Resumen de Ergodic properties of rational mappings with large topological degree

Vincent Guedj

• Let X be a projective manifold and f : X ¿¿ X a rational mapping with large topological degree, dt > ¿Ék.1(f) := the (k . 1)th dynamical degree of f. We give an elementary construction of a probability measure ¿Êf such that d.n t (fn).¿¿ ¿¿ ¿Êf for every smooth probability measure ¿¿ on X. We show that every quasiplurisubharmonic function is ¿Êf -integrable. In particular ¿Êf does not charge either points of indeterminacy or pluripolar sets, hence ¿Êf is f-invariant with constant jacobian f.¿Êf = dt¿Êf . We then establish the main ergodic properties of ¿Êf : it is mixing with positive Lyapunov exponents, preimages of ¿hmost¿h points as well as repelling periodic points are equidistributed with respect to ¿Êf . Moreover, when dimC X . 3 or when X is complex homogeneous, ¿Êf is the unique measure of maximal entropy.