  The non-ergodic behavior of the deterministic Fixed Energy Sandpile (DFES),
with Bak-Tang-Wiesenfeld (BTW) rule, is explained by the complete
characterization of a class of dynamical invariants (or toppling invariants).
The link between such constants of motion and the discrete Laplacian's
properties on graphs is algebraically and numerically clarified. In particular,
it is possible to build up an explicit algorithm determining the complete set
of independent toppling invariants. The partition of the configuration space
into dynamically invariant sets, and the further refinement of such a partition
into basins of attraction for orbits, are also studied. The total number of
invariant sets equals the graph's complexity. In the case of two dimensional
lattices, it is possible to estimate a very regular exponential growth of this
number vs. the size. Looking at other features, the toppling invariants exhibit
a highly irregular behavior. The usual constraint on the energy positiveness
introduces a transition in the frozen phase. In correspondence to this
transition, a dynamical crossover related to the halting times is observed. The
analysis of the configuration space shows that the DFES has a different
structure with respect to dissipative BTW and stochastic sandpiles models,
supporting the conjecture that it lies in a distinct class of universality.
