Speed of convergence towards attracting sets for endomorphisms of P^k

TL;DR

This paper investigates the convergence properties of attracting sets for holomorphic endomorphisms of complex projective space, establishing the existence of a unique invariant current and exponential convergence of certain currents.

Contribution

It proves the existence and uniqueness of an invariant positive closed current supported on the attracting set and shows exponential convergence of push-forward currents to this invariant.

Findings

Existence of a unique invariant positive closed current au supported on A.

Exponential convergence of push-forwards of currents supported near A to au.

The equilibrium measure on A is hyperbolic.

Abstract

Let f be a non-invertible holomorphic endomorphism of P^k having an attracting set A. We show that, under some natural assumptions, A supports a unique invariant positive closed current \tau, of the right bidegree and of mass 1. Moreover, if R is a current supported in a small neighborhood of A then its push-forwards by f^n converge to \tau exponentially fast. We also prove that the equilibrium measure on A is hyperbolic.