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A characterization of Lattès endomorphisms by their Green measure. (Une caractérisation des endomorphismes de Lattès par leur mesure de Green.) (French) Zbl 1079.37039
The authors uses a linearization method for showing a previous characterization by the regularity of the Green current. They show Lattés endomorphisms of the complex \(k\)-dimensional projective space whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a consequence, Lattés endomorphisms are also characterized by other extremal properties as the maximality of the Hausdorff dimension of their measure of maximal entropy or the minimality of their Lyapunov exponents. The object of that paper is a characterization of the dynamical system \((\mathbb{P}^k,f,\mu)\) (\(\mathbb{P}^k\) being the complex projective space, \(f\) a holomorphic endomorphism). Thus, it is an answer to a question by J. E. Fornaess and N. Sibony in [Publ. Math., Barc. 45, 527–547 (2001; Zbl 0993.32001)]. The subject is very actual and very interesting. The paper is very well written.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32U40 Currents
37C45 Dimension theory of smooth dynamical systems
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A05 Dynamical aspects of measure-preserving transformations
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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