Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

TL;DR

This paper proves that for certain rational functions over non-archimedean fields, the Berkovich Julia set is uniformly perfect if and only if the function has no potentially good reductions, linking geometric and algebraic properties.

Contribution

It establishes a characterization of uniform perfectness of Berkovich Julia sets in terms of the reduction type of rational functions over non-archimedean fields.

Findings

Berkovich Julia set is uniformly perfect iff no potentially good reductions.

Uniform regularity of the boundary of Berkovich Fatou components.

Connection between geometric properties of Julia sets and algebraic reduction types.

Abstract

We show that a rational function $f$ of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of $f$ is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of $f$ is also established.