the following:

\[theo3\] [@Aus] [@GlMeUs] Let $(\mathbb{X},T)$ be a minimal dynamical system over a compact metric space, with Abelian acting group. Then the following assertions are equivalents:

\(1) the dynamical system $(\mathbb{X},T)$ is equicontinuous.

\(2) $E(\mathbb{X},T)$ is a compact group acting by homeomorphisms on $\mathbb{X}$.

\(3) $E(\mathbb{X},T)$ is metrisable.

\(4) $E(\mathbb{X},T)$ has left-continuous product.

\(5) $E(\mathbb{X},T)$ is Abelian.

\(6) $E(\mathbb{X},T)$ is made of continuous transformations.

In this case one has $E(\mathbb{X},T)=\mathbb{X}$ as compact Abelian group.

Here the compact Abelian group structure of a compact minimal equicontinuous system $(\mathbb{X},T)$ with Abelian acting group is only determined by the choice (which is arbitrary) of one element $\mathfrak{e}\in \mathbb{X}$ which plays the role of unit, from which the group structure extends that of $T$ mapped on the dense orbit $\mathfrak{e}.T$. In this case the equality $E(\mathbb{X},T)=\mathbb{X}$ is performed by identifying a transformation $g\in E(\mathbb{X},T)$ with $\mathfrak{e}.g$ in $\mathbb{X}$.

Outside the scope of equicontinuous systems, The Ellis semigroup is a quite complicated object as it is formed of mappings neither necessarily continuous nor invertible, and is not commutative. However a general construction allows to attach to any compact dynamical system a particular factor:

\[equ\] Let $(\mathbb{X},T)$ be a compact dynamical system. There exist an 