$$$$

1. For the matrix $$B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$ describe the set $$Z=\bigcap _{n\in \mathbb{N}}\{x\in \{0,1{\}}^{\mathbb{N}}:B(x(n),x(n+1))=1\}$$ 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 $$Y=\bigcap _{n\in \mathbb{N}}\{x\in \{0,1{\}}^{\mathbb{N}}:x(n)=1\Rightarrow x(n+2)=0\}$$ determine the cylinders of length three that are disjoint from $Y$.

Fix probabilities $$p(0)+p(1)=1q(0,0)+q(0,1)=1=q(1,0)+q(1,1)$$ with $$\left[\begin{array}{cc}p(0)& p(1)\end{array}\right]\left[\begin{array}{cc}q(0,0)& q(0,1)\\ q(1,0)& q(1,1)\end{array}\right]=\left[\begin{array}{cc}p(0)& p(1)\end{array}\right]$$

5. For which values does the corresponding Markov measure $$\mu ([{ϵ}_1\cdots {ϵ}_r])=p({ϵ}_1)q({ϵ}_1,{ϵ}_2)\cdots q({ϵ}_{r-1},{ϵ}_r)$$ satisfy $\mu (Y)=1$? What are those measures?
6. Come up with a four-vertex graph and a Markov measure $\eta$ on $\{0,1,2,3{\}}^{\mathbb{N}}$ and a map $$\pi :\{0,1,2,3{\}}^{\mathbb{N}}\to \{0,1{\}}^{\mathbb{N}}$$ such that $\xi =\eta \circ {\pi }^{-1}$ is a measure on $\{0,1{\}}^{\mathbb{N}}$ with $\xi (Y)=1$.