We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C k (under a natural assumption on the dynamical degrees). We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps and to small pertuba-tions of regular polynomial automorphisms of C k .
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