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On equicontinuous factors of flows on locally path-connected compact spaces

Part of: Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  13 November 2020

NIKOLAI EDEKO
NIKOLAI EDEKO*
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
*
e-mail: nied@math.uni-tuebingen.de
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Abstract

We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \textrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$. For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc. 147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$.

Keywords

locally connectedmontoneequicontinuous

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3270 - 3287
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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