Chapter 7

Suppose \(\alpha / 2 \pi\) is irrational. Prove that every point of \(\mathbb{T}\) has a forward orbit for the transformation \(T_{\alpha}\) which is dense in \(\mathbb{T} .\)

Prove that if \(\alpha / 2 \pi\) is rational, then every point of \(\mathbb{T}\) is periodic for \(T_{\alpha}\), i.e., for each \(x \in \mathbb{T}\) there is an \(n>0\) such that \(T_{\alpha}^{n}(x)=x\)

Prove that if \(m \in \mathbb{N}\) and \(m>1\), then the function \(M: \mathbb{T} \rightarrow\) \(\mathbb{T}\) given by \(M\left(e^{i x}\right)=e^{i m x}\) preserves Lebesgue measure and is ergodic. That is, there is nothing special about the role of the natural number 2 in the ergodicity of \(D\).

Define \(T:[0,1] \rightarrow[0,1]\) by $$ T(x)= \begin{cases}2 x, & \text { if } x \in\left[0, \frac{1}{2}\right] \\\ 2-2 x, & \text { if } x \in\left(\frac{1}{2}, 1\right]\end{cases} $$ Prove that Lebesgue measure is \(T\)-invariant.

Suppose \(T:[0,1] \rightarrow[0,1]\) preserves Lebesgue measure. Prove that for almost all \(x \in[0,1]\) there is a sequence of positive integers \(\left\\{n_{i}\right\\}\) such that $$ \lim _{i \rightarrow \infty} T^{n_{i}}(x)=x. $$