40.1 Topological Dynamical Systems
Definition 40.1.1 (Topological Dynamical System).label Let $G$ be a group and $X$ be a (non-empty) compact Hausdorff space, then a $G$-flow/topological dynamical system is a homomorphism $\alpha: G \to \text{Aut}(X)$. In which case, the system is denoted by $G \curvearrowright X$.
Definition 40.1.2 (Invariant).label Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$, then the following are equivalent:
- (1)
$GA \subset A$.
- (2)
$gA = A$ for all $g \in G$.
If the above holds, then $A$ is an invariant subset of $G$.
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 $gA = A$.$\square$
Definition 40.1.3 (Subflow).label Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$ be closed[1] and invariant, then $G \curvearrowright A$ under the restricted action is a $G$-subflow.
Definition 40.1.4 (Orbit).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $Gx$ is the orbit of $x$ under $G$, and $\ol{Gx}$ is the orbit closure of $x$ under $G$.
Definition 40.1.5 (Transitive).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is topologically transitive with respect to $G$ if $Gx$ is dense in $X$.
The flow $G \curvearrowright X$ is point transitive if it admits a topologically transitive point.
Finally, $G \curvearrowright X$ is topologically transitive if for any $U, V \subset X$ non-empty and open, there exists $g \in G$ such that $gU \cap V \ne \emptyset$[2].
Proposition 40.1.6.label Let $G \curvearrowright X$ be a point transitive $G$-flow, then $G \curvearrowright X$ is also topologically transitive.
Proof. Let $x \in G$ be a topologically transitive point and $U, V \subset X$ be non-empty and open. Since $Gx$ is dense in $G$, $Gx \cap U \ne \emptyset$ and $Gx \cap V \ne \emptyset$. After translating $x$, assume without loss of generality that $x \in U$, then $GU \supset Gx \cap V \ne \emptyset$. Therefore there exists $g \in G$ such that $gU \cap V \ne \emptyset$.$\square$
Definition 40.1.7 (Periodic).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is periodic with respect to $G$ if $Gx$ is finite.
Definition 40.1.8 (Return Set).label Let $G \curvearrowright X$ be a $G$-flow, $x \in X$, and $\emptyset \ne U \subset X$, then the return set of $x$ to $U$ is
Definition 40.1.9 (Recurrence).label Let $G$ be an infinite group, $G \curvearrowright X$ be a $G$-flow, and $x \in X$, then $x$ is recurrent if for every $U \in \cn_{X}(x)$, $N(x, U)$ is infinite.
Definition 40.1.10 (Syndetic).label Let $G$ be a group and $S \subset G$, then $S$ is (left) syndetic if there exists $F \subset G$ finite $FS = G$.
Lemma 40.1.11.label 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.
Example 40.1.12.label Let $S \subset \integer$, then $S$ has bounded gaps if
The set $S$ has bounded gaps if and only if it is syndetic.
Definition 40.1.13 (Minimality).label Let $G \curvearrowright X$ be a $G$-flow, then the following are equivalent:
- (1)
$G \curvearrowright X$ admits no proper subflows.
- (2)
Every point in $X$ is topologically transitive.
- (3)
For every $x \in X$ and $\emptyset \ne U \subset X$ open, $N(x, U)$ is syndetic in $G$.
If the above holds, then $G \curvearrowright X$ is minimal.
Proof. (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 $g \in hN(x, U) \subset FN(x, U)$.$\square$
Lemma 40.1.14.label Let $G$ be an infinite group and $G \curvearrowright X$ be minimal $G$-flow, then every point in $X$ is recurrent.
Proof. Let $x \in X$ and $U \in \cn_{X}(x)$, then $N(x, U)$ is syndetic. As $G$ is infinite, so is $N(x, U)$.$\square$
Theorem 40.1.15 (Existence of Minimal Subflows).label Let $G \curvearrowright X$ be a $G$-flow, then there exists $Y \subset X$ closed such that $G \curvearrowright Y$ is minimal.
Corollary 40.1.16.label Let $G$ be an infinite group and $G \curvearrowright X$ be a $G$-flow, then $X$ admits a recurrent point.
Proof. By Theorem 40.1.15, there exists $Y \subset X$ closed such that $G \curvearrowright X$ is minimal. In which case, Lemma 40.1.14 implies that every point in $Y$ is recurrent.$\square$
Definition 40.1.17 (Bernoulli Shift).label Let $G$ be a group, $X = 2^{G}$ equipped with the product topology, and
Viewing $2^{G}$ as the power set of $G$, $\alpha_{g}(A) = gA$ for any $A \subset G$. The $G$-flow $G \curvearrowright X$ is the Bernoulli shift of $G$, and $G$-subflows of $X$ are subshifts of $G$.
Example 40.1.18 (The Shift).label Let $\integer \curvearrowright 2^{\integer}$ be the Bernoulli shift of $\integer$, then:
- (1)
For any $x, y \in 2^{\integer}$, $y \in \ol{\integer x}$ if and only if for each $A \subset \integer$, there exists $n \in \integer$ such that $y|_{A} = nx|_{A}$. In other words, $x$ contains a copy of any segment of $y$[3].
- (2)
$\integer \curvearrowright 2^{\integer}$ is point transitive, and hence topologically transitive[4].
- (3)
- So $A$ is also compact Hausdorff. keyboard_return
- Approximations of points may be translated into approximations of other points. keyboard_return
- Topological conditions in Bernoulli shifts translate into combinatorial information. keyboard_return
- It is easier to show topological transitivity: move the trees until they are disjointkeyboard_return
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