Irrational rotation

Example

Let $\alpha$ be an irrational number and let $\mathbb{T} = \R/\Z$ be the circle, identified with $[0, 1)$. The irrational rotation by $\alpha$ is $$R_\alpha \colon \mathbb{T} \to \mathbb{T}, \qquad R_\alpha(x) = x + \alpha \bmod 1.$$

It is a measure-preserving transformation for Lebesgue measure $\lambda$. The preimage of an arc is an arc of the same length, and arcs generate the Borel sets.

It is ergodic: let $f \in L^2(\lambda)$ with $f \circ R_\alpha = f$, with Fourier coefficients $c_k$. Then $c_k e^{2\pi i k \alpha} = c_k$ for all $k \in \Z$. Since $k \alpha \notin \Z$ for $k \ne 0$, one gets $c_k = 0$ for $k \ne 0$, so $f$ is constant almost everywhere.

It is even uniquely ergodic, and every orbit is dense in $\mathbb{T}$. By Weyl's equidistribution theorem every orbit $(x + n\alpha \bmod 1)_{n \ge 0}$ is equidistributed.

It is not weakly mixing: $e^{2\pi i x}$ is a nonconstant eigenfunction of the Koopman operator, with eigenvalue $e^{2\pi i \alpha}$. Its Kolmogorov–Sinai entropy is $0$. If $\alpha = p/q$ is rational in lowest terms, then $R_\alpha^q$ is the identity and $R_\alpha$ is not ergodic.