(The) first return time in polygonal billiards

Abstract

We consider a polygonal billiards map T. Let R$_n be the first return time of T to the ball of radius 2$^{-n}$ centered at an initial point. Kac``s lemma is the well known fact which relates to the first return time. In this work, we show, roughly, that (log$_2 R$_n)/n converges to 1 in case of the typical triangle and confirm this by computer simulations. We also find the average of (log$_2 R$_n)/n which is related to L$^1-convergence. We expect that they hold for general polygons.