On the structure of foliations on dilation surfaces
speaker: Sophie Schmidhuber (Universität Zürich)abstract: Consider a polygon in the plane with pairwise parallel edges, where two edges of a pair may have different lengths. Glueing each pair of edges using compositions of translations and dilations we obtain a dilation surface. If in addition we pick a direction θ in S1 and consider the set of parallel lines in the plane in direction θ, this descends to a directional foliation on the dilation surface. Due to the expanding or shrinking nature of our glueing maps, this directional foliation may exhibit very rich dynamical behaviour. In this talk, I will present a structure theorem that classifies all possible types of orbit structures of such a foliation. In particular, I will show that there exists a decomposition of the surface into finitely many subsurfaces on which the orbit structure is either completely periodic, Morse-Smale, minimal or Cantor-like. I will also explain in which of these cases we can find a segment transversal to the foliation on which the first return map is semi-conjugated to a minimal interval exchange transformation, and accompany these results with an explicit example of a dilation surface called the Disco surface.
timetable:
Tue 21 May, 17:30 - 18:00, Aula Dini
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