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MATH 4290 Homework Assignment 5
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Due on Thursday, October 25th, at the BEGINNING of class.\\

\noindent $\bullet$ If $(X, T)$ is a topological dynamical system and $X$ is a compact metric space with metric $d$, prove that the quantity $d_n$ defined by $d_n(x, y) := \max_{0 \leq i < n} d(T^i x, T^i y)$ is a metric for all $n \in \mathbb{N}$.\\

\noindent $\bullet$ Prove the inequalities in Lemma 2.5.1 of the textbook.\\

\noindent $\bullet$ For the one-sided full shift $(\{0,1\}^{\mathbb{N}}, \sigma)$, find, with proof, $\textrm{sep}(n, 2^{-k})$ and $\textrm{span}(n, 2^{-k})$ for every $n \in \mathbb{N}$ and $k \in \mathbb{N} \cup \{0\}$. (Reminder: the metric here is $d((x_n), (y_n)) := 2^{-\max\{n \geq 0 \ : \ x_i = y_i \ \forall i \leq n\}}$.)\\

\noindent $\bullet$ If $(X, T)$ factors onto $(Y, S)$, prove that $h(X, T) \geq h(Y, S)$. \\

\noindent $\bullet$ If $(X, T)$ is an isometry (i.e. $d(x, y) = d(Tx, Ty)$ for all $x,y \in X$), prove that $h(X, T) = 0$.




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