# Examples of differential groups: irrational flows on the torus

*1985*

#### Translator’s note

*This page is a translation into English of the following:*

Donato, P. and Iglesias, P. “Exemples de groupes differentiels: flots irrationnels sur le tore.” *Comptes rendus de l’Académie des sciences* **301** (1985), 127–130.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `94f6dce`

# 1 Introduction

We consider the standard torus *diffeological spaces*, initially developed by J.-M. Souriau for the study of groups of infinite dimension, but which applies to all quotients (possibly singular) of Lie groups.
The quotient

f\in\operatorname{DL}(\mathbb{R}^n,T_\alpha) if and only iff is defined on an open\Omega of\mathbb{R}^n with values inT_\alpha and satisfies the following condition: for allx in\Omega , there exists an open neighbourhoodV ofx , and a mapF\in\mathrm{C}^\infty(V,T^2) that liftsf , i.e. onV we have the relationP_\alpha\circ F = f. \tag{1}

In the above, * n-plots* of

The differentiable maps from

For every diffeological group and every homogeneous space (quotient of a diffeological group by an arbitrary subgroup) we can define the notion of connectivity, and thus simple connectivity. In the connected case, we can also define the universal covering and the fundamental group, which depend only on the diffeological structure ([4] and [8]).

We illustrate these techniques in the precise case of irrational windings of the torus.
The passage to the universal covering of these quotients allows us to give a complete diffeological classification;
we can also make explicit the group of diffeomorphisms of

# 2 Covering and fundamental group

We briefly describe the construction of the universal covering of a homogeneous diffeological space:

Let

G be a connected diffeological group, andp\colon\widetilde{G}\to G its universal covering. LetH be an arbitrary subgroup ofG ; set\widetilde{H}=p^{-1}(H) , and let\widetilde{H}_0 be its identity component. We have the diagram\begin{CD} \widetilde{G} @>>> \widetilde{G}/\widetilde{H}_0 \\@VVV @VVV \\G @>>> G/H \end{CD} \tag{2} and\widetilde{G}/\widetilde{H}_0 is then the universal covering ofG/H , and\widetilde{H}/\widetilde{H}_0 is its fundamental group.

Conversely, every connected covering of

In the particular case that interests us, writing

Connectivity coincides with connectivity by differentiable arcs, and an easy calculation then shows that

Some comments: the diffeological universal covering of

The number of sheets, when

# 3 Classification of the T_\alpha

Let

Conversely, we see that, if

Two irrational toruses

This theorem is trivial satisfied for *rational toruses*.

# 4 Diffeomorphisms of T_\alpha .

We have seen in the previous section that the only diffeomorphisms from one irrational torus to another are the projections of affine maps of the form

We define on

The action of

If

The identity component of the group

\mathbb{Z}_2 if\alpha is a non-quadratic irrational;\mathbb{Z}_2\times\mathbb{Z} if\alpha is a quadratic irrational.

This is, to our knowledge, the first time that a classification of the

*The discussions that we had with J. Bellissard and J.-M. Souriau were invaluable to us; we thank them.*

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