\[
\newcommand{\C}{\mathbb{C}}
\newcommand{\haar}{\mathsf{m}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\newcommand{\symdiff}{\mathop\triangle}
\DeclareMathOperator{\leb}{\mathsf{Leb}}
\DeclareMathOperator{\cont}{\mathsf{C}}
\DeclareMathOperator{\lpell}{\mathsf{L}}
\newcommand{\lp}[1]{\lpell^{\!\mathsf{#1}}}
\]
Week 8 Worksheet
- Fix $p \in \N$. Define $T : \Z/p\Z \to \Z/p\Z$ by $T(x) = x+1 \bmod p$. Let $\mu$ be normalized counting measure on $\Z / p\Z$.
- Verify that $\mu$ is $T$ invariant.
- Verify that $\mu$ is ergodic for $T$.
- Fix $r,s \in \N$. Put $X = \Z / r \Z$ and define $S : X \to X$ by
\[
S(x) = x + s \bmod
\]
for all $x \in X$. When is $S$ ergodic?
- Define $T : [0,1) \to [0,1)$ by $T(x) = 2x \bmod 1$. Verify that Lebesgue measure restricted to $[0,1)$ is $T$ invariant.
- Take $X = \R^2 / \Z^2$. Define $T : X \to X$ by
\[
T(x + \Z^2) = Ax + \Z^2
\]
for all $x \in \R^2$ where $A = [\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}]$.
- Check for every rectangle $R \subset X$ that $T^{-1}(R)$ has the same area as $R$.
- Given an example of a point $x \in X$ that does not have dense orbit for $T$.
- Write down a measure on $X$ that is $T$ invariant.
- Suppose $(X,\B,\mu,T)$ is ergodic. Fix a measurable function $f : X \to [0,\infty)$ with finite integral. Prove that if $f \circ T = f$ then $f$ is equal, on a set of full measure, to a constant function.