Une caracterisation des endomorphismes de Lattes par leur mesure de   Green
  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. 
Francois Berteloot (LEP), Christophe Dupont (Morsay)