Hausdorff dimension in quasiregular dynamics

TL;DR

The paper addresses the problem of prescribing the Hausdorff dimension of the fast escaping set $\mathcal{A}(f)$ for quasiregular maps of transcendental type in $\mathbb{R}^3$, showing that $\dim\mathcal{A}(f)$ can attain any value in $[1,3]$. The authors construct a 3D Zorich-type map using a beam partition controlled by a carefully chosen set $S$ and auxiliary maps $H$ and $G$, then obtain matching upper and lower bounds via a nested Cantor-type construction and distortion-control estimates. Under a growth condition on $M(r,f)$, they also prove a positive lower bound for the dimension of the Julia set, $\dim\mathcal{J}(f)\ge 1$, using capacity/modulus techniques. Altogether, the work extends planar results on escaping sets to three-dimensional quasiregular dynamics and highlights how capacity and modulus arguments, together with a geometric 3D Zorich construction, yield sharp dimension control for dynamical sets.

Abstract

It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of ${\mathbb R}^3$ can take any value in the interval $[1,3]$. The Hausdorff dimension of the Julia set of such a map is estimated under some growth condition.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof : Proof of Lemma \ref{['lemma12']}