex construction, filtration, and matrix reduction) to compute the GF-barcode with quantified error. The implementation for 2D radial Hamiltonians demonstrates practical applicability, yielding barcodes that closely approximate conjectured structures and validating the approach for concrete dynamical systems.

Abstract

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it.
Paper Structure (19 sections, 18 theorems, 132 equations, 8 figures, 1 table)

This paper contains 19 sections, 18 theorems, 132 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. From $\mathop{\mathrm{Ham}}\nolimits_c\left(\mathbb{R}^{2n},\omega_0\right)$ to exact Lagrangian submanifolds of $T^*\mathbb{S}^{2n}$
  4. Generating functions of Lagrangian submanifolds
  5. Generating function homology
  6. Generating function barcode
  7. The approximation algrorithm
  8. Composition formula
  9. Obtaining a GFQI
  10. GF homology on a compact domain
  11. Step 1 - Constructing a cell complex
  12. Step 2 - Filtering the complex
  13. Step 3 - Extracting the barcode
  14. Complexity analysis
  15. Implementation for 2-dimensional radial functions
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\varphi_1,\ldots,\varphi_N$ be compactly supported $C^1$ small Hamiltonian diffeomorphisms, and let $\varepsilon >0$. Then there exists a finite time algorithm that gets as input the values of the generating functions of $\varphi_1,\ldots,\varphi_N$ applied to an appropriate finite sample, and

Figures (8)

  • Figure 1: Theoretical construction of GF-barcode (left) and the approximation algorithm (right)
  • Figure 2: The profile function $h$ and its derivative $h^\prime$
  • Figure 3: The dynamics of $\psi^t\sharp\varphi^t$ including the periodic orbits for case $\textrm{II}$.
  • Figure : Case I: 1 non-trivial orbit
  • Figure : Case I: 1 non-trivial orbit
  • ...and 3 more figures

Theorems & Definitions (76)

  • Theorem 1.1
  • Lemma 2.1
  • Remark
  • Remark
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Remark
  • Theorem 2.5: Viterbo's Uniqueness Theorem
  • ...and 66 more