On the complexity of polygonal billiards

We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. Consider an aperiodic phase point $ z $ and the sequence of sides its orbit visits. The set of points whose first $ n $ reflections coincide with this sequence is open and converges to the point $ z $ as $ n \to \infty $. We establish a polynomial lower bound estimate on this convergence rate for almost every $ z $. This yields an upper bound on the upper metric complexity and upper slow entropy of polygonal billiards. We also prove significant improvements of this lower bound along subsequences. Finally we extend these results to higher dimensions as well as to arbitrary invariant measures.