A transcendental Hénon map with an oscillating wandering Short $${\mathbb {C}}^2$$

Abstract

Short \({\mathbb {C}}^2\)’s were constructed in [5] as attracting basins of a sequence of holomorphic automorphisms whose rate of attraction increases superexponentially. The goal of this paper is to show that such domains also arise naturally as autonomous attracting basins: we construct a transcendental Hénon map with an oscillating wandering Fatou component that is a Short \({\mathbb {C}}^2\). The superexponential rate of attraction is not obtained at single iterations, but along consecutive oscillations.