Dilation surfaces are surfaces modeled after the complex plane whose structure group is generated by the group of translations and dilations. Given a dilation surface, for any direction in S1 there exists a corresponding directional foliation on the surface. In this talk, we will study the four possible types of dynamical behaviour that such a foliation may have (i.e completely periodic, Morse-Smale, minimal or Cantor-like) and deduce a dynamical decomposition theorem for the directional foliation on dilation surfaces using results of C.J. Gardiner and G. Levitt from the 1980s.
In a second step, we study the first return map of the directional foliation on a dilation surface, which is a so-called affine interval exchange transformation (AIET). We introduce a powerful tool called Rauzy-Veech induction in order to develop a renormalization scheme which allows to find a decomposition of any given AIET into finite union of intervals which exhibit only one of the four types of dynamical behaviour. This provides an alternative, purely combinatorial approach to the decomposition results of Levitt and Gardiner and is joint work with Corinna Ulcigrai and Charles Fougeron.