2023/12/13 by Anna Sophie Schmidhuber, Schmidhuber, Anna Sophie
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2401.00951
openalex publication_date 2023/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 S1, 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.