Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

We compute the Lyapunov exponent, the generalized Lyapunov exponents, and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, are used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. The Lyapunov exponent for the corresponding map agrees with the conjectured limit lambda(map) = -2 log(R) + C + O(R) and we derive an approximate value for the constant C in good agreement with numerical simulations. We also find a diverging diffusion constant D(t) similar to log t and a phase transition for the generalized Lyapunov exponents.