Complexity of Polygonal Billiards

Abstract: Given a polygon in the plane, one can code a billiard trajectory by the sequence of sides its intersects. The complexity is the number of words of a given length we get this way starting from any point with any initial direction. It is a measure of the disorder of the dynamical system. Katok considered this problem as one of the most resistant one in dynamics. For rational polygons, following results of Howard Masur, one can get cubic lower and upper bounds. I will explain recent results obtained old and recent results on this topic.