Week 10 Tutorial Solutions

1. $R(x)=\mathtt{0010011010}\cdots$
2. $R(x)=\mathtt{1010100101}\cdots$
3. The rule does not make sense at the constant sequence $$x=\mathtt{11111111111111}\cdots$$ because no $x(n)$ is equal to 0. Define $R$ at this point to take the value $$\mathtt{000000000000}\cdots$$ and think of the odometer resetting.
4. $R^{-1}([1])=[0]$ because if $x(1)=0$ then $(R(x))(1)=1$ and if $(R(x))(1)=1$ it must have been the case that $\gamma (x)=1$ and therefore $x(1)=0$.
5. Let $\mu$ be the $(1-p,p)$ coin measure. Thus $\mu ([1])=p$ and $\mu ([0])=1-p$. If $\mu$ is $R$ invariant then $$p=\mu ([1])=\mu (R^{-1}([1]))=\mu ([0])=1-p$$ which is impossible unless $p=0.5$.
6. Since $R^{-1}([0])=[1]$ and $R^{-1}([1])=[0]$ we have $$f\circ R=1_{R^{-1}[0]}-1_{R^{-1}[1]}=1_{[1]}-1_{[0]}$$ and from $$1_{[1]}-1_{[0]}=-(1_{[0]}-1_{[1]})$$ we see it is an eigenfunction of eigenvalue $-1$.
7. Take $$\begin{array}{rl}{f}_1& =1_{[00]}+1\cdot 1_{[10]}+1\cdot 1_{[01]}+1\cdot 1_{[11]}\\ f_2& =1_{[00]}+i\cdot 1_{[10]}-1\cdot 1_{[01]}-i\cdot 1_{[11]}\\ f_3& =1_{[00]}-1\cdot 1_{[10]}+1\cdot 1_{[01]}-1\cdot 1_{[11]}\\ f_4& =1_{[00]}-i\cdot 1_{[10]}-1\cdot 1_{[01]}+i\cdot 1_{[11]}\end{array}$$ which are eigenfunctions of $R$ with eigenvalues $1$, $i$, $-1$ and $-i$ respectively. As their eigenvalues are distinct they are linearly independent.
8. From $$1_{[0]}-1_{[1]}=1_{[00]}+1_{[01]}-1_{[10]}-1_{[11]}$$ we see that $1_{[0]}-1_{[1]}$ and $f_3$ span the same subspace.
9. We can write $$\begin{array}{rl}{1}_{[00]}& =\frac{1}{4}(f_1+f_2+f_3+f_4)\\ 1_{[10]}& =\frac{1}{4}(f_1-i{f}_2-f_3+i{f}_4)\\ 1_{[01]}& =\frac{1}{4}(f_1-f_2+f_3-f_4)\\ 1_{[11]}& =\frac{1}{4}(f_1+i{f}_2-f_3-i{f}_4)\end{array}$$ by some linear algebra.
10. The collection $\mathcal{D}$ of cylinders (including the empty set) is a π-system so it suffices to check that $\mu (R^{-1}(C))=\mu (C)$ for every cylinder $C$. But for every cylinder $[ϵ(1)\cdots ϵ(r)]$ its inverse image under $R$ is another cylinder of the same length, defined by the symbols $\delta (1)\cdots \delta (r)$ that odometrically precede $ϵ(1)\cdots ϵ(r)$. With respect to the fair coin measure, all cylinders of length $r$ have the same measure. Thus $\mu$ is an invariant measure for $R$.
11. Mimicking the answer to 7 with longer cylinders, we can find for every $\eta \in \mathbb{C}$ with $|\eta |=1$ and ${\eta }^{2^n}=1$ for some $n\in \mathbb{N}$ an eigenfunction of $R$ with eigenvalue $\eta$. As, for every cylinder set $C$, we can write $1_C$ as a linear combination of such eigenfunctions, we can conclude that the eigenfunctions span ${\mathsf{L}}^{\mathsf{2}}(X,\mathcal{B},\mu )$. There are therefore no other eigenvalues of $R$ and $R$ has discrete spectrum.