Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T20:37:19.335Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Topological dynamics Connections with other structures, applications

Published online by Cambridge University Press:  13 November 2020

NIKOLAI EDEKO
NIKOLAI EDEKO*
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076Tübingen, Germany
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I. and Avez, A.. Ergodic Problems of Classical Mechanics. Benjamin, New York, 1968.Google Scholar
Auslander, J.. Minimal Flows and Their Extensions. North Holland, Amsterdam, 1988.Google Scholar
Blokh, A., Oversteegen, L. and Tymchatyn, E.. On minimal maps of 2-manifolds. Ergod. Th. & Dynam. Sys. 25(1) (2005), 41–57.CrossRefGoogle Scholar
Bronstein, I.. Extensions of Minimal Transformation Groups. Springer, Netherlands, 1979.CrossRefGoogle Scholar
Calcut, J., Gompf, R. and McCarthy, J.. On fundamental groups of quotient spaces. Topology Appl. 159(1) (2012), 322–330.CrossRefGoogle Scholar
Downarowicz, T.. Survey of odometers and Toeplitz flows. Contemp. Math. 385 (2005), 7–38.CrossRefGoogle Scholar
Eisner, T., Farkas, B., Haase, M. and Nagel, R.. Operator Theoretic Aspects of Ergodic Theory ( Graduate Texts in Mathematics , 272). Springer, Berlin, 2015.CrossRefGoogle Scholar
Glasner, E., Gutman, Y. and Ye, X.. Higher order regionally proximal equivalence relations for general minimal group actions. Adv. Math. 333 (2018), 1004–1041.CrossRefGoogle Scholar
Greschonig, G.. Real extensions of distal minimal flows and continuous topological ergodic decompositions. Proc. Lond. Math. Soc. 3(109) (2014), 213–240.CrossRefGoogle Scholar
Hauser, T. and Jäger, T.. Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc. 147 (2019), 4539–4554.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nilpotent Structures in Ergodic Theory ( Mathematical Surveys and Monographs , 236). American Mathematical Society, Providence, RI, 2018.CrossRefGoogle Scholar
Hofmann, K. and Morris, S.. The Structure of Compact Groups. De Gruyter, Berlin, 2013.CrossRefGoogle Scholar
Hurewicz, W.. Über oberhalb-stetige Zerlegungen von Punktmengen in Kontinua. Fund. Math. 15 (1930), 57–60.CrossRefGoogle Scholar
Ihrig, E. and McMahon, D.. On distal flows of finite codimension. Indiana Univ. Math. J. 33(3) (1984), 345–351.CrossRefGoogle Scholar
James, I. M.. Topologies and Uniformities. Springer, London, 1999.CrossRefGoogle Scholar
Kellendonk, J., Lenz, D. and Savinien, J., Eds. Mathematics of Aperiodic Order. Birkhäuser, Basel, 2015.CrossRefGoogle Scholar
Keynes, H. and Robertson, J.. Eigenvalue theorems in topological transformation groups. Trans. Amer. Math. Soc. 139 (1969), 359–369.CrossRefGoogle Scholar
Kolyada, S., Snoha, L. and Trofimchuk, S.. Noninvertible minimal maps. Fund. Math. 168(2) (2001), 141–163.CrossRefGoogle Scholar
Lee, J.. Introduction to Smooth Manifolds. Springer, New York, 2012.CrossRefGoogle Scholar
Lind, D. A.. Spectral invariants and smooth ergodic theory. Dynamical Systems, Theory and Applications. Springer, Berlin, 1975, pp. 296–308.CrossRefGoogle Scholar
McMahon, D. and Wu, T. S.. On the connectedness of homomorphisms in topological dynamics. Trans. Amer. Math. Soc. 217 (1976), 257–270.CrossRefGoogle Scholar
Rees, M.. On the structure of minimal distal transformation groups with topological manifolds as phase spaces. PhD Thesis, University of Warwick, 1977.Google Scholar
Sepanski, M. R.. Compact Lie Groups. Springer, New York, 2007.CrossRefGoogle Scholar
Willard, S.. General Topology. Dover, Mineola, 2004.Google Scholar
Zemke, C.. Dimension and decompositions. Fund. Math. 95(3) (1977), 157–165.CrossRefGoogle Scholar