On non-tameness of the Ellis semigroup

A tame dynamical system can be characterized by the cardinality of its enveloping (or Ellis) semigroup. Indeed, this cardinality is that of the power set of the continuum 2ᶜ if the system is non-tame. The semigroup admits a minimal bilateral ideal and this ideal is a union of isomorphic copies of a group H , called the structure group. For almost automorphic systems, the cardinality of H is at most c that of the continuum. We show a partial converse of this which holds for minimal systems for which the Ellis semigroup of their maximal equicontinuous factor acts freely, namely that the cardinality of H is 2^ c if the proximal relation is not transitive and the subgroup generated by products ξ ζ ⁻¹ of singular points ξ ,ζ in the maximal equicontinuous factor is not open. This refines the above statement about non-tame Ellis semigroups, as it locates a particular algebraic component of the latter which has such a large cardinality.