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.

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 Some Fixed Points Results of Quadratic Functions in Split Quaternions

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Young Chel Kwun ◽  
Mobeen Munir ◽  
Waqas Nazeer ◽  
Shin Min Kang
Keyword(s):  
Fixed Points ◽  
Quadratic Polynomial ◽  
Quadratic Functions ◽  
Split Quaternion ◽  
Quadratic Polynomials ◽  
Split Quaternions

We attempt to find fixed points of a general quadratic polynomial in the algebra of split quaternion. In some cases, we characterize fixed points in terms of the coefficients of these polynomials and also give the cardinality of these points. As a consequence, we give some simple examples to strengthen the infinitude of these points in these cases. We also find the roots of quadratic polynomials as simple consequences.

scholarly journals ON STABLE QUADRATIC POLYNOMIALS

2012 ◽  
Vol 54 (2) ◽  
pp. 359-369 ◽  
Author(s):  
OMRAN AHMADI ◽  
FLORIAN LUCA ◽  
ALINA OSTAFE ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Quadratic Polynomial ◽  
Quadratic Polynomials ◽  
Polynomials Over Finite Fields ◽  
Characteristic 2 ◽  
The Stability ◽  
Almost All

AbstractWe recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

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 Non-Computable Impressions of Computable External Rays of Quadratic Polynomials

2014 ◽  
Vol 335 (2) ◽  
pp. 739-757 ◽  
Author(s):  
Ilia Binder ◽  
Cristobal Rojas ◽  
Michael Yampolsky
Keyword(s):  
Quadratic Polynomials ◽  
External Rays

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 Julia Set with Trigonometric Function

2019 ◽  
Vol 8 (2S7) ◽  
pp. 115-119
Keyword(s):  
Complex Number ◽  
Complex Dynamics ◽  
Julia Set ◽  
Julia Sets ◽  
New Class ◽  
Iteration Function ◽  
Cubic Polynomials ◽  
Constant Complex ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Julia sets are generated by initializing a complex number z = x + yi where z is then iterated using the iteration function fc (z)= zn 2 + c, where n indicates the number of iteration and c is a constant complex number. Recently, study of cubic Julia sets was introduced in Noor Orbit (NO) with improved escape criterions for cubic polynomials. In this paper, we investigate the complex dynamics of different functions and apply the iteration function to generate an entire new class of Julia sets. Here, we introduce different types of orbits on cubic Julia sets with trigonometric functions. The two functions to investigate from Julia sets are sine and cosine functions.

scholarly journals CUBIC AND QUADRATIC POLYNOMIAL ON JULIA SET WITH TRIGONOMETRIC FUNCTION

2018 ◽  
Vol 18 (2) ◽  
pp. 103
Author(s):  
Jullia Titaley ◽  
Tohap Manurung ◽  
Henriette D Titaley
Keyword(s):  
Complex Number ◽  
Complex Dynamics ◽  
Polynomial Function ◽  
Julia Set ◽  
Trigonometric Function ◽  
Quadratic Polynomial ◽  
Cosine Function ◽  
New Class ◽  
Iterated Function ◽  
Iteration Function

CUBIC AND QUADRATIC POLYNOMIAL ON JULIA SET WITH TRIGONOMETRIC FUNCTIONABSTRACTJulia set are defined by iterating a function of a complex number and is generated from the iterated function . We investigate in this paper the complex dynamics of different functions and applied iteration function system to generate an entire new class of julia set. The purpose of this research is to make variation of Cubic and Quadratic polynomial on Julia Set and the two obvious to investigate from julia set are Sine and Cosine function. The results thus obtained are innovative and studies about different behavior of two basic trigonometry.Keywords : Julia Set, trigonometric function, polynomial function  POLINOMIAL  KUBIK DAN KUADRATIK PADA HIMPUNAN JULIA DENGAN FUNGSI TRIGONOMETRI ABSTRAKHimpunan Julia didefiniskan oleh fungsi iterasi dari bilangan kompleks dan dibangkitkan dari fungsi iterasi . Kami melakukan penelitian dalam penulisan ini tentang sistem dinamik kompleks dari fungsi yang berbeda dengan iterasi yang diterapkan untuk menghasilkan kelas baru dari himpunan Julia. Tujuan dari penelitian ini adalah untuk membuah kelas baru himpunan Julia dengan fungsi polinomial kubik dan kuadratik dengan fungsi sinus dan kosinus. Hasil akhir dari penelitian ini ada menemukan inovatif baru dari himpunan Julia dengan menggunakan dua fungsi trigonometri.Kata kunci: Julia set, fungsi trigonometri, fungsi polinomial

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