  We study the asymptotic behaviour of the Bak, Tang, Wiesenfeld sandpile
automata as a closed system with fixed energy.
  We explore the full range of energies characterizing the active phase. The
model exhibits strong non-ergodic features by settling into limit-cycles whose
period depends on the energy and initial conditions. The asymptotic activity
$\rho_a$ (topplings density) shows, as a function of energy density $\zeta$, a
devil's staircase behaviour defining a symmetric energy interval-set over which
also the period lengths remain constant. The properties of $\zeta$-$\rho_a$
phase diagram can be traced back to the basic symmetries underlying the model's
dynamics.
