Abstract: The notion of Poincaré rotation number for homeomorphisms of the unit circle is generalized to the case of homeomorphisms of a general compact abelian one–dimensional solenoidal group, which is also a one–dimensional foliated space; specifically, the theory is developed for the algebraic universal covering space of the circle (the adèle class group of the rationals). Poincaré’s dynamical classification theorem is also generalized to homeomorphisms of solenoids whose rotation element is an irrational element (i.e., monothetic generator) of the given group.