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

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.

On uniform perfectness in quasimetric spaces

The aim of this paper is to investigate the equivalence conditions for uniform perfectness of quasimetric spaces. We also obtain the invariance property of uniform perfectness under quasim?bius mappings in quasimetric spaces. In the end, two applications are given.

ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.