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

Abstract

Short $\mathbb{C}^2$'s were constructed in [F] 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 H\'enon 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.