Week 11 Coursework Test

For each $N \in N$ write down a partition $ξ$ of ${0, 1}^{N}$ such that $H (ξ) \geq N$ with respect to the fair coin measure.
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 $μ (Y) = 1$ .
2. For the Markov chain defined by the matrix $[\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}]$ calculate the maximum possible value of the entropy for an invariant measure.