Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps

Abstract

We give an asymptotic formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps f λ (z) = z d + λ/z d , where d ≥ 3 and λ is small.

Asymptotics of the Hausdorff dimensions of the Julia sets of McMullen maps with error bounds*

“…It is well-known that every component of J( f λ ) passing through a point in R λ is a quasicircle (refer to [20]), which has the Hausdorff dimension greater or equal to 1. In fact, many of components of J( f λ ) have the Hausdorff dimensions greater than 1 (see for example [25]). If we replace these quasi-circle components by round circles to obtain a Cantor circle conformally homeomorphic to J λ × S 1 , then it seems that the Hausdorff dimension of J( f λ ) should be greater than one of J λ × S 1 .…”

“…Note that when c is small, the Julia set of p c is a Jordan curve (actually a quasi-circle). For the calculation of asymptotic expansions of the Hausdorff dimensions of Julia sets of other rational maps which are Jordan curves, one may refer to [16,25,27]. When |c| is large enough, the Julia sets of p c are Cantor sets.…”

On the dynamics of a family of generated renormalization transformations

“…These IFS are defined in terms of the inverse branches of the iterations of the rational map. The original proof idea of Lemma 7.3 comes from [YW,Lemma 2.6] and the proof appeared here is an improved version.…”

The Hausdorff dimension of the Julia sets concerning renormalization transformation

“…Ones have studied some important and interesting results [6,11,13]. In this paper, we investigate the Hausdorff dimension HD(J(T n, λ )) of the Julia sets J(T n, λ ) and obtain the following result.…”