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 \[ \begin{align*} f_1 &{} = 1_{[00]} +1\cdot 1_{[10]} + 1\cdot 1_{[01]} +1\cdot 1_{[11]} \\ f_2 &{} = 1_{[00]} +i\cdot 1_{[10]} - 1\cdot1_{[01]} -i\cdot 1_{[11]} \\ f_3 &{} = 1_{[00]} -1\cdot 1_{[10]} + 1\cdot1_{[01]} -1\cdot 1_{[11]} \\ f_4 &{} = 1_{[00]} -i\cdot 1_{[10]} -1\cdot 1_{[01]} + i\cdot 1_{[11]} \end{align*} \] 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 \[ \begin{align*} 1_{[00]} &{} = \tfrac{1}{4} ( f_1 + f_2 + f_3 + f_4) \\ 1_{[10]} &{} = \tfrac{1}{4} ( f_1 - i f_2 - f_3 + i f_4) \\ 1_{[01]} &{} = \tfrac{1}{4} ( f_1 - f_2 + f_3 - f_4) \\ 1_{[11]} &{} = \tfrac{1}{4} ( f_1 + i f_2 - f_3 - i f_4) \end{align*} \] 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 $[\epsilon(1) \cdots \epsilon(r)]$ its inverse image under $R$ is another cylinder of the same length, defined by the symbols $\delta(1)\cdots\delta(r)$ that odometrically precede $\epsilon(1) \cdots \epsilon(r)$. 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 $\eta \in \C$ with $|\eta| = 1$ and $\eta^{2^n} = 1$ for some $n \in \N$ 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 $\mathsf{L}^{\!\mathsf{2}}(X,\mathscr{B},\mu)$. There are therefore no other eigenvalues of $R$ and $R$ has discrete spectrum.