Edit 1:This is a cross post on MSE. See math.stackexchange.com/q/289595/12952

Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor minimal action does not exist for finite groups. So I am asking for general discrete group actions. What I really want to know is some interesting examples of minimal actions (not just a single homeomorphism) on suitable nice topological space. I just read an article on Furstenburg transformation am I was guessing the construction could be generalized to give minimal actions.

For a n-torus $\mathbb{T}^n$, A Furstenburg transformation $\phi$ is defined by: $$ \phi(\xi_1,\xi_2,\dots,\xi_n)=(e^{2\pi i\theta}\xi_1, f_1(\xi_1)\xi_2,\dots,f_{n-1}(\xi_1,\dots,\xi_{n-1})\xi_n) $$ Where $\theta\in \mathbb{R}$ and for each $i$, $f_i$ is a real continuous function on $\mathbb{T}^i$.

It is known that when $\theta$ is irrational, (Edit3: and all functions $f_i$ are in suitable homotopy classes not containing the constant functions) Furstenburg transformation defines a minimal dynamic system.

My question is, are there any interesting examples of minimal actions of other discrete groups on n-torus? I am thinking something similar to Furstenburg transformation, but any other examples are welcome too.