Une caractérisation des endomorphismes de Lattès par leur mesure de Green

Abstract

We show that the Lattès 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, Lattès 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.