Short-period attractors and non-ergodic behavior in the deterministic fixed-energy sandpile model

We study the asymptotic behaviour of the Bak, Tang, Wiesenfeld sandpile au- tomata 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 ρa (top- plings density) shows, as a function of energy density ζ, a devil’s staircase behaviour defining a symmetric energy interval-set over which also the period lengths remain constant. The prop- erties of the ζ-ρa phase diagram can be traced back to the basic symmetries underlying the model’s dynamics.