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 ${\varphi }_t:M\to M$ is a separating flow without fixed points, then ${\varphi }_t$ has a quasi-trivial centralizer, that is, if a continuous flow ${\psi }_t$ commutes with ${\varphi }_t$, then there exists a continuous function $A:M\to \mathbb{R}$ which is invariant along the orbit of ${\varphi }_t$ such that ${\psi }_t(x)={\varphi }_{A(x)t}(x)$ holds for all $x\in M$. We also show that if M is a compact Riemannian manifold without boundary and ${\mathrm{\Phi }}_u$ is a homogenous separating $C^1$ ${\mathbb{R}}^m$-action on M, then ${\mathrm{\Phi }}_u$ has a quasi-trivial centralizer, that is, if ${\mathrm{\Psi }}_u$ is a ${\mathbb{R}}^m$-action on M commuting with ${\mathrm{\Phi }}_u$, then there is a continuous map $A:M\to {\mathcal{M}}_{m\times m}(\mathbb{R})$ which is invariant along orbit of ${\mathrm{\Phi }}_u$ such that ${\mathrm{\Psi }}_u(x)={\mathrm{\Phi }}_{A(x)u}(x)$ for all $x\in M$. These improve Theorem 1 of [M. Oka, Expansive flows and their centralizers, Nagoya Math. J. 64 (1976), pp. 1–15.] and Theorem 2 of [W. Bonomo, J. Rocha, and P. Varandas, The centralizer of Komuro-expansive flows and expansive ${\mathbb{R}}^d$-actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively.

Keywords:

• Quasi-trivial centralizer
• separating flow
• fixed point free flow
• homogenous ${\mathbb{R}}^d$-action

Mathematical subject classifications:

• 37B05
• 37C10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Key R&D Program of China [2022YFA1005801]; National Natural Science Foundation of China [12071018] and Fundamental Reseach Funds for the Central Universities.