Particle drift in self-similar billiards

We study a particle moving at unit speed in a channel made by connected self-similar billiard tables that grow in size by a factor r > 1 from left to right (this model was recently introduced in the physics literature by Barra, Gilbert and Romo). Let q(T) denote the position of the particle at time T. Our main result is the existence of an asymptotic distribution of q (T)/T as T -> infinity and {ln T /ln r}->rho for some 0 <=rho < 1.