Deterministic Bak-Sneppen model: Lyapunov spectrum and avalanches as return times

A deterministic version of the Bak-Sneppen model is studied. The role of the Lyapunov spectrum in the onset of scale-free behavior is established, as well as the measure-theoretic nature of the Bak-Sneppen self-organized state. Avalanches are interpreted as return times to a small measure set and the problem of accurate determination of the scaling exponents near the critical barrier is addressed.