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Week 10 Tutorial

Put $X = \{0,1\}^\N$. Given $x \in \{0,1\}^\N$ put \[ \gamma(x) = \min \{ n \in \N : x(n) = 0 \} \] and define $R : X \to X$ by \[ (R(x))(n) = \begin{cases} 0 & n \l \gamma(x) \\ 1 & n = \gamma(x) \\ x(n) & n \g \gamma(x) \end{cases} \] 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 $x \in \{0,1\}^\N$ 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 $p \ne \tfrac{1}{2}$.

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

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