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Centralizer of fixed point free separating flows.
- Source :
- Dynamical Systems: An International Journal; Jun2024, Vol. 39 Issue 2, p282-301, 20p
- Publication Year :
- 2024
-
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 $ \phi _t:M\to M $ ϕ t : M → M is a separating flow without fixed points, then $ \phi _t $ ϕ t has a quasi-trivial centralizer, that is, if a continuous flow $ \psi _t $ ψ t commutes with $ \phi _t $ ϕ t , then there exists a continuous function $ A: M\to \mathbb {R} $ A : M → R which is invariant along the orbit of $ \phi _t $ ϕ t such that $ \psi _t(x)=\phi _{A(x)t}(x) $ ψ t (x) = ϕ A (x) t (x) holds for all $ x\in M $ x ∈ M. We also show that if M is a compact Riemannian manifold without boundary and $ \Phi _u $ Φ u is a homogenous separating $ C^1 $ C 1 $ \mathbb {R}^m $ R m -action on M, then $ \Phi _u $ Φ u has a quasi-trivial centralizer, that is, if $ \Psi _u $ Ψ u is a $ \mathbb {R}^{ m} $ R m -action on M commuting with $ \Phi _u $ Φ u , then there is a continuous map $ A: M\to \mathcal {M}_{m\times m}(\mathbb {R}) $ A : M → M m × m (R) which is invariant along orbit of $ \Phi _u $ Φ u such that $ \Psi _{u}(x)=\Phi _{A(x)u}(x) $ Ψ u (x) = Φ A (x) u (x) for all $ x\in M $ 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 $ \mathbb {R}^d $ R d -actions, Math. Z. 289(3–4) (2018), pp. 1059–1088.] respectively. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14689367
- Volume :
- 39
- Issue :
- 2
- Database :
- Complementary Index
- Journal :
- Dynamical Systems: An International Journal
- Publication Type :
- Academic Journal
- Accession number :
- 176897027
- Full Text :
- https://doi.org/10.1080/14689367.2023.2290045