Lifting the regionally proximal relation and characterizations of distal extensions
Authors: Cao Kai , Dai Xiongping 
Published: December 2021

<p style='text-indent:20px;'>We consider a commutative diagram (CD) of flows with discrete phase group <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> and extensions as follows:</p><p style='text-indent:20px;'><inline-formula id="DCDS-A-1078-0947_2022_5_2103-M5432"><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="DCDS-A-1078-0947_2022_5_2103-M5432.jpg"/></inline-formula></p><p style='text-indent:20px;'>By <inline-formula><tex-math id="M2">\begin{document}$ {\text{RP}}_{\!\phi} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ {\text{RP}}_{\!\psi} $\end{document}</tex-math></inline-formula> we denote the relativized regionally proximal relations in <inline-formula><tex-math id="M4">\begin{document}$ X $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ Y $\end{document}</tex-math></inline-formula>, respectively. We mainly prove, among other things, the following:</p><p style='text-indent:20px;'>1. <inline-formula><tex-math id="M6">\begin{document}$ $\end{document}</tex-math></inline-formula>If <inline-formula><tex-math id="M7">\begin{document}$ X $\end{document}</tex-math></inline-formula> is topologically transitive <inline-formula><tex-math id="M8">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>-distal, then <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula> is minimal.</p><p style='text-indent:20px;'>2. <inline-formula><tex-math id="M10">\begin{document}$ (\pi\times\pi) {\text{RP}}_{\!\phi} = {\text{RP}}_{\!\psi} $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>3. If <inline-formula><tex-math id="M11">\begin{document}$ Y $\end{document}</tex-math></inline-formula> is locally <inline-formula><tex-math id="M12">\begin{document}$ \psi $\end{document}</tex-math></inline-formula>-Bronstein, then <inline-formula><tex-math id="M13">\begin{document}$ {\text{RP}}_{\!\psi}\circ {\text{P}}_{\!\psi} $\end{document}</tex-math></inline-formula> is an equivalence relation, and, <inline-formula><tex-math id="M14">\begin{document}$ \bar{y}\in {\text{RP}}_{\!\psi} $\end{document}</tex-math></inline-formula> whenever <inline-formula><tex-math id="M15">\begin{document}$ \bar{y}\in {\text{RP}}_{\!\psi}\circ {\text{RP}}_{\!\psi} $\end{document}</tex-math></inline-formula> is almost periodic.</p><p style='text-indent:20px;'>4. If <inline-formula><tex-math id="M16">\begin{document}$ Y_d $\end{document}</tex-math></inline-formula> is the maximal distal extension of <inline-formula><tex-math id="M17">\begin{document}$ Z $\end{document}</tex-math></inline-formula> below <inline-formula><tex-math id="M18">\begin{document}$ Y $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ Z^d $\end{document}</tex-math></inline-formula> is the universal minimal distal extension of <inline-formula><tex-math id="M20">\begin{document}$ Z $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M21">\begin{document}$ Y\perp_{Y_d}Z^d $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>5. (a) <inline-formula><tex-math id="M22">\begin{document}$ Y $\end{document}</tex-math></inline-formula> is locally <inline-formula><tex-math id="M23">\begin{document}$ \psi $\end{document}</tex-math></inline-formula>-Bronstein iff <inline-formula><tex-math id="M24">\begin{document}$ F^d&lt;F^\prime A $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M25">\begin{document}$ A,F,F^d $\end{document}</tex-math></inline-formula> are respectively the Ellis groups of <inline-formula><tex-math id="M26">\begin{document}$ Y,Z, Z^d $\end{document}</tex-math></inline-formula>. (b) If <inline-formula><tex-math id="M27">\begin{document}$ Y $\end{document}</tex-math></inline-formula> is locally <inline-formula><tex-math id="M28">\begin{document}$ \psi $\end{document}</tex-math></inline-formula>-Bronstein and <inline-formula><tex-math id="M29">\begin{document}$ K $\end{document}</tex-math></inline-formula> a <inline-formula><tex-math id="M30">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>-closed group with <inline-formula><tex-math id="M31">\begin{document}$ F^\prime A&lt;K&lt;F $\end{document}</tex-math></inline-formula>, then there is a unique <inline-formula><tex-math id="M32">\begin{document}$ Y_{\!K} $\end{document}</tex-math></inline-formula> which is <inline-formula><tex-math id="M33">\begin{document}$ \psi_{\!K}^\prime $\end{document}</tex-math></inline-formula>-equicontinuous and has the Ellis group <inline-formula><tex-math id="M34">\begin{document}$ K $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also prove the above theorems in the case <inline-formula><tex-math id="M35">\begin{document}$ T $\end{document}</tex-math></inline-formula> is a semigroup. Moreover, we show the following in minimal semiflows:</p><p style='text-indent:20px;'>6. <inline-formula><tex-math id="M36">\begin{document}$ $\end{document}</tex-math></inline-formula><inline-formula><tex-math id="M37">\begin{document}$ Y $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M38">\begin{document}$ \psi $\end{document}</tex-math></inline-formula>-distal iff <inline-formula><tex-math id="M39">\begin{document}$ \psi $\end{document}</tex-math></inline-formula> has a DE-tower iff there is a least group-like extension <inline-formula><tex-math id="M40">\begin{document}$ X $\end{document}</tex-math></inline-formula> via <inline-formula><tex-math id="M41">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> (i.e., <inline-formula><tex-math id="M42">\begin{document}$ \phi^{-1}\phi x = {\text{Aut}}_\phi(T,X)x $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M43">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>).</p><p style='text-indent:20px;'>7. <inline-formula><tex-math id="M44">\begin{document}$ \psi $\end{document}</tex-math></inline-formula> is group-like iff <inline-formula><tex-math id="M45">\begin{document}$ \psi $\end{document}</tex-math></inline-formula> has a G-tower that consists of group extensions and inverse limits.</p>