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

### Jeffrey Diller

University of Notre Dame, USA### Romain Dujardin

École Polytechnique, Palaiseau, France### Vincent Guedj

Université Paul Sabatier, Toulouse, France

## 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.

## Cite this article

Jeffrey Diller, Romain Dujardin, Vincent Guedj, Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure. Comment. Math. Helv. 86 (2011), no. 2, pp. 277–316

DOI 10.4171/CMH/224