Ergodic properties of rational mappings with large topological degree

Abstract

Let X be a projective manifold and f:XX a rational mapping with large topological degree, dt>λk1(f):= the (k1)th dynamical degree of f. We give an elementary construction of a probability measure μf such that dnt(fn)Θμf for every smooth probability measure Θ on X. We show that every quasiplurisubharmonic function is μf-integrable. In particular μf does not charge either points of indeterminacy or pluripolar sets, hence μf is f-invariant with constant jacobian fμf=dtμf. We then establish the main ergodic properties of μf: it is mixing with positive Lyapunov exponents, preimages of “most” points as well as repelling periodic points are equidistributed with respect to μf. Moreover, when dimCX3 or when X is complex homogeneous, μf is the unique measure of maximal entropy.

Authors

Vincent Guedj

Laboratoire Emile Picard, Université Paul Sabatier, 31062 Toulouse, France