[Paper Review] On the Closing Lemma problem for vector fields of bounded type on the torus

S. P. Lloyd|Nov 7, 2008

Advanced Differential Equations and Dynamical Systems10 references3 citations

TL;DR

This paper establishes the $C^r$ Closing Lemma for $C^r$ vector fields of bounded type on the 2-torus with zero divergence at saddle points, using a twist perturbation method. It proves that any non-trivially recurrent point becomes periodic under arbitrarily small $C^r$ perturbations, relying on a new non-existence result for semi-wandering intervals in order-preserving circle maps with bounded type rotation number.

ABSTRACT

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$. The proof relies on a new result in 1-dimensional dynamics on the non-existence of semi-wandering intervals of smooth maps of the circle.

Motivation & Objective

• To resolve the $C^r$ Closing Lemma problem for bounded type vector fields on the 2-torus, a case previously considered obstructed by negative results.
• To extend the $C^r$ Closing Lemma to vector fields with zero divergence at saddle points and bounded type rotation number, completing a complementary case to prior work on unbounded type.
• To establish conditions under which twist perturbations can close recurrent orbits, particularly in the presence of co-directed black cells and no grey cells.
• To prove the non-existence of forward or backward semi-wandering intervals in order-preserving circle maps with bounded type rotation number and finite variation of log-derivative.
• To provide a structural characterization of vector fields on the torus with bounded type rotation number, showing absence of grey cells and co-directed black cells under zero divergence at saddles.

Proposed method

• Analyzes the induced return map on a transverse loop $\Sigma$ to the vector field $X$, extending it to a continuous map when black cells are co-directed.
• Applies a new result on order-preserving circle maps: if the rotation number is of bounded type and the log-derivative has finite variation, then no forward or backward semi-wandering intervals exist.
• Uses a 1-parameter family of $C^r$ twist perturbations along a transverse loop $\Sigma$, ensuring the perturbed vector field remains $C^r$-close to $X$.
• Employs the Intermediate Value Theorem on iterates of the perturbed return map to locate periodic points arbitrarily close to the recurrent point $p$.
• Constructs a loop $\Sigma$ disjoint from repellor basins so that the return map extends continuously, enabling application of perturbation theory.
• Relies on $C^r$ linearizing coordinates at saddle points (assumed for $r \geq 4$) to ensure smoothness of the perturbation and return map.

Experimental results

Research questions

• RQ1 Can the $C^r$ Closing Lemma be established for bounded type vector fields on the 2-torus, given that previous results failed for this class?
• RQ2 Do twist perturbations suffice to close recurrent orbits in bounded type vector fields with zero divergence at saddle points?
• RQ3 Can the non-existence of semi-wandering intervals be proven for order-preserving circle maps with bounded type rotation number and finite variation of log-derivative?
• RQ4 What structural properties do vector fields on the torus with bounded type rotation number and zero divergence at saddles exhibit?
• RQ5 Under what conditions on the return map does a $C^r$ twist perturbation yield a periodic orbit arbitrarily close to a recurrent point?

Key findings

• The $C^r$ Closing Lemma holds for $C^r$ vector fields on the 2-torus with bounded type rotation number, finitely many hyperbolic singularities, and zero divergence at each saddle, for $r \geq 4$.
• For such vector fields, any non-trivially recurrent point $p$ becomes periodic under a $C^r$ twist perturbation arbitrarily close to the original vector field.
• The induced return map on a suitable transverse loop $\Sigma$ extends to a continuous map when black cells are co-directed, enabling the use of perturbation techniques.
• A new non-existence result is proven: order-preserving circle maps with bounded type rotation number and finite variation of log-derivative possess neither forward nor backward semi-wandering intervals.
• The absence of grey cells and the co-directedness of black cells are established as structural properties of bounded type vector fields with zero divergence at saddles.
• The result remains valid for $C^2$ vector fields if $C^2$ linearizing coordinates exist at saddle points, extending the applicability beyond $C^4$.