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MATH 4290 Homework Assignment 2
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Due on Thursday, September 27th, at the BEGINNING of class.\\

\noindent $\bullet$ We can define the two-sided full shift $(\{0,1\}^{\mathbb{Z}}, \sigma)$ in almost the same way as the (one-sided) full shift $(\{0,1\}^{\mathbb{N}}, \sigma)$ from class. (It's convenient to view $\{0,1\}^{\mathbb{Z}}$ as a metric space with the metric 
\[
d((x_n), (y_n)) = 2^{-\max\{i \in \mathbb{N} \ : \ x_j = y_j \textrm{ for all } j \textrm{ satisfying } |j| < |i|\}}.)
\]
Prove that $\sigma$ is a homeomorphism on $\{0,1\}^{\mathbb{Z}}$, that the two-sided full shift factors onto the one-sided full shift, and that the two-sided full shift is NOT conjugate to the one-sided full shift.\\

\noindent $\bullet$ A dynamical system $(X, T)$ is called \textbf{topologically transitive} if it has a transitive point, i.e. a point with an orbit which is dense in $X$. Prove that the following definition is equivalent: $(X, T)$ is topologically transitive if for all nonempty open sets $A, B$, there exists $n$ so that $A \cap T^n B \neq \varnothing$.\\

\noindent $\bullet$
Show that the map $\phi: \{0,1\}^{\mathbb{N}} \rightarrow [0,1)$ defined by $\phi(x_1 x_2 \ldots) = \sum_{n = 1}^{\infty} x_i 2^{-i}$ is continuous, with the product topology on $\{0,1\}^{\mathbb{N}}$ and the usual (Borel) topology on $[0,1)$.\\

\noindent $\bullet$ For the full shift $(\{0,1\}^{\mathbb{N}}, \sigma)$, prove that the set of points with dense orbits is uncountable. (\textbf{Optional challenge question:} show that this set is residual, i.e. that its complement is a countable union of nowhere dense sets.)\\

\noindent $\bullet$
Show that if $\frac{\alpha}{\beta} \notin \mathbb{Q}$, there does not exist a factor map from $(\mathbb{T}, T_{\alpha})$ to $(\mathbb{T}, T_{\beta})$. (Hint: you may use the fact that $T_{\alpha, \beta}$ is minimal for this problem, even though we haven't finished a proof in class.)\\

\noindent $\bullet$
If $x$ is a uniformly recurrent point in a dynamical system $(X, T)$, and $X$ is a compact metric space, prove that $(\overline{\mathcal{O}(x)}, T)$ is a minimal system.\\

\noindent $\bullet$ \textbf{Optional challenge question:} Find $x$ in the full shift which is uniformly recurrent but not periodic. This will yield an infinite minimal subsystem $(\overline{\mathcal{O}(x)}, \sigma)$ within the full shift.

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