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arXiv · 2501.17961

Local fields, iterated extensions, and Julia Sets

Pui Hang Lee·Michelle Manes·Nha Xuan Truong
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Abstract

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $\alpha \in K$, let $K_\infty$ be the extension obtained by adjoining all iterated preimages of $\alpha$ under a unicritical polynomial $f_c(z)=z^\ell - c \in K[z]$. We study the extension $K_\infty/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$.

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Pui Hang Lee, Michelle Manes, Nha Xuan Truong. 2025-01-29. Local fields, iterated extensions, and Julia Sets. https://arxiv.org/abs/2501.17961

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