Geometry of currents, intersection theory and dynamics of horizontal-like maps. (English) Zbl 1089.37036

Dinh, Tien-Cuong; Sibony, Nessim

Ann. Inst. Fourier 56, No. 2, 423-457 (2006).

Summary: 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.

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U40	Currents
37B40	Topological entropy
37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)

Keywords:

topological entropy; intersection of currents; Green currents; equilibrium measure; horizontal-like mappings; extremality properties; mixing

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