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Lyapunov 1-forms on orbifolds

Fabricio Valencia

TL;DR

The paper develops a framework for Lyapunov 1-forms on compact orbifolds, modeled via proper étale groupoids, and proves equivalences between the existence of such forms in a prescribed cohomology class $\xi$ and dynamical/ topological data including asymptotic cycles and chain-recurrence. It extends the manifold theory of Lyapunov 1-forms to orbifolds by leveraging groupoid cohomology, transverse measures, and $G$-path integrals, and shows how negativity conditions on cycle pairings certify existence. The main theorem provides practical criteria: a Lyapunov 1-form in $\xi$ exists iff certain cycle and measure pairings are non-positive (or negative on non- Lyapunov regions), aligning the dynamical behavior outside the chain-recurrent set with the chosen cohomology class. The paper also supplies concrete orbifold examples, including the pillowcase, and a general construction demonstrating how to engineer the relevant invariants to realize Lyapunov 1-forms.

Abstract

We introduce and analyze a notion of smooth Lyapunov 1-form for flows generated by vector fields on orbifolds. Using asymptotic cycles and chain-recurrent sets, we establish topological conditions that guarantee the existence of a Lyapunov 1-form, lying in a prescribed cohomology class, for a given vector field on a compact orbifold.

Lyapunov 1-forms on orbifolds

TL;DR

The paper develops a framework for Lyapunov 1-forms on compact orbifolds, modeled via proper étale groupoids, and proves equivalences between the existence of such forms in a prescribed cohomology class and dynamical/ topological data including asymptotic cycles and chain-recurrence. It extends the manifold theory of Lyapunov 1-forms to orbifolds by leveraging groupoid cohomology, transverse measures, and -path integrals, and shows how negativity conditions on cycle pairings certify existence. The main theorem provides practical criteria: a Lyapunov 1-form in exists iff certain cycle and measure pairings are non-positive (or negative on non- Lyapunov regions), aligning the dynamical behavior outside the chain-recurrent set with the chosen cohomology class. The paper also supplies concrete orbifold examples, including the pillowcase, and a general construction demonstrating how to engineer the relevant invariants to realize Lyapunov 1-forms.

Abstract

We introduce and analyze a notion of smooth Lyapunov 1-form for flows generated by vector fields on orbifolds. Using asymptotic cycles and chain-recurrent sets, we establish topological conditions that guarantee the existence of a Lyapunov 1-form, lying in a prescribed cohomology class, for a given vector field on a compact orbifold.

Paper Structure

This paper contains 10 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Vector fields, differential forms, and measures
  3. Basic vector fields and differential forms
  4. Transverse measures
  5. $G$-path integrals
  6. Lyapunov basic 1-forms
  7. Asymptotic cycles
  8. Chain-recurrent sets
  9. Proof of Theorem \ref{['Main Thm']}
  10. Examples

Key Result

Theorem 1.2

Let $\overline{v}$ be a vector field on a compact orbifold $X$ and let $\overline{\Phi}:\mathbb{R}\times X\to X$ denote the flow generated by $\overline{v}$. Let $\xi\in H^1(X)$ be a cohomology class such that the restriction $\xi|_{R_\xi}$, viewed as a $\Check{C}$ech cohomology class $\xi|_{R_\xi}\

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 10 more