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\begin{example}[The Shift]
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\label{example:bernoulli-shift-z}
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Let $\integer \curvearrowright 2^\integer$ be the Bernoulli shift of $G$, then:
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Let $\integer \curvearrowright 2^\integer$ be the Bernoulli shift of $\integer$, then:
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\begin{enumerate}
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\item 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 $y$\footnote{Topological conditions in Bernoulli shifts translate into combinatorial information. }.
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\item $\integer \curvearrowright 2^\integer$ is point transitive, and hence topologically transitive\footnote{It is easier to show topological transitivity: move the trees until they are disjoint}.
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