  We show that the Lattes endomorphisms are the only holomorphic endomorphisms
of the complex k-dimensional projective space whose measure of maximal entropy
is absolutely continuous with respect to the Lebesgue measure. As a
consequence, Lattes endomorphisms are also characterized by other extremal
properties as the maximality of the Hausdorff dimension of their measure of
maximal entropy or the minimality of their Liapounov exponents. Our proof uses
a linearization method which is of independant interest and a previous
characterization by the regularity of the Green current.
