Ergodic properties of rational mappings with large topological degree

Abstract

Let $X$ be a projective manifold and $f:X \rightarrow X$ a rational mapping with large topological degree, $d_t>\lambda_{k-1}(f):=$ the $(k-1)^{\rm th}$ dynamical degree of $f$. We give an elementary construction of a probability measure $\mu_f$ such that $d_t^{-n}(f^n)^* \Theta \rightarrow \mu_f$ 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 $\dim_{\mathbb{C}} X \leq 3$ or when $X$ is complex homogeneous, $\mu_f$ is the unique measure of maximal entropy.