Week 11 Worksheet

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1. Compute the entropy of the shift map on $\{0,1,2{\}}^{\mathbb{N}}$.
2. Fix $p\in \mathbb{N}$. Prove that $T(x)=x+1mod1$ on $\mathbb{Z}/p$ has zero entropy.
3. Prove that the number $w_n$ of length $n$ strings of zeroes and ones with no consecutive ones satisfies the recurrence $$w_1=2w_2=3w_{n+2}=w_{n+1}+w_n$$ for all $n\in \mathbb{N}$.
4. Fix $0<p\le 1$. Assuming the shift map on $\{0,1{\}}^{\mathbb{N}}$ is ergodic, use the pointwise ergodic theorem to prove that $$Y=\bigcap _{n\in \mathbb{N}}\{x\in \{0,1{\}}^{\mathbb{N}}:x(n)=1\Rightarrow x(n+1)=0\}$$ has zero measure for the $(1-p,p)$ coin measure.
5. Let $\nu$ be the fair measure on $\{0,1,2{\}}^{\mathbb{N}}$. Prove that the shift on $\{0,1,2{\}}^{\mathbb{N}}$ equipped with $\nu$ is not isomorphic to the shift on $\{0,1{\}}^{\mathbb{N}}$ equipped with any fair coin measure.
6. Determine the Markov measure on $$\{x\in \{0,1,2{\}}^{\mathbb{N}}:x(n)=1\Rightarrow x(n+1)\ne 1\}$$ and calculate its entropy.