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

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 199-202 ◽  
Author(s):  
Milos Arsenovic ◽  
Vesna Manojlovic
Keyword(s):  
Unit Ball ◽  
Modulus Of Continuity ◽  
Lipschitz Continuity ◽  
Quasiregular Mapping ◽  
Quasiregular Mappings

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]).

scholarly journals WEIGHTED LIPSCHITZ CONTINUITY AND HARMONIC BLOCH AND BESOV SPACES IN THE REAL UNIT BALL

2005 ◽  
Vol 48 (3) ◽  
pp. 743-755 ◽  
Author(s):  
Guangbin Ren ◽  
Uwe Kähler
Keyword(s):  
Unit Ball ◽  
Besov Spaces ◽  
Lipschitz Continuity ◽  
Bloch Space ◽  
The Real

AbstractThe characterization by weighted Lipschitz continuity is given for the Bloch space on the unit ball of $\mathbb{R}^n$. Similar results are obtained for little Bloch and Besov spaces.

scholarly journals Hausdorff measure of the singular set of quasiregular maps on Carnot groups

2003 ◽  
Vol 2003 (35) ◽  
pp. 2203-2220 ◽  
Author(s):  
Irina Markina
Keyword(s):  
Vector Space ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Quasiregular Mapping ◽  
Carnot Groups ◽  
Singular Set ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Dimensional Hausdorff Measure ◽  
Homogeneous Dimension

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Letνstand for the homogeneous dimension of a Carnot group and letmbe the index of the last vector space of the corresponding Lie algebra. We prove that the(ν−m−1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimates of the local index of quasiregular mappings are also obtained.

scholarly journals Quasiregular mapping that preserves plane

2016 ◽  
Vol 12 (9) ◽  
pp. 6603-6607
Author(s):  
Anila Duka ◽  
Ndriçim Sadikaj
Keyword(s):  
Analytic Functions ◽  
Quasiregular Mapping ◽  
Higher Dimensions ◽  
Natural Generalization ◽  
Quasiconformal Mappings ◽  
Quasiregular Mappings

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.

scholarly journals On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

2017 ◽  
Vol 220 (1) ◽  
pp. 453-478 ◽  
Author(s):  
Jiaolong Chen ◽  
Peijin Li ◽  
Swadesh Kumar Sahoo ◽  
Xiantao Wang
Keyword(s):  
Lipschitz Continuity ◽  
Quasiregular Mappings

scholarly journals Julia sets of uniformly quasiregular mappings are uniformly perfect

2011 ◽  
Vol 151 (3) ◽  
pp. 541-550 ◽  
Author(s):  
ALASTAIR N. FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Hausdorff Dimension ◽  
Analogous Result ◽  
Julia Set ◽  
Julia Sets ◽  
Quasiregular Mapping ◽  
Higher Dimensions ◽  
Ring Domain ◽  
Rational Map ◽  
Quasiregular Mappings ◽  
Uniformly Perfect

AbstractIt 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.

scholarly journals Superattracting fixed points of quasiregular mappings

2014 ◽  
Vol 36 (3) ◽  
pp. 781-793 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Fixed Point ◽  
Spherical Shell ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Basin Of Attraction ◽  
Quasiregular Mapping ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Polynomial Type ◽  
The Basin Of Attraction

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.

scholarly journals Quasiregular self-mappings of manifolds and word hyperbolic groups

2007 ◽  
Vol 143 (6) ◽  
pp. 1613-1622 ◽  
Author(s):  
Martin Bridson ◽  
Aimo Hinkkanen ◽  
Gaven Martin
Keyword(s):  
Hyperbolic Group ◽  
Quasiregular Mapping ◽  
Hyperbolic Groups ◽  
Quasiregular Mappings ◽  
Word Hyperbolic Group ◽  
Word Hyperbolic Groups ◽  
Mostow Rigidity ◽  
Negative Sectional Curvature ◽  
Free Word ◽  
Torsion Free

AbstractAn extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature ($n\neq 4$), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

scholarly journals WEIGHTED LIPSCHITZ CONTINUITY, SCHWARZ–PICK'S LEMMA AND LANDAU–BLOCH'S THEOREM FOR HYPERBOLIC-HARMONIC MAPPINGS IN ℂN

2013 ◽  
Vol 18 (1) ◽  
pp. 66-79 ◽  
Author(s):  
Shaolin Chen ◽  
Saminathan Ponnusamy ◽  
Xiantao Wang
Keyword(s):  
Unit Ball ◽  
Harmonic Functions ◽  
Lipschitz Continuity ◽  
Type Theorem ◽  
Harmonic Mappings ◽  
Bloch Spaces ◽  
Bloch Constant ◽  
Bloch’S Theorem ◽  
Hyperbolic Harmonic Functions ◽  
The Relationship

In this paper, we discuss some properties on hyperbolic-harmonic functions in the unit ball of ℂ n . First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz–Pick type theorem for hyperbolic-harmonic functions and apply it to prove the existence of Landau-Bloch constant for functions in α-Bloch spaces.

scholarly journals Mappings of Lp-integrable distortion: regularity of the inverse

2016 ◽  
Vol 146 (3) ◽  
pp. 647-663 ◽  
Author(s):  
Jani Onninen ◽  
Ville Tengvall
Keyword(s):  
Unit Ball ◽  
Modulus Of Continuity ◽  
Distortion Function ◽  
Open Set ◽  
Local Modulus

Let be an open set in ℝn and suppose that is a Sobolev homeomorphism. We study the regularity of f–1 under the Lp-integrability assumption on the distortion function Kf. First, if is the unit ball and p > n – 1, then the optimal local modulus of continuity of f–1 is attained by a radially symmetric mapping. We show that this is not the case when p ⩽ n – 1 and n ⩾ 3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for ∣Df–1∣ in terms of the Lp-integrability assumptions of Kf.

Dynamics of quasiregular mappings with constant complex dilatation

2014 ◽  
Vol 36 (2) ◽  
pp. 514-549 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
ROB FRYER
Keyword(s):  
Complex Dynamics ◽  
Quasiregular Mapping ◽  
Quadratic Polynomial ◽  
Complex Dilatation ◽  
Quasiregular Mappings ◽  
Quadratic Polynomials ◽  
Linear Map ◽  
External Rays ◽  
Constant Complex

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.

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