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 ϕt:M→M is a separating flow without fixed points, then ϕt has a quasi-trivial centralizer, that is, if a continuous flow ψt commutes with ϕt, then there exists a continuous function A:M→R which is invariant along the orbit of ϕt such that ψt(x)=ϕA(x)t(x) holds for all x∈M. We also show that if M is a compact Riemannian manifold without boundary and Φu is a homogenous separating C1 Rm-action on M, then Φu has a quasi-trivial centralizer, that is, if Ψu is a Rm-action on M commuting with Φu, then there is a continuous map A:M→Mm×m(R) which is invariant along orbit of Φu such that Ψu(x)=ΦA(x)u(x) for all x∈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 Rd-actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively.