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

Abstract

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

40 pages, 1 figure

References in corpus (1)

• Continuity of the Green function in meromorphic families of polynomials