On the structure of foliations on dilation surfaces

Abstract

Dilation surfaces are geometric surfaces modelled after the complex plane whose structure group is generated by the groups of translations and dilations. For any dilation surface, for any direction $θ$ in $S^1$, there exists a foliation on the surface called the directional foliation in direction $θ$. In this Thesis, we prove a structure theorem for the directional foliations on dilation surfaces using a decomposition theorem established by C.J. Gardiner in the 1980s. We show that given a directional foliation on any dilation surface, there exists a decomposition of the surface into finitely many subsurfaces on which the foliation structure is in one of four possible cases: completely periodic, Morse-Smale, minimal or Cantor-like. We further prove that in the last two cases, the first return map on a segment transversal to the foliation is semi-conjugated to a minimal interval exchange transformation. As a corollary, we obtain an analogous result for affine interval exchange transformations. Throughout the thesis, we accompany our results with an explicit example of a dilation surface called the Disco surface. We analyze the directional foliations on the Disco surface that exhibit non-trivially recurrent behaviour and explain geometrically why these foliations accumulate to a Cantor set.