  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 and
avalanches are interpreted as return times to a zero-measure set. The problem
of accurate determination of the scaling exponents near the critical barrier is
addressed using a characteristic function approach. A general equation for
return times to a small measure set is established and used to obtain
information on the nature of Bak-Sneppen dynamics.
