[Paper Review] On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis

Sylvain Crovisier, Dawei Yang|arXiv (Cornell University)|Apr 21, 2014

Mathematical Dynamics and Fractals13 references3 citations

TL;DR

This paper proves a conjecture by Palis stating that in three-dimensional $C^1$ vector fields, singular hyperbolicity or homoclinic tangencies are dense in the $C^1$ topology. The authors extend Mañé and Pujals-Sambarino's theorems on uniform contraction of one-dimensional dominated bundles to local fibered flows, using compactified rescaled Poincaré flows and identification maps to handle time-continuous dynamics and non-symmetric bundle structures, ultimately establishing $C^1$-density of singular hyperbolic vector fields.

ABSTRACT

In this note we announce a result for vector fields on three-dimensional manifolds: those who are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of $C^1$-vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibered flows of Mañé and Pujals-Sambarino's theorems about the uniform contraction of one-dimensional dominated bundles. Sur la densité de l'hyperbolicité singulière pour les champs de vecteurs en dimension trois : une conjecture de Palis Dans cette note, nous annonçons un résultat portant sur les champs de vecteurs des variétés de dimension $3$ : ceux qui vérifient l'hyperbolicité singulière ou qui possèdent une tangence homocline forment un sous-ensemble dense de l'espace des champs de vecteurs $C^1$. Ceci répond à une conjecture de Palis. La démonstration utilise une généralisation pour les flots fibrés locaux des théorèmes de Mañé et Pujals-Sambarino traitant de la contraction uniforme de fibrés unidimensionnels dominés.

Motivation & Objective

• To resolve Palis' conjecture on the $C^1$-density of singular hyperbolic vector fields or those with homoclinic tangencies in three-dimensional manifolds.
• To extend the Pujals-Sambarino theorem on uniform contraction of one-dimensional dominated bundles to continuous-time dynamics via local fibered flows.
• To overcome challenges in continuous-time systems, such as shear along orbits and lack of global symmetry, by constructing compatible identification maps and Markovian boxes.
• To establish a $C^1$-dense dichotomy in 3D vector fields: either singular hyperbolicity or homoclinic tangency, closing a long-standing conjecture in dynamical systems.

Proposed method

• The authors construct a compactified rescaled sectional Poincaré flow $\widehat{P}^*_t$ on a bundle $\widehat{\mathcal{N}}$ over a compactified space $\widehat{K}$, which preserves the 0-section and allows for $C^{r-1}$-regularity along fibers.
• They introduce a compatibility condition via local identification maps $\pi_{x,y}$ between nearby fibers, ensuring consistency of the flow across the base space $\widehat{K}$.
• A dominated splitting $T\widehat{\mathcal{N}} = \widehat{E} \oplus \widehat{F}$ is assumed for the 0-section, with $\widehat{E}$ uniformly contracted along periodic orbits and base dynamics avoiding conjugacy to irrational rotations.
• The proof uses induced hyperbolic returns and local Markovian boxes to analyze long-term behavior despite the local nature of the flow and non-symmetric bundle roles.
• The key technical innovation is extending the Pujals-Sambarino contraction argument to continuous-time, fibered dynamics by leveraging the $C^2$-regularity along fibers and the identification structure.
• The argument relies on a recent result by Crovisier, Pujals, and Sambarino to control the dynamics in non-uniformly hyperbolic regions.

Experimental results

Research questions

• RQ1 Can singular hyperbolic vector fields be $C^1$-dense in the space of all $C^1$ vector fields on a 3-manifold, as conjectured by Palis?
• RQ2 Does the Pujals-Sambarino theorem on uniform contraction of one-dimensional dominated bundles extend to continuous-time, local fibered flows with non-trivial bundle geometry?
• RQ3 How can the shear effect of flows along orbits be controlled to ensure uniform contraction in time-continuous settings?
• RQ4 What structural conditions on the base dynamics prevent the existence of non-contracting invariant sets in the compactified flow?
• RQ5 Can a $C^1$-dense dichotomy between singular hyperbolicity and homoclinic tangency be established in 3D vector fields?

Key findings

• The space of $C^1$ vector fields on a 3-dimensional compact manifold is $C^1$-dense in the union of singular hyperbolic vector fields and those with a homoclinic tangency.
• The uniform contraction of a one-dimensional dominated bundle $\widehat{E}$ in the compactified rescaled Poincaré flow $\widehat{P}^*_t$ implies a dominated splitting $T_K M = E \oplus F$ for the tangent flow $D\varphi_t$ with $\dim(E) = 1$ and $\mathbb{R}.X \subset F$.
• The existence of compatible identification maps $\pi_{x,y}$ between nearby fibers ensures consistency of the flow structure across the base space, enabling local analysis.
• The absence of base dynamics conjugate to irrational rotations prevents the formation of non-contracting repelling sets, which is essential for the uniform contraction result.
• The proof establishes that if $\widehat{E}$ is contracted along periodic orbits and the compatibility and regularity conditions hold, then $\widehat{E}$ is uniformly contracted by the linearized flow $\widehat{P}^*_t$.
• The result confirms that singular hyperbolicity is $C^1$-dense in 3D, resolving a key conjecture in the $C^1$-dynamics of vector fields.