\[ \newcommand{\C}{\mathbb{C}} \newcommand{\cont}{\operatorname{\mathsf{C}}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\sph}{\mathsf{S}^\mathsf{1}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\orb}{\mathop{\mathsf{orb}}} \newcommand{\symdiff}{\mathop\triangle} \newcommand{\lp}[1][2]{\operatorname{\mathscr{L}^{\mathsf{#1}}}} \newcommand{\Lp}[1][2]{\operatorname{\mathsf{L}^{\!\mathsf{#1}}}} \newcommand{\ent}{\operatorname{\mathsf{H}}} \]

1. For each $N \in \N$ write down a partition $\xi$ of $\{0,1\}^\N$ such that \[ \ent(\xi) \ge N \] with respect to the fair coin measure.
2. 1. Describe the sequences that belong to \[ Y = \{ x \in \{0,1\}^\N : x(n) = 0 \Rightarrow x(n+1) = 0 \} \] and calculate the entropy of the shift map with respect to every shift-invariant probability measure on $\{0,1\}^\N$ with the property that $\mu(Y) = 1$.
   2. For the Markov chain defined by the matrix \[ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \] calculate the maximum possible value of the entropy for an invariant measure.