# Interesting examples of minimal action on torus

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.

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No: finite groups have finite (and hence closed) orbits. – Victor Protsak Jan 29 '13 at 18:28
You should mention when you crosspost a question (math.stackexchange.com/q/289595/12952) to MSE so that when it is answered on one site, people on the other site will know about it. – Alexander Gruber Jan 29 '13 at 18:55
If the functions $f_1,\ldots,f_{n-1}$ are all constant and equal to the identity then $\phi$ is not minimal. – Lee Mosher Jan 29 '13 at 22:43
Take a minimal foliation of the torus given by some action of a connected Lie group $G$ (e.g. the Kronecker foliation). Take $\Gamma$ any dense subgroup in $G$. Restrict the action from $G$ to $\Gamma$, and get a minimal action of $\Gamma$. – Alain Valette Jan 30 '13 at 0:22
When considering the one-dimensional torus, you have lots of interesting examples. The firsts coming to my mind are Fuchsian groups (of first kind) or the action of Thompson's group $G$. In the differentiable world there are many beautiful results and conjectures, as you can read in a paper by Deroin, Klepstyn and Navas (arxiv.org/pdf/0806.1974.pdf) – Michele Triestino Jan 30 '13 at 12:24