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Anti-classification for flows on two-tori

Nataliya Goncharuk

TL;DR

This work shows that orbital topological classification of vector fields on the two-torus admits no complete Borel numerical invariant, for both smooth and real-analytic cases. It achieves this via a Borel-reduction framework: constructing a smooth family $v_{\rho,\phi}$ whose parameters encode a non-smooth circle equivalence $E_\phi$, so that $v_{\rho_1,\phi_1}\sim v_{\rho_2,\phi_2}$ forces $\phi_1=\phi_2$ and $\rho_1\equiv \rho_2 \pmod{\phi}$ up to a lattice shift. The paper then proves the sharp connection both ways: parameters related by $\rho_1=\rho_2+n\phi$ (with suitable exclusions) yield orbital equivalence, while otherwise they do not, thereby establishing nonclassifiability. A stronger, generic-codimension result shows these nonclassifiable vector fields arise in a codimension-7 submanifold and hence in generic 7-parameter families. The analytic extension confirms the robustness of the phenomenon in real-analytic settings, using an explicit Hamiltonian family with an explicit Denjoy-type reduction mechanism.

Abstract

We prove that the classification of real-analytic vector fields on the two-torus up to orbital topological equivalence does not admit a complete numerical invariant that is a Borel function. Moreover, smooth vector fields that are difficult to classify appear in generic smooth 7-parameter families. In dimension 2, this improves the recent result of Gorodetski and Foreman (arXiv:2206.09322) for non-classifiability of smooth diffeomorphisms up to continuous conjugacy.

Anti-classification for flows on two-tori

TL;DR

This work shows that orbital topological classification of vector fields on the two-torus admits no complete Borel numerical invariant, for both smooth and real-analytic cases. It achieves this via a Borel-reduction framework: constructing a smooth family whose parameters encode a non-smooth circle equivalence , so that forces and up to a lattice shift. The paper then proves the sharp connection both ways: parameters related by (with suitable exclusions) yield orbital equivalence, while otherwise they do not, thereby establishing nonclassifiability. A stronger, generic-codimension result shows these nonclassifiable vector fields arise in a codimension-7 submanifold and hence in generic 7-parameter families. The analytic extension confirms the robustness of the phenomenon in real-analytic settings, using an explicit Hamiltonian family with an explicit Denjoy-type reduction mechanism.

Abstract

We prove that the classification of real-analytic vector fields on the two-torus up to orbital topological equivalence does not admit a complete numerical invariant that is a Borel function. Moreover, smooth vector fields that are difficult to classify appear in generic smooth 7-parameter families. In dimension 2, this improves the recent result of Gorodetski and Foreman (arXiv:2206.09322) for non-classifiability of smooth diffeomorphisms up to continuous conjugacy.

Paper Structure

This paper contains 7 sections, 10 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Borel reduction
  3. Construction of the family $v_{\rho,\phi}$.
  4. Equivalent vector fields have $E_\phi$-equivalent parameters
  5. Vector fields with $E_\phi$-equivalent parameters are equivalent
  6. Genericity of nonclassifiable vector fields
  7. Analytic vector fields

Key Result

Theorem 2

Orbital topological equivalence in $\mathcal{V}^2(T^2)$ has no complete Borel numerical invariant: there is no Borel function $g \colon \mathcal{V}^2(T^2) \to Y$ with $Y$ a Polish space such that for all $v, w \in X$, we have $v \sim w$ if and only if $g(v) = g(w)$.

Figures (3)

  • Figure 1: Vector field $v_{\rho, \phi}$.
  • Figure 2: Domains $V_1$ (light-gray), $V_2$ (dark-gray) and $V_3$ (white) for vector fields $v_{\rho_1}$ (left) and $v_{\rho_2}$ (right) for $n=3$.
  • Figure 3: Level curves of $u_{\phi, b,c,d}$ (phase curves of the vector fields $v_{\phi, b,c,d}$) for $\phi=1/3$, $c=1, b=2, d=1.0016$. Black square has sides of length $2\pi$. Separatrices of the saddle $s_1$ are shown in orange thick; separatrices of $s_2, s_3$ are shown in blue thick. Zeros of $v_{\phi, b,c,d}$ are shown in red. Black dashed lines are level curves $u_{\phi, b,c,d}=1,u_{\phi, b,c,d}=5$.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • proof
  • Definition 9
  • ...and 14 more