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On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

Robert Cardona, Julian Chaidez, Francisco Torres de Lizaur

TL;DR

This work analyzes the invariants of coadjoint orbits for 3D volume-preserving diffeomorphisms, focusing on helicity as a central but not exclusive invariant. It constructs a framework showing that helicity’s uniqueness can fail ubiquitously in $C^{1}$ on integral homology spheres: either the Ruelle invariant or the topological entropy must vary within fixed-helicity sets, depending on whether non-Anosov dynamics are present or all flows are Anosov, respectively. On the 3-sphere, the Ruelle invariant is actually independent of helicity, yielding a strong negative answer to Arnold–Khesin’s question about somewhere density of coadjoint orbits in helicity level sets; the authors extend a similar negative conclusion to arbitrary 3-manifolds using local invariants. The paper develops a toolkit of fixed-helicity perturbations (including a Franks-type lemma for orbit pieces), toric flow tubes for explicit invariant computations, and local invariants based on zeroes and short periodic orbits to establish nowhere-dense coadjoint orbits and to illuminate the subtle interplay between helicity and dynamical invariants.

Abstract

The helicity, or asymptotic linking number, is a functional of exact volume-preserving vector fields on 3-manifolds, invariant under volume-preserving transformations. It is known to exhibit remarkable uniqueness properties: many invariant functionals reduce to functions of helicity. We examine how severely this uniqueness can fail. On integral homology spheres with the $C^{1}$-topology, the failure is extreme: for every $C^{1}$-open set of nonvanishing exact fields of fixed helicity, some other global dynamical invariant is continuous and non-constant in that set; the Ruelle invariant if some field is non-Anosov, and topological entropy otherwise. In particular, on the three-sphere, the Ruelle invariant is everywhere independent of helicity. This implies, in a very strong sense, a negative answer to a question of Arnold and Khesin on the somewhere density of coadjoint orbits of 3D volume-preserving diffeomorphisms, when considered for the $C^1$-topology. On arbitrary three-manifolds, we also answer the question in the negative using local rather than global invariants.

On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

TL;DR

This work analyzes the invariants of coadjoint orbits for 3D volume-preserving diffeomorphisms, focusing on helicity as a central but not exclusive invariant. It constructs a framework showing that helicity’s uniqueness can fail ubiquitously in on integral homology spheres: either the Ruelle invariant or the topological entropy must vary within fixed-helicity sets, depending on whether non-Anosov dynamics are present or all flows are Anosov, respectively. On the 3-sphere, the Ruelle invariant is actually independent of helicity, yielding a strong negative answer to Arnold–Khesin’s question about somewhere density of coadjoint orbits in helicity level sets; the authors extend a similar negative conclusion to arbitrary 3-manifolds using local invariants. The paper develops a toolkit of fixed-helicity perturbations (including a Franks-type lemma for orbit pieces), toric flow tubes for explicit invariant computations, and local invariants based on zeroes and short periodic orbits to establish nowhere-dense coadjoint orbits and to illuminate the subtle interplay between helicity and dynamical invariants.

Abstract

The helicity, or asymptotic linking number, is a functional of exact volume-preserving vector fields on 3-manifolds, invariant under volume-preserving transformations. It is known to exhibit remarkable uniqueness properties: many invariant functionals reduce to functions of helicity. We examine how severely this uniqueness can fail. On integral homology spheres with the -topology, the failure is extreme: for every -open set of nonvanishing exact fields of fixed helicity, some other global dynamical invariant is continuous and non-constant in that set; the Ruelle invariant if some field is non-Anosov, and topological entropy otherwise. In particular, on the three-sphere, the Ruelle invariant is everywhere independent of helicity. This implies, in a very strong sense, a negative answer to a question of Arnold and Khesin on the somewhere density of coadjoint orbits of 3D volume-preserving diffeomorphisms, when considered for the -topology. On arbitrary three-manifolds, we also answer the question in the negative using local rather than global invariants.
Paper Structure (31 sections, 39 theorems, 272 equations, 1 figure)

This paper contains 31 sections, 39 theorems, 272 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results.
  3. Background
  4. Perturbations with fixed helicity
  5. Helicity corrections
  6. Kupka-Smale flows
  7. The $C^1$-closing lemma
  8. Elliptic orbits and a Franks-type lemma
  9. A Franks' lemma for pieces of orbit.
  10. Sketch of Theorem \ref{['thm:fixedhelliptic']}.
  11. Ruelle invariant near non-Anosov vector fields
  12. The Ruelle invariant
  13. Helicity in manifolds with boundary
  14. Invariants of a toric flow tube
  15. Construction of toric flow tubes
  16. ...and 16 more sections

Key Result

Theorem 2

Let $M$ be an integral homology sphere. Fix any $h\in \mathbb{R}$ and any $C^1$-open set $\mathcal{U}$ in the set of non-vanishing exact fields with helicity $h$. According to two cases, the following holds:

Figures (1)

  • Figure :

Theorems & Definitions (92)

  • Theorem 2
  • Remark 3
  • Remark 4
  • Conjecture 5: Arnold-Khesin 1998
  • Theorem 6
  • Remark 7
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 82 more