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\begin{center}
{\large MATH497C Assignment 3} \\
{\bf Due: September 21, 2012}
\end{center}
\begin{problem}
Let $G$ be an abelian group, and $d$ be a metric on $G$ such that $d(gh_1,gh_2) = d(h_1,h_2)$ for every $g,h_1,h_2 \in G$ (such a metric is {\it translation invariant}). Show that if $H$ is a closed subgroup of $G$ (equipped with the metric d), the function:
\[d([g_1],[g_2]) = \inf \{ d(g_1h_1,g_2h_2) : h_1,h_2 \in H \} \]
is a translation invariant metric on $G/H$, and that with this metric, the projection $p : G \to G/H$ is continuous. What would happen if $H$ was not closed? Calculate this metric in the case when $G = \mathbb{R}$ has the usual metric, $H = \mathbb{Z}$, and we think of $\mathbb{R} / \mathbb{Z}$ as the set of points $[0,1)$. Does this remind you of anything?
\end{problem}
\begin{problem}
Let $T:X \to X$ be an invertible dynamical system, and suppose that $\mathcal O(x)$ is a compact set. Show that $x$ is periodic.
\end{problem}
\begin{problem}
Let $T : X \to X$ be an invertible dynamical system on a compact metric space $X$, and $x \in X$. Show that if $\omega(x)$ can be written as $\omega(x) = C_1 \cup C_2$ such that $C_1$ and $C_2$ are disjoint, closed, invariant sets, then either $C_1 = \emptyset$ or $C_2 = \emptyset$.
\end{problem}
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