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arXiv · 2209.01715

The wandering domain problem for attracting polynomial skew products

Zhuchao Ji·Weixiao Shen
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Abstract

Wandering Fatou components were recently constructed by Astorg et al for higher-dimensional holomorphic maps on projective spaces. Their examples are polynomial skew products with a parabolic invariant line. In this paper, we study this wandering domain problem for polynomial skew product $f$ with an attracting invariant line $L$ (which is the more common case). We show that if $f$ is unicritical (in the sense that the critical curve has a unique transversal intersection with $L$), then every Fatou component of $f$ in the basin of $L$ is an extension of a one-dimensional Fatou component of $f|_L$. As a corollary, there is no wandering Fatou component. We will also discuss the multicritical case under additional assumptions.

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Zhuchao Ji, Weixiao Shen. 2022-09-05. The wandering domain problem for attracting polynomial skew products. https://arxiv.org/abs/2209.01715

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