\[ \newcommand{\C}{\mathbb{C}} \newcommand{\cont}{\operatorname{\mathsf{C}}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\sph}{\mathsf{S}^\mathsf{1}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\orb}{\mathop{\mathsf{orb}}} \newcommand{\symdiff}{\mathop\triangle} \newcommand{\lp}[1][2]{\operatorname{\mathscr{L}^{\mathsf{#1}}}} \newcommand{\Lp}[1][2]{\operatorname{\mathsf{L}^{\!\mathsf{#1}}}} \]

1. Fix $\alpha \in \Q$. Define $T : [0,1) \to [0,1)$ by $T(x) = x + \alpha$ mod 1. Describe $\orb(0,T)$ in terms of $\alpha$.
2. Fix $\alpha$ irrational and define \[ T(x) = x + \alpha \bmod 1 \] on $[0,1)$. Let $\mu$ be a Borel measure on $[0,1)$ with \[ \mu([0,1)) = 1 \] and the property that \[ \int f \intd \mu = \int f \circ T \intd \mu \] for all continuous functions $f : [0,1) \to \C$. Recall that $\psi_k(x) = \exp(2 \pi i k x)$ for all $k \in \Z$.
   1. Verify that \[ \int \psi_k \intd \mu = \int \dfrac{1}{N} \sum_{n=0}^{N-1} \psi_k \circ T^n \intd \mu \] for all $k \in \Z$ and all $N \in \N$.
   2. Apply the dominated convergence theorem to prove that \[ \int \psi_k \intd \mu = 0 \] for all $k \ne 0$.