JournalscmhVol. 86, No. 2pp. 277–316

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

We continue our study of the dynamics of meromorphic mappings with small topological degree λ2(f)<λ1(f)\lambda_2(f)<\lambda_1(f) on a compact Kähler surface XX. 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 XX has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of C2\mathbb{C}^2. They are new even in the birational case (λ2(f)=1\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