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This is a basic introduction to the ergodic theory of translation surfaces and interval exchange transformations. The study of translation surfaces, initially motivated by billiards in polygons, is a rich field which has bloomed in the last decades, developing deep connections with areas of number theory (multi-dimensional continued fraction algorithms and Diophantine approximation) and geometry (moduli spaces and Teichmuller geodesic flow).
No specific prior knowledge is assumed, although familiarity with some ergodic theory will be an advantage. We will briefly cover some basic ergodic theory material at the beginning (some measure theory knowledge is encouraged), with examples mainly focused on rotations, continued fractions and interval exchange transformations.
We will introduce:
polygonal billiards;
Interval exchange transformations;
Translation Surfaces;
The Teichmuller flow on the moduli space of translation surfaces.
We will then explore connections with arithmetic and Diophantine properties of interval exchange maps, developing theory of:
Rauzy-Veech induction (a multi-dimensional continued fraction algorithm).
As an application of material developed, we will prove:
We plan to give two different proofs one more combinatorial, using the Rauzy-Veech algorithm, one more geometric, using the Teichmuller geodesic flow and Masur's criterion. Time permitting, we might cover more recent developments, as dynamics on lattice surfaces and Lyapunov exponents.
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