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

We continue our study of the dynamics of meromorphic mappings with small topological degree ?2(f)<?1(f) on a compact Kahler 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 C2. They are new even in the birational case (?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.

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