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Dynamics of horizontal-like maps in higher dimension. (English) Zbl 1149.37025

Summary: We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in \(\mathbb C^k\), 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, to regular polynomial automorphisms of \(\mathbb C^k\) and to their small perturbations.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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