Introduction to Translation Surfaces and Interval Exchange Transformations
Abstract
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15.04.2026, 15:15 (Seminargebäude 3-14)
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This course provides an introduction to the dynamics of billiards in polygons and their connections with modern geometric structures.
We begin with billiards in simple polygons, such as the square and rational triangles, and use them as guiding examples throughout the course. These systems naturally lead to the notion of translation surfaces, which offer a geometric framework for understanding polygonal billiards. Translation surfaces can be described from several complementary perspectives, and we discuss the equivalence between these viewpoints.
A central theme of the course is the relation between translation flows and interval exchange transformations (IETs), which are specific functions from an interval to itself. Every directional flow on a translation surface can be encoded by the iterations of an IET, revealing how certain two-dimensional dynamical systems can be studied through one-dimensional systems. Conversely, IETs can be geometrically realized on translation surfaces. Keane's theorem provides a criterion for minimality of IETs and will be one of the main results discussed in the course.
Renormalization is a key idea in dynamical systems: it is about transforming the initial dynamical system into another (in the same class) to get new information on the initial one. We introduce Rauzy induction as a renormalization procedure acting on the parameter space of interval exchange transformations. In parallel, we introduce parameter spaces of translation surfaces and the $GL^{+}(2,\mathbb{R})$-action, which gives a renormalization process for the directional flows. These two viewpoints provide complementary approaches to the study of translation surfaces, and we discuss how these actions can be used to investigate their dynamical behavior.
The aim of the course is to present a coherent picture connecting polygonal billiards, flat geometry, and dynamical systems, while preparing the ground for further study in Teichmüller dynamics and related areas.
Polygonal billiards; translation surfaces; interval exchange transformations; dynamical systems.
Prerequisites
Basic background in complex analysis and topology. Familiarity with holomorphic functions, oriented two-dimensional manifolds, differential 1-forms, and basic notions of measure theory is assumed. Some familiarity with differential geometry or dynamical systems is helpful but not required. Necessary concepts will be introduced as needed.
