Geometry of currents, intersection theory and dynamics of   horizontal-like maps
  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. 
Tien-Cuong Dinh and Nessim Sibony