Growth rates for geometric complexities and counting functions in polygonal billiards

Abstract

AbstractWe introduce a new method for estimating the growth of various quantities arising in dynamical systems. Applying our method to polygonal billiards on surfaces of constant curvature, we obtain estimates almost everywhere on direction complexities and position complexities. As a byproduct, we recover the power bounds of Boshernitzan on the number of billiard orbits between almost all pairs of points in a planar polygon [M. Boshernitzan. Private letter to H. Masur (1986)].