From 1e76f55d6861e5fa142a6b412752b9ababb1f061 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 16 Sep 2026 13:25:08 -0400 Subject: [PATCH] Housekeeping after class. --- src/dyn/index.tex | 1 + src/dyn/notation.tex | 12 ++++++++ src/dyn/top/def.tex | 66 +++++++++++++++++++++++++++++++++++++++++++- 3 files changed, 78 insertions(+), 1 deletion(-) create mode 100644 src/dyn/notation.tex diff --git a/src/dyn/index.tex b/src/dyn/index.tex index e147363..603b238 100644 --- a/src/dyn/index.tex +++ b/src/dyn/index.tex @@ -2,3 +2,4 @@ \label{part:dynamics} \input{./top/index.tex} +\input{./notation.tex} diff --git a/src/dyn/notation.tex b/src/dyn/notation.tex new file mode 100644 index 0000000..0984cda --- /dev/null +++ b/src/dyn/notation.tex @@ -0,0 +1,12 @@ +\chapter{Notations} +\label{chap:dyn-notations} + +\begin{tabular}{lll} + \textbf{Notation} & \textbf{Description} & \textbf{Source} \\ + \hline + + $G \curvearrowright X$ & Topological dynamical system. & \autoref{definition:topological-dynamical-system}\\ + $N(x, U)$ & Return set of $x$ to $U$. & \autoref{definition:tds-return-set} + +\end{tabular} + diff --git a/src/dyn/top/def.tex b/src/dyn/top/def.tex index 511fdb7..8cbddbe 100644 --- a/src/dyn/top/def.tex +++ b/src/dyn/top/def.tex @@ -3,7 +3,7 @@ \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 $G \curvearrowright X$. + Let $G$ be a group and $X$ be a (non-empty) 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] @@ -53,12 +53,48 @@ 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} +\begin{definition}[Return Set] +\label{definition:tds-return-set} + Let $G \curvearrowright X$ be a $G$-flow, $x \in X$, and $\emptyset \ne U \subset X$, then the \textbf{return set} of $x$ to $U$ is + \[ + N(x, U) = \bracsn{g \in G|gx \in U} + \] +\end{definition} + +\begin{definition}[Recurrence] +\label{definition:tds-recurrence} + Let $G$ be an infinite group, $G \curvearrowright X$ be a $G$-flow, and $x \in X$, then $x$ is \textbf{recurrent} if for every $U \in \cn_X(x)$, $N(x, U)$ is infinite. +\end{definition} + +\begin{definition}[Syndetic] +\label{definition:group-syndetic} + Let $G$ be a group and $S \subset G$, then $S$ is \textbf{(left) syndetic} if there exists $F \subset G$ finite $FS = G$. +\end{definition} + +\begin{lemma} +\label{lemma:subgroup-syndetic} + Let $G$ be a group and $H \subset G$ be a subgroup, then $H$ is syndetic if and only if the index of $H$ in $G$ is finite. +\end{lemma} +% Proof omitted. + +\begin{example} + \label{example:z-syndetic} + Let $S \subset \integer$, then $S$ has \textbf{bounded gaps} if + \[ + \sup_{n \in \integer}d(n, S) < \infty + \] + + The set $S$ has bounded gaps if and only if it is syndetic. +\end{example} +% Proof omitted. + \begin{definition}[Minimality] \label{definition:tds-minimal} Let $G \curvearrowright X$ be a $G$-flow, then the following are equivalent: \begin{enumerate} \item $G \curvearrowright X$ admits no proper subflows. \item Every point in $X$ is topologically transitive. + \item For every $x \in X$ and $\emptyset \ne U \subset X$ open, $N(x, U)$ is syndetic in $G$. \end{enumerate} @@ -68,6 +104,34 @@ (1) $\Rightarrow$ (2): Let $x \in X$, then $\ol{Gx}$ with the restricted action is a $G$-subflow of $X$. As $\emptyset \ne \ol{Gx}$ and $G \curvearrowright X$ admits no proper subflows, $\ol{Gx} = x$. (2) $\Rightarrow$ (1): Let $G \curvearrowright Y$ be a $G$-subflow of $G \curvearrowright X$. For any $x \in Y$, $Y \supset \ol{Gx} = X$, so $Y = X$. + + (2) $\Rightarrow$ (3): Let $\emptyset \ne U \subset X$ be open. Since $G \curvearrowright X$ is minimal, $Gx \cap U \ne \emptyset$ for all $x \in X$. Thus $X \subset G^{-1}U = GU$. By compactness of $X$, there exists $F \subset G$ finite such that $FU = X$. + + Let $x \in X$ be arbitrary, then for each $g \in G$, $gx \in X = FU$. Thus there exists $h \in F$ with $gx \in hU$ and $h^{-1}gx \in U$. Hence $h^{-1}g \in N(x, U)$, and $g \in hN(x, U) \subset FN(x, U)$. +\end{proof} + +\begin{lemma} +\label{lemma:tds-minimal-recurrent} + Let $G$ be an infinite group and $G \curvearrowright X$ be minimal $G$-flow, then every point in $X$ is recurrent. +\end{lemma} +\begin{proof} + Let $x \in X$ and $U \in \cn_X(x)$, then $N(x, U)$ is syndetic. As $G$ is infinite, so is $N(x, U)$. +\end{proof} + + + +\begin{theorem}[Existence of Minimal Subflows] +\label{theorem:tds-minimal-existence} + Let $G \curvearrowright X$ be a $G$-flow, then there exists $Y \subset X$ closed such that $G \curvearrowright Y$ is minimal. +\end{theorem} +% Proof omitted. + +\begin{corollary} +\label{corollary:tds-recurrent-existence} + Let $G$ be an infinite group and $G \curvearrowright X$ be a $G$-flow, then $X$ admits a recurrent point. +\end{corollary} +\begin{proof} + By \autoref{theorem:tds-minimal-existence}, there exists $Y \subset X$ closed such that $G \curvearrowright X$ is minimal. In which case, \autoref{lemma:tds-minimal-recurrent} implies that every point in $Y$ is recurrent. \end{proof}