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Approximation and billiards

This survey is based on a series of talks I gave at the conference "Dynamical systems and diophantine approximation" at l'Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundance of periodic trajectories in certain irrational polygons.