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

TL;DR

This paper investigates the dynamics of rational maps over finite extensions of p-adic fields, establishing conditions under which the Julia set over such fields aligns with the complex p-adic Julia set and characterizing the dynamics as a Markov shift.

Contribution

It proves the equivalence of $K$-Julia sets with the restriction of $C_p$-Julia sets under certain conditions and shows that geometrically finite maps exhibit Markov shift dynamics.

Findings

$K$-Julia set equals the restriction of $C_p$-Julia set under well-behaved critical orbits.

Dynamics of geometrically finite maps on the $K$-Julia set are a countable state Markov shift.

Results extend understanding of p-adic Julia sets to finite extensions of $Q_p$.

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.