Attracting current and equilibrium measure for attractors on P^k

TL;DR

This paper constructs an attracting current and an equilibrium measure for attracting sets of holomorphic endomorphisms on projective space, revealing their laminar structure and dynamical properties.

Contribution

It introduces a method to build attracting currents and equilibrium measures with specific laminar and extremal properties for holomorphic endomorphisms on P^k.

Findings

Attracting current is weakly laminar and extremal in the cone of invariant currents.

Equilibrium measure is mixing and has maximal entropy.

Provides a new framework for understanding attractors in complex dynamics.

Abstract

Let f be a holomorphic endomorphism of P^k having an attracting set A. We construct an attracting current and an equilibrium measure associated to A. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy.