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

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 ${\mathbb{CP}}^k$.

21 pages, 5 figures