Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure

Abstract

We continue our study of the dynamics of meromorphic mappings with small topological degree $\lambda_2(f)<\lambda_1(f)$ on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and has a natural geometric description.

Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbb{C}^2$. They are new even in the birational case ($\lambda_2(f)=1$). We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.