Commentarii Mathematici Helvetici


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Volume 86, Issue 2, 2011, pp. 277–316
DOI: 10.4171/CMH/224

Published online: 2011-02-27

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

Jeffrey Diller[1], Romain Dujardin[2] and Vincent Guedj[3]

(1) University of Notre Dame, USA
(2) École Polytechnique, Palaiseau, France
(3) Université Paul Sabatier, Toulouse, France

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.

Keywords: Complex dynamics, meromorphic mappings, geometric currents, intersection of positive closed currents

Diller Jeffrey, Dujardin Romain, Guedj Vincent: Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure. Comment. Math. Helv. 86 (2011), 277-316. doi: 10.4171/CMH/224