From 16a35e6e2d60330a824c031878699c24da3648c8 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Thu, 10 Sep 2026 14:13:29 -0400
Subject: [PATCH] Added semester notes.

---
 document.tex                  |  1 +
 src/dyn/index.tex             |  4 +++
 src/dyn/top/def.tex           | 47 +++++++++++++++++++++++++++++++++++
 src/dyn/top/index.tex         |  4 +++
 src/op/system/definitions.tex |  9 +++++++
 src/op/system/index.tex       |  4 +++
 6 files changed, 69 insertions(+)
 create mode 100644 src/dyn/index.tex
 create mode 100644 src/dyn/top/def.tex
 create mode 100644 src/dyn/top/index.tex
 create mode 100644 src/op/system/definitions.tex
 create mode 100644 src/op/system/index.tex

diff --git a/document.tex b/document.tex
index 4f1fe74..2ebe390 100644
--- a/document.tex
+++ b/document.tex
@@ -16,6 +16,7 @@ Occasionally, I make up some definitions to play with. These definition blocks w
 \input{./src/measure/index}
 \input{./src/dg/index}
 \input{./src/op/index}
+\input{./src/dyn/index}
 %\input{./src/process/index}
 
 \bibliographystyle{alpha} % We choose the "plain" reference style
diff --git a/src/dyn/index.tex b/src/dyn/index.tex
new file mode 100644
index 0000000..e147363
--- /dev/null
+++ b/src/dyn/index.tex
@@ -0,0 +1,4 @@
+\part{Dynamical Systems}
+\label{part:dynamics}
+
+\input{./top/index.tex}
diff --git a/src/dyn/top/def.tex b/src/dyn/top/def.tex
new file mode 100644
index 0000000..053dc53
--- /dev/null
+++ b/src/dyn/top/def.tex
@@ -0,0 +1,47 @@
+\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}
+
+
+
+
+
diff --git a/src/dyn/top/index.tex b/src/dyn/top/index.tex
new file mode 100644
index 0000000..38b357e
--- /dev/null
+++ b/src/dyn/top/index.tex
@@ -0,0 +1,4 @@
+\chapter{Topological Dynamical Systems}
+\label{chap:topological-dynamics}
+
+\input{./def.tex}
diff --git a/src/op/system/definitions.tex b/src/op/system/definitions.tex
new file mode 100644
index 0000000..9e06806
--- /dev/null
+++ b/src/op/system/definitions.tex
@@ -0,0 +1,9 @@
+\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}
+
+
diff --git a/src/op/system/index.tex b/src/op/system/index.tex
new file mode 100644
index 0000000..c36bfb5
--- /dev/null
+++ b/src/op/system/index.tex
@@ -0,0 +1,4 @@
+\chapter{Operator Systems}
+\label{chap:opertaor-systems}
+
+\input{./definitions.tex}