Entropy of meromorphic maps and dynamics of birational maps
Mémoires de la Société Mathématique de France, no. 122 (2010) , 103 p.

We study the dynamics of meromorphic maps for a compact Kähler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.

On étudie la dynamique des applications méromorphes sur les variétés kählériennes compactes. Plus précisément, on donne un critère simple qui permet de produire des mesures d’entropie maximale. On peut appliquer ce résultat pour borner les exposants de Lyapounov. Ensuite, on étudie le cas particulier d’une famille générique d’applications birationnelles de k pour laquelle on construit les courants de Green et la mesure d’équilibre. On utilise pour cela la théorie des super-potentiels. On montre que la mesure est mélangeante et qu’elle n’a pas de masse sur les ensembles pluripolaires. En utilisant le critère on obtient que la mesure est d’entropie maximale. Cela implique finalement que la mesure est hyperbolique.

DOI: 10.24033/msmf.434
Classification: 37Fxx,  32H04,  32Uxx,  37A35,  37Dxx
Keywords: Complex dynamics, meromorphic maps, super-potentials, currents, entropy, hyperbolic measure
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De Thélin, Henry; Vigny, Gabriel. Entropy of meromorphic maps and dynamics of birational maps. Mémoires de la Société Mathématique de France, Serie 2, , no. 122 (2010), 103 p. doi : 10.24033/msmf.434. http://numdam.org/item/MSMF_2010_2_122__1_0/

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