A continuity result on quadratic matings with respect to parameters of odd denominator rationals

2018 ◽  
Vol 167 (02) ◽  
pp. 369-388
Author(s):  
LIANGANG MA
Keyword(s):  
Mandelbrot Set ◽  
Hyperbolic Polynomial ◽  
Hyperbolic Polynomials ◽  
Hyperbolic Component ◽  
Continuity Result

AbstractIn this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one, is(1/π)logn. This result is true for allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

Very hyperbolic polynomials in one variable

2005 ◽  
Vol 135 (4) ◽  
pp. 833-844 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Real Variable ◽  
Real Polynomial ◽  
Hyperbolic Polynomial ◽  
Geometric Properties ◽  
Hyperbolic Polynomials ◽  
Real Roots

A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters ai for which the polynomial xn + a1xn−1 + … + an is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a1 = 0, a2 ≥ −1}.

scholarly journals Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2519
Author(s):  
Young-Hee Geum ◽  
Young-Ik Kim
Keyword(s):  
Computational Geometry ◽  
Euclidean Plane ◽  
Theoretical Background ◽  
Mandelbrot Set ◽  
Boundary Line ◽  
Computational Results ◽  
Boundary Equation ◽  
Geometrical Properties ◽  
Cubic Equation ◽  
Hyperbolic Component

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical properties. The budding point of the period-3k component, which is born on the boundary of the period-3 component, and its relevant period-3k points are theoretically obtained by means of Cardano’s formula for the cubic equation. In addition, computational results are presented in tables and figures to support the theoretical background of this paper.

Very hyperbolic polynomials in one variable

2005 ◽  
Vol 135 (4) ◽  
pp. 833-844
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Real Variable ◽  
Real Polynomial ◽  
Hyperbolic Polynomial ◽  
Geometric Properties ◽  
Hyperbolic Polynomials ◽  
Real Roots

A real polynomial in one real variable is called hyperbolic if it has only real roots. The polynomial f is called a primitive of order ν of the polynomial g if f(ν) = g. A hyperbolic polynomial is called very hyperbolic if it has hyperbolic primitives of all orders. In the paper we prove some geometric properties of the set D of values of the parameters ai for which the polynomial xn + a1xn−1 + … + an is very hyperbolic. In particular, we prove the Whitney property (the curvilinear distance to be equivalent to the Euclidean one) of the set D ∩{a1 = 0, a2 ≥ −1}.

Describing quadratic Cremer point polynomials by parabolic perturbations

1998 ◽  
Vol 18 (3) ◽  
pp. 739-758 ◽  
Author(s):  
DAN ERIK KRARUP SØRENSEN
Keyword(s):  
Infinite Order ◽  
Julia Set ◽  
Main Idea ◽  
Mandelbrot Set ◽  
Basic Tool ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
External Rays ◽  
External Ray ◽  
Single Comb

We describe two infinite-order parabolic perturbation procedures yielding quadratic polynomials having a Cremer fixed point. The main idea is to obtain the polynomial as the limit of repeated parabolic perturbations. The basic tool at each step is to control the behaviour of certain external rays.Polynomials of the Cremer type correspond to parameters at the boundary of a hyperbolic component of the Mandelbrot set. In this paper we concentrate on the main cardioid component. We investigate the differences between two-sided (i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove the existence of polynomials having an explicitly given external ray accumulating both at the Cremer point and at its non-periodic preimage. We think of the Julia set as containing a ‘topologist's double comb’.In the one-sided case we prove a weaker result: the existence of polynomials having an explicitly given external ray accumulating at the Cremer point, but having in the impression of the ray both the Cremer point and its other preimage. We think of the Julia set as containing a ‘topologist's single comb’.By tuning, similar results hold on the boundary of any hyperbolic component of the Mandelbrot set.

Accumulation theorems for quadratic polynomials

1996 ◽  
Vol 16 (3) ◽  
pp. 555-590 ◽  
Author(s):  
Dan Erik Krarup Sørensen
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Hyperbolic Component ◽  
Non Local ◽  
External Ray ◽  
The One ◽  
Image Position ◽  
Symmetrical Points

AbstractWe consider the one-parameter family of quadratic polynomials:i.e. monic centered quadratic polynomials with an indifferent fixed point αtand prefixed point −αt. LetAt, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to aGδ-dense subset ofthere is an explicitly given external ray accumulating onAt. In the caseAt= {±αt} the theorem is known as theDouady accumulation theorem.Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the denseGδsets arising from the first two possible choices ofAtare the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αtand –αtmay be joined by an arc in the Julia set.

scholarly journals Quadrature domains and the real quadratic family

2021 ◽  
Vol 25 (6) ◽  
pp. 104-125
Author(s):  
Kirill Lazebnik
Keyword(s):  
Congruence Subgroup ◽  
Mandelbrot Set ◽  
Quadrature Domain ◽  
Symmetric Polynomials ◽  
Schwarz Function ◽  
Content Type ◽  
Quadrature Domains ◽  
Hyperbolic Component ◽  
Simply Connected ◽  
Real Quadratic

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of P S L ( 2 , Z ) \mathrm {PSL}(2,\mathbb {Z}) , and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

Julia sets converging to filled quadratic Julia sets

2012 ◽  
Vol 34 (1) ◽  
pp. 171-184 ◽  
Author(s):  
ROBERT T. KOZMA ◽  
ROBERT L. DEVANEY
Keyword(s):  
Singular Perturbations ◽  
Julia Set ◽  
Julia Sets ◽  
Quadratic Polynomial ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Hausdorff Topology ◽  
Quadratic Map ◽  
Hyperbolic Component

AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

scholarly journals Equivalence between subshrubs and chaotic bands in the Mandelbrot set

2006 ◽  
Vol 2006 ◽  
pp. 1-25 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
G. Alvarez ◽  
D. Arroyo ◽  
F. Montoya
Keyword(s):  
Canonical Form ◽  
Mandelbrot Set ◽  
One Dimensional ◽  
Transformation Rules ◽  
Hyperbolic Component ◽  
Inductive Study ◽  
The One ◽  
Do So

We study in depth the equivalence between subshrubs and chaotic bands in the Mandelbrot set. In order to do so, we introduce the rules for chaotic bands and the rules for subshrubs, as well as the transformation rules that allow us to interchange them. From all the denominations of a chaotic band, we show the canonical form; that is, the one associated to the hyperbolic component that generates such a chaotic band. Starting from the study of the one-dimensional route, we fulfil an inductive study that gives a generalization of the shrub concept.

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