Local (sub) Finslerian geometry for the maximum norms in dimension 2 *

International audienceWe consider specific sub-Finslerian structures in the neighborhood of 0 in R 2 , defined by fixing a familly of vector fields (F1, F2) and considering the norm defined on the non constant rank distribution ∆ = vect{F1, F2} by |G| = inf u {max{|u1|, |u2|} | G = u1F1 + u2F2}. If F1 and F2 are not proportionnal at p then we obtain a Finslerian structure; if not, the structure is sub-Finslerian on a distribution with non constant rank. We are interested in the study of the local geometry of these Finslerian and sub-Finslerian structures: generic properties, normal form, short geodesics, cut locus, switching locus and small spheres