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arXiv · 2111.01579

Julia sets and geometrically finite maps over finite extensions of the $p$-adic field

Shilei Fan·Lingmin Liao·Hongmin Nie·Yuefei Wang
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Abstract

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $\phi\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $\phi$ is the natural restriction of $\mathbb{C}_p$-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that $\phi$ is geometrically finite, we prove that the dynamics on the $K$-Julia set of $\phi$ is a countable state Markov shift.

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Shilei Fan, Lingmin Liao, Hongmin Nie, Yuefei Wang. 2021-11-02. Julia sets and geometrically finite maps over finite extensions of the $p$-adic field. https://arxiv.org/abs/2111.01579

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