On the Uniform Perfectness of the Boundary of Multiply Connected Wandering Domains

Abstract

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.2010 Mathematics subject classification: primary 37F10; secondary 30D05.

Multiply connected wandering domains of entire functions

“…We note that U n = f n (U ), for n ∈ N, since U is bounded and thus f n : U → U n is proper; see [19,28,Corollary 1] or the discussion in Section 3. Other examples of transcendental entire functions having multiply connected wandering domains with various additional properties were given in [7,12,16,18,31,32], the last of which will be discussed later in this paper.…”

“…It follows from Theorem A that the Julia set of an entire function f with a multiply connected wandering domain is not uniformly perfect; see [31,51]. Recently [16], criteria have been given to determine whether the boundary of a multiply connected wandering domain of an entire function is uniformly perfect.…”

“…Earlier [7], Baker constructed an example which has infinite (eventual) connectivity. It has recently been shown [16] that Baker's original example [5] of a multiply connected wandering domain also has infinite connectivity.…”

“…We also characterize those multiply connected wandering domains U which are infinitely connected but for which the infinite connectivity occurs either near the outer boundary component of U or near the inner boundary component of U . Examples of multiply connected wandering domains with these connectivity properties were given in [16].…”

Julia sets of uniformly quasiregular mappings are uniformly perfect

“…Further, any transcendental entire function with multiply-connected Fatou components is a counterexample by a result of Zheng [24]. See also [5] for results in this direction. However, as far as the authors are aware, all known examples of uniformly quasiregular mappings of R n , for n ≥ 3, extend to mappings of S n .…”