Enveloping semigroups of affine skew products and Sturmian-like systems

Let (X,Γ) be a topological dynamical system, meaning that X is a compact Hausdorff space, and Γ is a group of continuous maps from X to itself. The enveloping semigroup E(X,Γ) of the system (X,Γ) is defined to be the closure of Γ in X^X equipped with the product topology. We consider distal actions of groups generated by uinpotent affine transformations on a finite dimensional torus and we investigate the structure of the arising enveloping semigroup. It is known that in this case the enveloping semigroup is a group. We show that this group is necessarily nilpotent and find bounds on its nilpotency class. Moreover, if Γ is generated by a single transformation T of the aforementioned form we are able to determine precisely how the nilpotency class depends on T.

We also compute the enveloping semigroups of Sturmian and Sturmian-like systems enlarging the collection of existing explicit computations of these objects.