Geometry of currents, intersection theory and dynamics of horizontal-like maps

Dinh, Tien-Cuong; Sibony, Nessim

Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 423-457.

Nous introduisons une géométrie sur le cône des courants positifs fermés de bidegré $(p,p)$ et nous l’utilisons pour définir l’intersection de tels courants. Nous construisons et étudions aussi les courants de Green et la mesure d’équilibre pour les applications d’allure horizontale, en toute dimension. Les courants de Green vérifient certaines propriétés d’extrémalité. La mesure d’équilibre est invariante, mélangeante et d’entropie maximale. Elle est égale à l’intersection des courants de Green associés à l’application et à son inverse.

We introduce a geometry on the cone of positive closed currents of bidegree $(p,p)$ and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.

Classification : 37F, 32H50, 32U40
Keywords: Structural discs of currents, Green current, equilibrium measure, mixing, entropy.
Mot clés : disque structurel de courants, courant de Green, mesure d’équilibre, mélange, entropie

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