Dynamics of horizontal-like maps in higher dimension

TL;DR

This paper investigates the properties of Green currents and equilibrium measures for horizontal-like maps in higher-dimensional complex spaces, analyzing their regularity, convergence, decay of correlations, and hyperbolicity.

Contribution

It provides new results on the regularity, convergence speed, and hyperbolic nature of equilibrium measures for higher-dimensional horizontal-like maps, including Henon-like maps.

Findings

Green currents are unique invariant positive closed currents.

The equilibrium measure is hyperbolic.

Results apply to Henon-like maps and polynomial automorphisms.

Abstract

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 Henon-like maps, to regular polynomial automorphisms of C^k and to their small pertubations.