(Non-)Tameness among Toeplitz shifts

Given a dynamical system (X,T) [where X is compact metric and T is a self-homeo on X] its Ellis semigroup is defined as the closure of the collection {T^n} in the space of self-maps on X. The Ellis semigroup is a cornerstone of the algebraic theory of topological dynamics. Unfortunately, quite often, it is quite nasty. This talk is about when the Ellis semigroup of Toeplitz shifts is well-behaved (or: tame). There will be few to no proofs but rather a discussion of the involved techniques (most notably Brattelli-Vershik representations of minimal Cantor systems). Joint work with Johannes Kellendonk and Reem Yassawi.