Ergodic properties of rational mappings with large topological degree

Vincent Guedj

doi:10.4007/annals.2005.161.1589

Abstract

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

Highlights

Let X be a projective algebraic manifold and ω a Hodge form on X normalized so that X ωk = 1, k = dimC X
We let In particular μf (If) denote the indeterminacy locus of f: this is an algebraic subvariety of codimension ≥ 2
We let dt denote the topological degree of f : this is the number of preimages of a generic point

Summary

Introduction

We let dt denote the topological degree of f : this is the number of preimages of a generic point. Define f ∗ωk to be the trivial extension through If of (f|X\If )∗ω ∧ · · · ∧ (f|X\If )∗ω This is a Radon measure of total mass dt. In particular μf does not charge pluripolar sets This answers a question raised by Russakovskii and Shiffman [RS 97] which was addressed by several authors (see [HP 99], [FG 01], [G 02], [Do 01], [DS 02]). This shows that μf is an invariant measure with positive entropy ≥ log dt > 0. When dimC X ≤ 3 or when the group of automorphisms Aut(X) acts transitively on X, μf is the unique measure of maximal entropy (Theorem 4.1)

Numerical invariants

First ergodic properties of μf

Uniqueness of the measure of maximal entropy