# Interesting examples of minimal action on torus

Edit 1:This is a cross post on MO. See http://mathoverflow.net/questions/120236/interesting-examples-of-minimal-action-on-torus

Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor(on MO) 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 and I was guessing the construction could be generalized to give minimal actions.

For a n-torus $\mathbb{T}^n$, A Furstenberg 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, Furstenberg transformation defines a minimal dynamic system.

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

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You should mention when you crosspost a question to MO so that when it is answered on one site, people on the other site will know about it. – Alexander Gruber Jan 29 '13 at 18:54
It's Furstenberg, not Furstenburg – Martin Jan 29 '13 at 22:18
@Martin Corrected, Thanks！ – Qingyun Wang Jan 31 '13 at 16:06