  This article presents a generalization of the notion of \emph{Poincar\'e
rotation set} to homeomorphisms of the ad\`ele class group
$\mathbb{A}/\mathbb{Q}$ of the rational numbers $\mathbb{Q}$, which is a
connected compact abelian group which can be identified with the
one-dimensional universal solenoid $\mathbf{S}$, \ie the algebraic universal
covering of the circle. The definition is first introduced in general for
homeomorphisms of $\mathbf{S}$ which are isotopic to a translation, and then
specializing in homeomorphisms of $\mathbf{S}$ isotopic to the identity, in
which case the rotation set is a closed interval contained in the base leaf
(the connected component of the identity). If in the latter case the rotation
interval reduces to a single element $\rho$ and $\rho$ is irrational (\ie it is
a monothetic generator of $\mathbf{S}$), we show that the homeomorphism is
semiconjugate to the translation $z\mapsto\rho{z}$, like in the classical
theory of Poincar\'e. This theory is valid for any general compact abelian one
dimensional solenoidal group $\mathbf{S}_G$, which are Pontryagin duals of
dense subgroups $G$ of the rational numbers with the discrete topology. These
solenoidal groups are one-dimensional laminations which are locally
homeomorphic to the product of a Cantor set by an interval so they behave very
much like a ``diffuse'' version of the circle. Our approach differs from others
because we use Pontryagin duality of compact abelian groups to define the
rotation sets.
  \end{abstract}
