Added semester notes.

src/dyn/index.tex

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+\part{Dynamical Systems}
+\label{part:dynamics}
+
+\input{./top/index.tex}

src/dyn/top/def.tex

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+\section{Topological Dynamical Systems}
+\label{section:topological-dynamical-systems}
+
+\begin{definition}[Topological Dynamical System]
+\label{definition:topological-dynamical-system}
+    Let $G$ be a group and $X$ be a compact Hausdorff space, then a \textbf{$G$-flow/topological dynamical system} is a homomorphism $\alpha: G \to \text{Aut}(X)$. In which case, the system is denoted by $G \curvearrowright X$. 
+\end{definition}
+
+\begin{definition}[Invariant]
+\label{definition:tds-invariant}
+    Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$, then the following are equivalent:
+    \begin{enumerate}
+        \item $GA \subset A$.
+        \item $gA = A$ for all $g \in G$.
+    \end{enumerate}
+    
+    If the above holds, then $A$ is an \textbf{invariant} subset of $G$. 
+\end{definition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): Let $g \in G$, then $gA \subset A$ and $g^{-1}A \subset A$ by (1). In which case, $A = gg^{-1}A \subset gA$, so $gA = A$. 
+\end{proof}
+
+
+\begin{definition}[Subflow]
+\label{definition:tds-sub}
+    Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$ be closed\footnote{So $A$ is also compact Hausdorff. } and invariant, then $G \curvearrowright A$ under the restricted action is a \textbf{$G$-subflow}. 
+\end{definition}
+
+\begin{definition}[Orbit]
+\label{definition:tds-orbit}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $Gx$ is the \textbf{orbit} of $x$ under $G$, and $\ol{Gx}$ is the \textbf{orbit closure} of $x$ under $G$. 
+\end{definition}
+
+\begin{definition}[Transitive]
+\label{definition:tds-transitive}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{topologically transitive} with respect to $G$ if $Gx$ is dense in $X$. 
+\end{definition}
+
+\begin{definition}[Periodic]
+\label{definition:tds-periodic}
+    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{periodic} with respect to $G$ if $Gx$ is finite. 
+\end{definition}
+
+
+
+
+

src/dyn/top/index.tex

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+\chapter{Topological Dynamical Systems}
+\label{chap:topological-dynamics}
+
+\input{./def.tex}

src/op/system/definitions.tex

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+\section{Operator Systems}
+\label{section:operator-systems}
+
+\begin{definition}[Operator System]
+\label{definition:operator-system}
+    Let $A$ be a $C^*$-algebra and $S \subset A$, then $S$ is an \textbf{operator system} if $S$ is a linear subspace of $A$ and $S = S^*$. 
+\end{definition}
+
+

src/op/system/index.tex

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+\chapter{Operator Systems}
+\label{chap:opertaor-systems}
+
+\input{./definitions.tex}