On the size of attractors in $\mathbb{CP}^k$

TL;DR

This paper investigates the size of attracting sets in complex projective spaces, showing that under certain conditions, these sets support positive currents and are not pluripolar, with abundant examples provided.

Contribution

It introduces a framework for non-algebraic attracting sets in $ ext{CP}^k$ and proves they support positive currents with bounded quasi-potential, answering a question from T.C. Dinh.

Findings

Attracting sets support closed positive currents with bounded quasi-potential.

Under dimensional conditions, these sets are not pluripolar.

Examples of such sets are abundant in $ ext{CP}^k$.

Abstract

Let $f$ be a holomorphic endomorphism of $\mathbb{CP}^k$ having an attracting set $A$. In this paper, we address the question of the "size" of $A$ in a pluripolar sense. We introduce a conceptually simple framework to have non-algebraic attracting sets. We prove that adding a dimensional condition, these sets support a closed positive current with bounded quasi-potential (which answers a question from T.C. Dinh). Therefore, they are not pluripolar. Moreover, the examples are abundant on $\mathbb{CP}^k$.