Week 5 Coursework Test

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