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1. For the matrix \[ B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \] describe the set \[ Z = \bigcap_{n \in \N} \{ x \in \{0,1\}^\N : B(x(n),x(n+1)) = 1 \} \] defined by only allowing transitions from $B$.
2. What is the unique shift-invariant probability measure $\nu$ on $Z$?
3. What is the entropy of $\nu$?
4. Now for \[ Y = \bigcap_{n \in \N} \{ x \in \{0,1\}^\N : x(n) = 1 \Rightarrow x(n+2) = 0 \} \] determine the cylinders of length three that are disjoint from $Y$.

Fix probabilities \[ p(0) + p(1) = 1 \qquad q(0,0) + q(0,1) = 1 = q(1,0) + q(1,1) \] with \[ \begin{bmatrix} p(0) & p(1) \end{bmatrix} \begin{bmatrix} q(0,0) & q(0,1) \\ q(1,0) & q(1,1) \end{bmatrix} = \begin{bmatrix} p(0) & p(1) \end{bmatrix} \]

5. For which values does the corresponding Markov measure \[ \mu([\epsilon_1 \cdots \epsilon_r]) = p(\epsilon_1) q(\epsilon_1,\epsilon_2) \cdots q(\epsilon_{r-1},\epsilon_r) \] satisfy $\mu(Y) = 1$? What are those measures?
6. Come up with a four-vertex graph and a Markov measure $\eta$ on $\{0,1,2,3\}^\N$ and a map \[ \pi : \{0,1,2,3\}^\N \to \{0,1\}^\N \] such that $\xi = \eta \circ \pi^{-1}$ is a measure on $\{0,1\}^\N$ with $\xi(Y) = 1$.