$$$$

Put $X=\{0,1{\}}^{\mathbb{N}}$. Given $x\in \{0,1{\}}^{\mathbb{N}}$ put $$\gamma (x)=min\{n\in \mathbb{N}:x(n)=0\}$$ and define $R:X\to X$ by $$(R(x))(n)=\{\begin{array}{ll}0& n<\gamma (x)\\ 1& n=\gamma (x)\\ x(n)& n>\gamma (x)\end{array}$$ which is the 2-adic odometer.

1. Calculate $R(x)$ for $x=\mathtt{1100011010}\cdots$
2. Calculate $R(x)$ for $x=\mathtt{0010100101}\cdots$
3. There is one $x\in \{0,1{\}}^{\mathbb{N}}$ where the above definition does not apply. How would you define $R$ at that point?
4. What is $R^{-1}([1])$?
5. Use your answer to 4. to show the $(1-p,p)$ coin measure is not an invariant measure for $R$ if $p\ne \frac{1}{2}$.

Write $\mu$ for the fair coin measure on $X$. Take for granted that $R$ is a measure-preserving transformation for $\mu$.

6. Put $f=1_{[0]}-1_{[1]}$. What function is $f\circ R$? Is $f$ an eigenfunction of $R$?
7. Find four linearly independent eigenfunctions using linear combinations of $1_{[00]}$, $1_{[10]}$, $1_{[01]}$ and $1_{[11]}$.
8. Is one of your eigenfunctions from 7. in the subspace spanned by $1_{[0]}-1_{[1]}$?
9. Write each of $1_{[00]}$, $1_{[10]}$, $1_{[01]}$ and $1_{[11]}$ as a linear combination of your eigenfunctions.
10. Check that the fair coin measure $\mu$ is an invariant measure for $R$.
11. Prove that $R$ has discrete spectrum.