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.

LIMITING DYNAMICS FOR THE COMPLEX STANDARD FAMILY

1995 ◽  
Vol 05 (03) ◽  
pp. 673-699 ◽  
Author(s):  
NÚRIA FAGELLA
Keyword(s):  
Cross Sections ◽  
Julia Set ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Dimensional Parameter ◽  
Quadratic Polynomials ◽  
New Family ◽  
Standard Family ◽  
The Family ◽  
Characteristic Features

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

scholarly journals Multicorns are not path connected

2014 ◽  
Author(s):  
John Hamal Hubbard ◽  
Dierk Schleicher
Keyword(s):  
Julia Set ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Unicritical Polynomials ◽  
Path Connected ◽  
Special Case ◽  
Connectedness Locus

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

scholarly journals A landing theorem for entire functions with bounded post-singular sets

2020 ◽  
Vol 30 (6) ◽  
pp. 1465-1530
Author(s):  
Anna Miriam Benini ◽  
Lasse Rempe
Keyword(s):  
Entire Function ◽  
Periodic Point ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Polynomial ◽  
Landing Point ◽  
External Rays ◽  
Singular Sets ◽  
External Ray ◽  
Polynomial Dynamics

AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
Author(s):  
A. CONNES ◽  
E. MCDONALD ◽  
F. SUKOCHEV ◽  
D. ZANIN
Keyword(s):  
Hausdorff Dimension ◽  
Jordan Curve ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Trace Theorem ◽  
Quadratic Polynomials ◽  
Full Proof

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

scholarly journals A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
M. Romera ◽  
G. Pastor ◽  
A. B. Orue ◽  
A. Martin ◽  
M.-F. Danca ◽  
...  
Keyword(s):  
Computer Programs ◽  
Mandelbrot Set ◽  
Floating Point ◽  
Drawing Tool ◽  
Floating Point Arithmetic ◽  
External Rays ◽  
External Ray ◽  
Floating Point Computation ◽  
Point Arithmetic ◽  
Drawing Program

The external rays of the Mandelbrot set are a valuable graphic tool in order to study this set. They are drawn using computer programs starting from the Böttcher coordinate. However, the drawing of an external ray cannot be completed because it reaches a point from which the drawing tool cannot continue drawing. This point is influenced by the resolution of the standard for floating-point computation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmetic is the most widely used standard for floating-point computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 754-2008 Standard in order to draw external rays. When the drawing is not possible, due to a lack of resolution of this standard, we introduce a method to draw external rays based on the escape lines and Bézier curves.

scholarly journals Total disconnectedness of Julia sets of random quadratic polynomials

2021 ◽  
pp. 1-17
Author(s):  
KRZYSZTOF LECH ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical System ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Large Disc ◽  
Fatou Sets ◽  
Autonomous Version ◽  
Composition Of Functions ◽  
Totally Disconnected

Abstract For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

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 $z^2 + c$.

Bifurcations of dynamic rays in complex polynomials of degree two

1992 ◽  
Vol 12 (3) ◽  
pp. 401-423 ◽  
Author(s):  
Pau Atela
Keyword(s):  
Fixed Points ◽  
Julia Set ◽  
Period Doubling ◽  
Mandelbrot Set ◽  
Complex Polynomials ◽  
Circle Maps ◽  
External Rays ◽  
The Family ◽  
Parameter Values ◽  
Dynamic Plane

AbstractIn the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M, the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M. We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M, in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.

scholarly journals FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

2011 ◽  
Vol 21 (01) ◽  
pp. 77-99 ◽  
Author(s):  
YI-CHIUAN CHEN ◽  
TOMOKI KAWAHIRA ◽  
HUA-LUN LI ◽  
JUAN-MING YUAN
Keyword(s):  
Differential Equations ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Continuous Family ◽  
Coupled Differential Equations ◽  
External Rays ◽  
Period Orbit ◽  
External Angle ◽  
Integral Curves

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

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