Monodromy Representations and Lyapunov Exponents of Origamis [Elektronische Ressource] / André Kappes. Betreuer: G. Weitze-Schmithüsen

Monodromy Representations andLyapunov Exponents of OrigamisZur Erlangung des akademischen Grades einesDoktors derNaturwissenschaftenvon der Fakultät für Mathematik desKarlsruher Instituts für TechnologiegenehmigteDissertationvonDipl.-Math. André Kappesaus Landau in der PfalzTag der mündlichen Prüfung: 25. Mai 2011Referent: JProf. Dr. Gabriela Weitze-SchmithüsenKorreferenten: Prof. Dr. Frank HerrlichProf. Dr. Martin Möller, Goethe-Universität FrankfurtPrefaceAn origami is a compact Riemann surface X, which is tiled by finitely many Eu-clidean unit squares. An example is given in Figure 0.1. Away from the vertices, thetiling provides a particular atlas forX: Locally, its transition maps are translationsz7→z +c, c∈C. More generally, any finite collection of polygons in the Euclideanplane, whose sides can be paired by translations, gives rise to a compact Riemannsurface with such a translation structure ω. The pair (X,ω) is called a translationsurface.If we apply the linear action of A∈ SL (R) to the collection of polygons, we still2can pair the sides accordingly, but the translation structure and even the complexstructure of the deformed Riemann surface will usually differ from the original one.However, it may happen that there is a way to cut and reglue the polygons to obtainthe original collection. In this case, we can find a homeomorphismf :X→X, whichis affine w.r.t. to the translation structure and whose matrix part is A.