  We describe in this article the dynamics of a $1$-parameter family of affine
interval exchange transformations. It amounts to studying the directional
foliations of a particular affine surface, the Disco surface. We show that this
family displays various dynamical behaviours: it is generically dynamically
trivial, but for a Cantor set of parameters the leaves of the foliations
accumulate to a (transversely) Cantor set. s study is achieved through the
analysis the dynamics of the Veech group of this surface combined a modified
version of Rauzy induction in the context of affine interval exchange
transformations.
