Centralizer of fixed point free separating flows

In this paper, we study the centralizer of a separating continuous flow without fixed points. We show that if $M$ is a compact metric space and $\phi_t:M\to M$ is a separating flow without fixed points, then $\phi_t$ has a quasi-trivial centralizer, that is, if a continuous flow $\psi_t$ commutes with $\phi_t$, then there exists a continuous function $A: M\to\mathbb{R}$ which is invariant along the orbit of $\phi_t$ such that $\psi_t(x)=\phi_{A(x)t}(x)$ holds for all $x\in M$. We also show that if $M$ is a compact Riemannian manifold without boundary and $\Phi_u$ is a separating $C^1$ $\mathbb{R}^d$-action on $M$, then $\Phi_u$ has a quasi-trivial centralizer, that is, if $\Psi_u$ is a $\mathbb{R}^d$-action on $M$ commuting with $\Phi_u$, then there is a continuous map $A: M\to\mathcal{M}_{d\times d}(\mathbb{R})$ which is invariant along orbit of $\Phi_u$ such that $\Psi_{u}(x)=\Phi_{A(x)u}(x)$ for all $x\in M$. These improve Theorem 1 of \cite{O} and Theorem 2 of \cite{BRV} respectively.