Ergodic properties of rational mappings with large topological degree

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