On the centralizers of rescaling separating differentiable vector fields

Abstract

We introduce a new version of expansiveness similar to separating property for flows. Let $M$ be a compact Riemannian manifold without boundary and $X$ be a $C^1$ vector field on $M$ that generates a flow $\varphi_t$ on $M$. We call $X$ {\it rescaling separating} on a compact invariant set $\Lambda$ of $X$ if there is $\delta>0$ such that, for any $x,y\in \Lambda$, if $d(\varphi_t(x), \varphi_{t}(y))\le \delta\|X(\varphi_t(x))\|$ for all $t\in \mathbb R$, then $y\in{\rm Orb}(x)$. We prove that if $X$ is rescaling separating on $\Lambda$ and every singularity of $X$ in $\Lambda$ is hyperbolic, then for any $C^1$ vector field $Y$, if the flow generated by $Y$ is commuting with $\varphi_t$ on $\Lambda$, then $Y$ is collinear to $X$ on $\Lambda$. As applications of the result, we show that the centralizer of a rescaling separating $C^1$ vector field without nonhyperbolic singularity is quasi-trivial and there is is an open and dense set $\mathcal{U}\subset\mathcal{X}^1(M)$ such that for any star vector field $X\in\mathcal{U}$, the centralizer of $X$ is collinear to $X$ on the chain recurrent set of $X$.