\[
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\newcommand{\B}{\mathscr{B}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\newcommand{\symdiff}{\mathop\triangle}
\DeclareMathOperator{\leb}{\mathsf{Leb}}
\DeclareMathOperator{\cont}{\mathsf{C}}
\DeclareMathOperator{\lpell}{\mathsf{L}}
\newcommand{\lp}[1]{\lpell^{\!\mathsf{#1}}}
\DeclareMathOperator{\LpL}{\mathsf{L}}
\newcommand{\Lp}[1]{\LpL^{{\!\mathsf{#1}}}}
\]
Week 11 Worksheet
- Compute the entropy of the shift map on $\{0,1,2\}^\N$.
- Fix $p \in \N$. Prove that $T(x) = x+1 \bmod 1$ on $\Z/p$ has zero entropy.
- Prove that the number $w_n$ of length $n$ strings of zeroes and ones with no consecutive ones satisfies the recurrence
\[
w_1 = 2
\qquad
w_2 = 3
\qquad
w_{n+2} = w_{n+1} + w_n
\]
for all $n \in \N$.
- Fix $0 \l p \le 1$. Assuming the shift map on $\{0,1\}^\N$ is ergodic, use the pointwise ergodic theorem to prove that
\[
Y = \bigcap_{n \in \N} \{ x \in \{0,1\}^\N : x(n) = 1 \Rightarrow x(n+1) = 0 \}
\]
has zero measure for the $(1-p,p)$ coin measure.
- Let $\nu$ be the fair measure on $\{0,1,2\}^\N$. Prove that the shift on $\{0,1,2\}^\N$ equipped with $\nu$ is not isomorphic to the shift on $\{0,1\}^\N$ equipped with any fair coin measure.
- Determine the Markov measure on
\[
\{ x \in \{0,1,2\}^\N : x(n) = 1 \Rightarrow x(n+1) \ne 1 \}
\]
and calculate its entropy.