Quasi-isometricity and equivalent moduli of continuity of planar 1/|ω|2-harmonic mappings

In this paper, we prove that 1/|?|2-harmonic quasiconformal mapping is bi-Lipschitz continuous with respect to quasihyperbolic metric on every proper domain of C\{0}. Hence, it is hyperbolic quasi-isometry in every simply connected domain on C\{0}, which generalized the result obtained in [14]. Meanwhile, the equivalent moduli of continuity for 1/|?|2-harmonic quasiregular mapping are discussed as a by-product.

Quasiregular mapping that preserves plane

Quasiregular mappings are a natural generalization of analytic functions to higher dimensions. Quasiregular mappings have many properties. Our work in this paper is to prove the following theorem: If f  a b is a quasiregular mapping which maps the plane onto the plane, then f is a bijection. We do this by finding the connection between quasiregular and quasiconformal mappings.

Dynamics of quasiregular mappings with constant complex dilatation

The study of quadratic polynomials is a foundational part of modern complex dynamics. In this article, we study quasiregular counterparts to these in the plane. More specifically, let $h:\mathbb{C}\rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear map and consider the quasiregular mapping $H=g\circ h$, where $g$ is a quadratic polynomial. By studying $H$ and via the Böttcher-type coordinate constructed in A. Fletcher and R. Fryer [On Böttcher coordinates and quasiregular maps. Contemp. Math.575 (2012), 53–76], we are able to obtain results on the dynamics of any degree-two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. We also show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.

Superattracting fixed points of quasiregular mappings

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

Julia sets of uniformly quasiregular mappings are uniformly perfect

It is well known that the Julia set J(f) of a rational map f: ℂ → ℂ is uniformly perfect; that is, every ring domain which separates J(f) has bounded modulus, with the bound depending only on f. In this paper we prove that an analogous result is true in higher dimensions; namely, that the Julia set J(f) of a uniformly quasiregular mapping f: ℝn → ℝn is uniformly perfect. In particular, this implies that the Julia set of a uniformly quasiregular mapping has positive Hausdorff dimension.

On the modulus of continuity of harmonic quasiregular mappings on the unit ball in Rn

We show that, for a class of moduli functions ?(?), 0 ? ? ? 2, the property ??(?) - ?(?) ?? ? (?? - ??), ?, ?? Sn-1 implies the corresponding property ?u(x)-u(y) ?? C?(?x-y?) x, y ? Bn; for u = P[?], provided u is a quasiregular mapping. Our class of moduli functions includes ?(?) = ?? (0 < ? ? 1), so our result generalizes earlier results on H?lder continuity (see [1]) and Lipschitz continuity (see [2]).