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.

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.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

2016 ◽  
Vol 26 (01) ◽  
pp. 1630002 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Parameter Space ◽  
Complex Structure ◽  
Three Dimensional ◽  
Periodic Points ◽  
Mandelbrot Set ◽  
Three Dimensions ◽  
Quadratic Mapping ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Nilpotent Elements

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

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 ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES?

2008 ◽  
Vol 23 (22) ◽  
pp. 3613-3684 ◽  
Author(s):  
V. DOLOTIN ◽  
A. MOROZOV
Keyword(s):  
Phase Transition ◽  
Forest Structure ◽  
Qualitative Difference ◽  
Three Dimensional ◽  
Geometric Approach ◽  
Mandelbrot Set ◽  
The Family ◽  
Mandelbrot Sets ◽  
Special Case

Direct look at the celebrated "chaotic" Mandelbrot Set (in Fig. 1) immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific forest structure. In the paper arXiv:hep-th/0501235, a systematic algebro-geometric approach was developed to the study of generic Mandelbrot sets, but emergency of nearly ideal circles in the special case of the family x2 + c was not fully explained. In the present, paper, the shape of the elementary constituents of Mandelbrot Set is explicitly calculated, and difference between the shapes of root and descendant domains (cardioids and circles respectively) is explained. Such qualitative difference persists for all other Mandelbrot sets: descendant domains always have one less cusp than the root ones. Details of the phase transition between different Mandelbrot sets are explicitly demonstrated, including overlaps between elementary domains and dynamics of attraction/repulsion regions. Explicit examples of three-dimensional sections of Universal Mandelbrot Set are given. Also a systematic small-size approximation is developed for evaluation of various Feigenbaum indices.

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.

Equilibria, stability and bifurcations of rotating columns of fluid subjected to planar disturbances

1991 ◽  
Vol 433 (1887) ◽  
pp. 81-99 ◽  
Keyword(s):  
Three Dimensional ◽  
Power Series Method ◽  
Element Analysis ◽  
Hydrodynamic Analysis ◽  
Drop Shape ◽  
Axisymmetric Mode ◽  
Polar Representation ◽  
New Family ◽  
The Family ◽  
Stability And Bifurcations

Long gyrostatically rotating cylindrical drops held together by surface tension are amenable to conventional bifurcation analysis as well as newer, computer-aided methods of analysis, and so are useful prototypes of three-dimensional drops. Instability to Rayleigh’s axisymmetric mode is set aside and effects of translationally symmetric (planar) disturbances are investigated. The shapes and stability of drops on and near the family of perfect cylinders are determined by means of the power series method of Millman & Keller. Nonlinearly distorted shapes and their stability are calculated by Galerkin or finite-element analysis, using either ( a ) a polar or ( b ) a composite polar and cartesian representation of drop shape. The disadvantage of using single-coordinate representation of drop shapes near break-up is brought out. The new results show that a family of symmetric two-lobed shapes bifurcates from the main family of perfectly cylindrical shapes when the rotation rate attains a critical value, in accord with the linearized hydrodynamic analysis of Hocking. Moreover, a new family of asymmetric two-lobed shapes is uncovered that bifurcates from and rejoins the family of symmetric two-lobed shapes, using a polar representation of drop shape. Plainly, the new shape family is the two-dimensional analog of the family of three-dimensional capillary peanuts discovered by Brown & Scriven, who used a spherical polar representation of drop shape. By way of contrast, the results obtained using a composite polar and cartesian representation of drop shape show that the gyrostatic family of symmetric two-lobed shapes does not exchange stability with a family of asymmetric shapes and eventually breaks up by becoming a family of self-intersecting shapes.

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.

CLASSIFICATION OF DENSITY MATRICES FOR A FOUR-STATE SYSTEM

2009 ◽  
Vol 07 (01) ◽  
pp. 323-348 ◽  
Author(s):  
ISTOK MENDAŠ
Keyword(s):  
Cross Sections ◽  
Complex Structure ◽  
Three Dimensional ◽  
Monte Carlo Sampling ◽  
State Density ◽  
State System ◽  
Density Matrices ◽  
Dimensional Parameter ◽  
Parameter Case

Properties and structure of the 15-dimensional parameter space of four-state density matrices are examined using the SU(4) generator expansion. Appropriate classification of one-, two- and three-parameter density matrices is obtained, based on the sameness of the characteristic polynomial of density matrices belonging to a given type. It is found that in the one parameter case of 15 different density matrices only three distinct types exist, while in the two parameter case 105 different density matrices group into 11 distinct types. In the three parameter case appropriate classification of 455 different density matrices into 44 types is determined. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are illustrated by randomly drawing matrices for several types of density matrices, providing some insight into the intricate and complex structure of the space of density matrices for a four-state system. Positions of the representative points corresponding to the pure states are found for all types. Global properties of observables are determined by generating, by the Monte Carlo sampling method, and averaging over nearly all density matrices pertaining to a given type.

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