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

Let $\pi \text{\hspace{0.17em}⁣}:\mathcal{X}\to \mathcal{Y}$ be an extension of minimal compact metric flows such that ${\mathtt{\text{R}}}_{\pi }≠{\mathrm{\Delta }}_X$. A subflow of ${\mathtt{\text{R}}}_{\pi }$ 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 ${\mathrm{\Delta }}_X$ is the unique M-flow containing ${\mathrm{\Delta }}_X$ in ${\mathtt{\text{R}}}_{\pi }$. (2) If $\pi$ is not PI, then there exists a canonical Li-Yorke chaotic M-flow in ${\mathtt{\text{R}}}_{\pi }$. 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.