Let $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ be the $d$-dimensional torus with normalized Haar measure $\mu_1$ and let $U(2)$ be the group of $2\times2$ unitary matrices with normalized Haar measure $\mu_2$. Let $\alpha:=(\alpha_1,\ldots,\alpha_d)$ be $d$ rationally independent real numbers, so that the translation $$ \mathbb T^d\ni x\mapsto x+\alpha\in\mathbb T^d $$ is ergodic with respect to $\mu_1$. Take a $C^1$-function $\phi\in C^1\big(\mathbb T^d,U(2)\big)$ and let $$ T_\phi:\mathbb T^d\times U(2)\to\mathbb T^d\times U(2) $$ be the skew product given by $$ T_\phi(x,g):=\big(x+\alpha,g\,\phi(x)\big). $$
From what I understand, there exist $C^1$-functions $\phi$ such that the map $T_\phi$ is ergodic with respect to $\mu_1\times\mu_2$ (the paper of M. G. Nerurkar "On the construction of smooth ergodic skew-products. Ergodic Theory and Dynamical Systems (1988)" gives general results in this direction).
Therefore, my questions are:
(i) Are there simple (explicit) examples of functions $\phi$ such that $T_\phi$ is ergodic?
(ii) If it is the case, can someone explain me how to construct such functions $\phi$?
Thanks a lot.
