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

Abstract: Let $\pi\colon\mathscr{X}\rightarrow\mathscr{Y}$ be an extension of minimal compact metric flows such that $\texttt{R}\pi\not=\Delta_X$. A subflow of $\texttt{R}\pi$ 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 $\texttt{R}\pi$. (2) If $\pi$ is not PI, then there exists a canonical Li-Yorke chaotic M-flow in $\texttt{R}\pi$. 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.