Week 7 Worksheet

Home | Assessment | Notes | Worksheets | Tutorials

1. 1. Let $(T(x))(n)=x(n+1)$ be the shift map on $X=\{0,1{\}}^{\mathbb{N}}$. Describe a point $x\in \{0,1{\}}^{\mathbb{N}}$ with dense orbit.
   2. Let $(T(x))(n)=x(n+1)$ be the shift map on $X=\{0,1,2{\}}^{\mathbb{N}}$. Describe a point $x\in \{0,1,2{\}}^{\mathbb{N}}$ with dense orbit.
2. Let $(T(x))(n)=x(n+1)$ be the shift map on $X=\{0,1{\}}^{\mathbb{N}}$. Describe a point $x\in \{0,1{\}}^{\mathbb{N}}$ with
   1. ${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)={\displaystyle \frac{1}{2}}}$
   2. ${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)={\displaystyle \frac{1}{4}}}$
   3. ${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)={\displaystyle \frac{1}{3}}}$
   4. ${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)={\displaystyle \frac{1}{\sqrt{2}}}}$
   5. ${\displaystyle \underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}x(n)}$ not existing.
3. Let $\mu$ be the fair coin measure on $\{0,1{\}}^{\mathbb{N}}$. Put $$Y=\bigcap _{n\in \mathbb{N}}\{x\in \{0,1{\}}^{\mathbb{N}}:x(n)=1\Rightarrow x(n+1)=0\}$$
   1. Describe what it means for $x\in \{0,1{\}}^{\mathbb{N}}$ to belong to $Y$.
   2. Calculate $\mu (Y)$.
   3. Does $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}y(n)={\displaystyle \frac{1}{2}}$$ for some point $y\in Y$?
   4. Does $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^{N}y(n)y(n+1)={\displaystyle \frac{1}{4}}$$ for some point $y\in Y$?
   5. Describe a measure $\nu$ on $\{0,1{\}}^{\mathbb{N}}$ with $\nu (Y)=1$.
4. Define $f:\{0,1{\}}^{\mathbb{N}}\to [0,1]$ by $$f(x)=\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{x(n)}{2^n}}$$ for all $x\in \{0,1{\}}^{\mathbb{N}}$.
   1. Check that $f$ is measurable.
   2. Let $\mu$ be the fair coin measure on $\{0,1{\}}^{\mathbb{N}}$. Define $\nu$ on $\mathsf{Bor}([0,1])$ by $\nu (B)=\mu (f^{-1}(B))$. Check that $\nu$ is a measure.
   3. What does the strong law of large numbers tell you about $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{1}_{[0,\frac{1}{2}]}(2^{n}xmod1)$$ for $\nu$ almost-every $x\in [0,1]$?