Class is over.

src/dyn/top/stone-cech.tex

@@ -1,4 +1,4 @@
-\section{$G \curvearrowright \beta G$}
+\section{The Universal Flow}
 \label{section:stone-cech-flow}
 
 \begin{definition}[Greatest Ambit]
@@ -52,3 +52,97 @@
     for all $g \in G$. Finally, since $\beta G$ is compact and $X$ is Hausdorff, $\pi(\beta G)$ is \hyperref[also compact]{proposition:compact-extensions} and \hyperref[hence closed]{proposition:compact-closed}. As $(X, x_0)$ is an ambit and $\pi(\beta G) \supset Gx_0$, $\pi(\beta G) \supset \ol{Gx_0} = X$. 
 \end{proof}
 
+\begin{definition}
+\label{definition:stone-cech-semigroup}
+    Let $G$ be a discrete group. For each $\fU, \fV \in \beta G$, let $\fU \cdot \fV = \lim_{g \to \fU}g\fV$, then
+    \begin{enumerate}
+        \item For each $A \subset G$, $A \in \fU \cdot \fV$ if and only if $\bracsn{g \in G|g^{-1}A \in \fV} \in \fU$. 
+        \item $\beta G$ equipped with the composition law $(\fU, \fV) \mapsto \fU \cdot \fV$ is a semigroup with identity $1_G$. 
+        \item (\textbf{Right Topological}) For each $\fV \in \beta G$, the mapping $\beta G \to \beta G$ defined by $\fU \mapsto \fU \cdot \fV$ is continuous. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    (3): By (U2) of the \hyperref[Stone-Čech compactification]{definition:stone-cech}. 
+\end{proof}
+
+\begin{definition}[Induced Action]
+\label{definition:stone-cech-semigroup-induced-action}
+    Let $G$ be a group, $(X, x_0)$ be a pointed $G$-flow, and $p_x: G \to X$ be the orbit map, then the mapping
+    \[
+    \pi: \beta G \to C(X; X) \quad \pi_\fU x  = \fU x = \lim_{g \to \fU}gx
+    \]
+
+    is a semigroup action of $\beta G$ on $X$, known as the \textbf{induced action} of $\beta G$ on $X$. 
+\end{definition}
+
+
+
+\begin{proposition}
+\label{proposition:tds-return-set-stone-cech}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$. For each $y \in X$ and $\fU \in \beta G$, then $y = \fU x$ if and only if $N(x, U) \in \fU$.
+\end{proposition}
+\begin{proof}
+    By definition of filter limits, $y = \fU x$ if and only if $p_x^{-1}(U) \in \fU$ for all $U \in \cn_X(y)$. Since $p_x^{-1}(U) = N(x, U)$, $y = \fU x$ if and only if $N(x, U) \in \fU$. 
+\end{proof}
+
+\begin{proposition}
+\label{proposition:z-stone-cech-fixed-point}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then the following are equivalent:
+    \begin{enumerate}
+        \item For each $U \in \cn_X(x)$, $N(x, U)$ is infinite. 
+        \item There exists $\fU \in \beta G \setminus G$ such that $px = x$. 
+    \end{enumerate}
+\end{proposition}
+% "obvious"
+
+
+\begin{definition}[Left Ideal]
+\label{definition:left-ideal-semigroup}
+    Let $G$ be a semigroup and $\emptyset \ne I \subset G$, then $I$ is a \textbf{left ideal} if for any $g \in G$ and $h \in I$, $gh \in I$. The left ideal $I$ is \textbf{minimal} if there exists no left ideal $J \subsetneq I$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:stone-cech-ideal-subflow}
+    Let $G$ be a group and $\emptyset \ne I \subset \beta G$. If $I$ is closed, then the following are equivalent:
+    \begin{enumerate}
+        \item $L$ is a $G$-subflow of $\beta G$. 
+        \item $L$ is a closed left ideal of $\beta G$. 
+    \end{enumerate}
+
+    Moreover, for arbitrary $\emptyset \ne I \subset \beta G$, the following are equivalent
+    \begin{enumerate}[label=(M\arabic*)]
+        \item $L$ is a minimal $G$-subflow of $\beta G$.
+        \item $L$ is a minimal left ideal of $\beta G$. 
+    \end{enumerate}
+    
+    In particular, minimal left ideals of $\beta G$ are closed. 
+\end{proposition}
+
+\begin{corollary}
+\label{corollary:stone-cech-minimal-ideal-existence}
+    Let $G$ be a group, then $\beta G$ admits a minimal ideal. 
+\end{corollary}
+\begin{proof}
+    By \autoref{theorem:tds-minimal-existence} and \autoref{proposition:stone-cech-ideal-subflow}. 
+\end{proof}
+
+\begin{definition}[Universal Minimal Flow]
+\label{definition:universal-minimal-flow}
+    Let $G$ be a group, then there exists a flow $(M, \alpha)$ such that:
+    \begin{enumerate}
+        \item $(M, \alpha)$ is a minimal $G$-flow. 
+        \item[(U)] For any minimal $G$-flow $(X, \beta)$, $x_0 \in X$, and $m_0 \in M$, there exists a unique continuous map $\Phi \in C(M; X)$ such that $\Phi(m_0) = x_0$ and the following diagram commutes
+        \[
+        \xymatrix{
+        M \ar@{->}[r]^{\Phi} \ar@{->}[d]_{\alpha(g)} & X \ar@{->}[d]^{\beta(g)} \\
+        M \ar@{->}[r]_{\Phi} & X
+        }
+        \]
+
+        for all $g \in G$.
+    \end{enumerate}
+\end{definition}
+
+
+
+