43.1 Topological Dynamical Systems

Definition 43.1.1 (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 $G \curvearrowright X$.

Let $G \curvearrowright X$ and $G \curvearrowright$ be $G$-flows, then a morphism of $G$-flows is a continuous map $\pi: X \to Y$ such that the following diagram commutes

for all $g \in G$.

Definition 43.1.2 (Invariant).label Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$, then the following are equivalent:

1. (1)

   $GA \subset A$.

2. (2)

   $gA = A$ for all $g \in G$.

If the above holds, then $A$ is an invariant subset of $G$.

Definition 43.1.3 (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 43.1.4 (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 43.1.5 (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 43.1.6.label Let $G \curvearrowright X$ be a point transitive $G$-flow, then $G \curvearrowright X$ is also topologically transitive.

Example 43.1.7.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 43.1.8 (Pointed Flow).label Let $G$ be a group, then a pointed $G$-flow is a pair $(X, x_{0})$, where $X$ is a $G$-flow, and $x_{0} \in X$ is a distinguished point. The mapping $\rho_{x}: G \to X$ defined by $g \mapsto gx_{0}$ is the orbit map of $(X, x_{0})$.

Let $(G \curvearrowleft X, x_{0})$ and $(G \curvearrowright Y, y_{0})$ be pointed $G$-flows, then a morphism of pointed $G$-flows is a morphism of $G$-flows $\pi: X \to Y$ such that $\pi(x_{0}) = y_{0}$.

Definition 43.1.9 (Ambit).label Let $G$ be a group and $(G \curvearrowright X, x_{0})$ be a pointed $G$-flow, then $(G \curvearrowright X, x_{0})$ is an ambit if $x_{0}$ is a topologically transitive point.

Definition 43.1.10 (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 43.1.11 (The Shift).label Let $\integer \curvearrowright 2^{\integer}$ be the Bernoulli shift of $\integer$, then:

1. (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. (2)

   $\integer \curvearrowright 2^{\integer}$ is point transitive, and hence topologically transitive^[4].

3. (3)