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

TL;DR

This paper investigates the structure of extensions of minimal compact metric flows, establishing conditions for PI extensions, existence of Li-Yorke chaos, and properties of M-flows related to sensitivity and almost periodic points.

Contribution

It characterizes PI extensions via M-flows, proves the existence of Li-Yorke chaos in non-PI extensions, and explores properties of unbounded and syndetically distal flows.

Findings

PI extensions have unique M-flows containing the diagonal

Non-PI extensions contain Li-Yorke chaotic M-flows

Unbounded or non-minimal M-flows are sensitive to initial conditions

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.