On M-dynamics and Li-Yorke chaos of extensions of minimal dynamics

Let π:X→Y be an extension of minimal compact flows such that Rπ≠ΔX. A subflow of Rπ is called an M-flow if it is T.T. and contains a dense set of a.p. points. In this paper we mainly prove the following:(1)π is PI iff ΔX is the unique M-flow containing ΔX in Rπ.(2)If X is a compact metric space and...

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Bibliographic Details
Published in:Journal of Differential Equations 2023-06, Vol.359, p.152-182
Main Author: Dai, Xiongping
Format: Article
Language:English
Subjects:
Li-Yorke chaos
M-flow
Minimal flow
PI-extension
Syndetic distability
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Summary:Let π:X→Y be an extension of minimal compact flows such that Rπ≠ΔX. A subflow of Rπ is called an M-flow if it is T.T. and contains a dense set of a.p. points. In this paper we mainly prove the following:(1)π is PI iff ΔX is the unique M-flow containing ΔX in Rπ.(2)If X is a compact metric space and π is not PI, then there exists a canonical Li-Yorke chaotic M-flow in Rπ. In particular, an Ellis weak-mixing non-proximal metric extension is non-PI and so Li-Yorke chaotic. In addition, we show(3)a unbounded or non-minimal M-flow, not necessarily compact, is sensitive; and(4)every syndetically distal compact flow is pointwise Bohr a.p.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.02.033