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Sand Pile Models are discrete dynamical systems emphasizing the phenomenon ofSelf-Organized Criticality. From a configuration composed of a finite number ofstacked grains, we apply on every possible positions (in parallel) two grainmoving transition rules. The transition rules permit one grain to fall to itsright or left (symmetric) neighboring column if the difference of heightbetween those columns is larger than 2. The model is nondeterministic andgrains always fall downward. We propose a study of the set of fixed pointsreachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparisonwith the Symmetric Sand Pile Model (SSPM) on which rules are applied once ateach iteration, we get a continuity property. This property states that withinPSSPM we can't reach every fixed points of SSPM, but a continuous subsetaccording to the lexicographic order. Moreover we define a successor relationto browse exhaustively the sets of fixed points of those models.

On the set of Fixed Points of the Parallel Symmetric Sand Pile Model