On the centralizer of vector fields: criteria of triviality and genericity results

TL;DR

This paper studies the size and structure of the centralizer of vector fields on compact manifolds, providing criteria for triviality and showing that generically, the centralizer is small or trivial, especially under hyperbolicity conditions.

Contribution

It introduces new criteria for small and trivial centralizers of vector fields, including generic and hyperbolic cases, and establishes equivalences between collinearity and triviality in higher regularity.

Findings

Generic vector fields have small, collinear centralizers.

Hyperbolic singularities lead to quasi-trivial centralizers.

In higher regularity, collinearity and triviality are equivalent.

Abstract

In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which guarantees that the centralizer of a $C^1$-generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a $C^1$ \textit{separating} flow has a collinear $C^1$-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the $C^1$-centralizer of a $C^1$-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a $C^1$-generic vector field, which includes $C^1$-generic Axiom A (or sectional Axiom A) vector fields and $C^1$-generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the $C^d$-centralizer are equivalent properties for a generic vector field in the $C^d$ topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity $C^2$, the $C^1$-centralizer is trivial.