Billiards with countably many scatterers under no eclipse

We consider a billiard flow with countably infinitely many scatterers on the plane without eclipse. We show that the non-wandering set of the billiard flow is in one-to-one correspondence with a two-sided topological Markov shift with countably many states. We also give a sufficient condition for the Euler product formula for the zeta function with respect to the billiard flow.