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

Abstract

Let $\pi :\mathcal{X}\to \mathcal{Y}$ be an extension of minimal compact metric flows such that ${\text{R}}_{\pi }\ne {\Delta }_X$. A subflow of ${\text{R}}_{\pi }$ 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) $\pi$ is PI iff ${\Delta }_X$ is the unique M-flow containing ${\Delta }_X$ in ${\text{R}}_{\pi }$. (2) If $\pi$ is not PI, then there exists a canonical Li-Yorke chaotic M-flow in ${\text{R}}_{\pi }$. In particular, an Ellis weak-mixing non-proximal extension is non-PI and so Li-Yorke chaotic. (3) A unbounded or non-minimal M-flow, not necessarily compact, is sensitive on initial conditions. (4) every syndetically distal flow is pointwise Bohr a.p.

28 pages and to appear in JDE