  We propose a new point of view on multidimensional continued fraction
algorithms inspired by Rauzy induction. The generic behaviour of such an
algorithm is described here as a random walk on a graph that we call simplicial
system. These systems provide a family of examples for random walks with memory
recorded by a finite dimensional vector.
  We introduce a general criterion on these graphs that induces ergodicity
together with a bundle of many other dynamical properties. In particular, after
computing the representation of Brun, Selmer and Arnoux-Rauzy-Poincare
algorithm in this formalism, it provides a unified proof of ergodicity for
these classical examples as well as new results such as uniqueness of the
measure of maximal entropy on a canonical suspension.
  These objects also bring a new perspective to some fractal sets such as Rauzy
gaskets. We show general explicit upper bound on Hausdorff dimensions of
fractals described in this formalism as well as a construction of their measure
of maximal entropy. This implies in particular that the Rauzy gasket in all
dimensions has Hausdorff dimension strictly smaller than its ambient space, as
well as sharper bounds on the dimension and an asymptotic result.
