  Dual maps have been introduced as a generalization to higher dimensions of
word substitutions and free group morphisms. In this paper, we study the action
of these dual maps on particular discrete planes and surfaces -- namely stepped
planes and stepped surfaces. We show that dual maps can be seen as
discretizations of toral automorphisms. We then provide a connection between
stepped planes and the Brun multi-dimensional continued fraction algorithm,
based on a desubstitution process defined on local geometric configurations of
stepped planes. By extending this connection to stepped surfaces, we obtain an
effective characterization of stepped planes (more exactly, stepped
quasi-planes) among stepped surfaces.
