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MATH 4290 Homework Assignment 6
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Due on Thursday, November 1st, at the BEGINNING of class.\\

\noindent $\bullet$ Define $S$ to be the shift with alphabet $\{0,1\}$ consisting of all bi-infinite sequences in which the number of $0$s between any closest $1$s is even, and the number of $1$s between any closest $0$s is a multiple of $3$. Prove that $S$ is sofic and mixing, and find $h(S, \sigma)$ by using techniques similar to those used to treat the even shift in class.\\

\noindent $\bullet$ Define $A = \{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\}$, $X = A^{\mathbb{N}}$, and the metric $d$ on $X$ by
\[
d((x_n), (y_n)) = \sum_{n \in \mathbb{N}} 2^{-n} |x_n - y_n|,
\]
where $|x_n - y_n|$ is defined by the usual absolute value in $\mathbb{R}$. (You do NOT have to show that $d$ is a metric.) If $\sigma$ is the usual left shift on $X$, prove that $h(X, \sigma) = \infty$.\\

\noindent $\bullet$ Prove that if $X$ is a mixing $1$-step SFT (i.e. the forbidden list is given by pairs of adjacent letters) and $|X| > 1$, then $h(X, \sigma) > 0$.\\

\noindent $\bullet$ Define a directed graph $G$ so that the associated edge shift $X(G)$ is mixing and has $h(X(G), \sigma) = \log(\sqrt[3]{2} + 1)$. \\

\noindent $\bullet$ Prove that it is impossible to find $G$ as in the previous problem with $h(X(G)), \sigma) = \log(3 - \sqrt{2})$.



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