Centralizer of fixed point free separating flows

Abstract

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 $\ddot{I}{•}_t:M\to M$ is a separating flow without fixed points, then $\ddot{I}{•}_t$ has a quasi-trivial centralizer, that is, if a continuous flow $\ddot{I}{ˆ}_t$ commutes with $\ddot{I}{•}_t$, then there exists a continuous function $A:M\to \mathbb{R}$ which is invariant along the orbit of $\ddot{I}{•}_t$ such that $\ddot{I}{ˆ}_t(x)=\ddot{I}{•}_{A(x)t}(x)$ holds for all $x\in M$. We also show that if $M$ is a compact Riemannian manifold without boundary and $\hat{I}{¦}_u$ is a separating $C^1$ ${\mathbb{R}}^d$-action on $M$, then $\hat{I}{¦}_u$ has a quasi-trivial centralizer, that is, if $\hat{I}{¨}_u$ is a ${\mathbb{R}}^d$-action on $M$ commuting with $\hat{I}{¦}_u$, then there is a continuous map $A:M\to {\mathcal{M}}_{d\times d}(\mathbb{R})$ which is invariant along orbit of $\hat{I}{¦}_u$ such that $\hat{I}{¨}_u(x)=\hat{I}{¦}_{A(x)u}(x)$ for all $x\in M$. These improve Theorem 1 of \cite{O} and Theorem 2 of \cite{BRV} respectively.