scholarly journals The parameter space of cubic laminations with a fixed critical leaf

2016 ◽  
Vol 37 (8) ◽  
pp. 2453-2486
Author(s):  
ALEXANDER BLOKH ◽  
LEX OVERSTEEGEN ◽  
ROSS PTACEK ◽  
VLADLEN TIMORIN
Keyword(s):  
Parameter Space ◽  
Mandelbrot Set ◽  
Quadratic Invariant ◽  
The Family ◽  
Combinatorial Model ◽  
Cubic Polynomials ◽  
Quadratic Case

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

THE PARAMETER SPACE OF A QUADRATIC FAMILY OF POLYNOMIAL MAPS OF ℂ2

2008 ◽  
Vol 19 (05) ◽  
pp. 541-556
Author(s):  
ALINA ANDREI
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Mandelbrot Set ◽  
General Context ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Regular Maps ◽  
The Family ◽  
The One ◽  
Polynomial Family

In this paper, we study the parameter space of the quadratic polynomial family fλ,μ(z, w) = (λz + w2, μw + z2), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family fλ,μ, we prove that the sum of the Lyapunov exponents is continuous.

ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

2003 ◽  
Vol 13 (10) ◽  
pp. 2825-2844 ◽  
Author(s):  
SHIZUO NAKANE ◽  
DIERK SCHLEICHER
Keyword(s):  
Parameter Space ◽  
Bifurcation Diagrams ◽  
The Family ◽  
Real Analytic ◽  
Cubic Polynomials

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

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 Smooth Decomposition of Generalized Fatou Set Explains Smooth Structure in Generalized Mandelbrot Set

1988 ◽  
Vol 43 (1) ◽  
pp. 14-16 ◽  
Author(s):  
J. Peinke ◽  
J. Parisi ◽  
B. Röhricht ◽  
O. E. Rössler ◽  
W. Metzler
Keyword(s):  
Parameter Space ◽  
Initial Conditions ◽  
Logistic Map ◽  
Mandelbrot Set ◽  
Fatou Set ◽  
Smooth Structure ◽  
Mandelbrot Sets ◽  
Generalized Mandelbrot Sets

Abstract Generalized Mandelbrot sets arise in perturbed (non-analytic) versions of the complex logistic map. Numerically, it contains smooth portions as shown previously. To exclude that this result is specific to particular initial conditions only, the structure of the analogue to the Fatou set is looked at in the region in question. The set of non-divergent points is being "eaten up" by a smooth invading boundary. Therefore, the same type of decomposition applies independent of position in parameter space, in the region in question.

scholarly journals The iteration of cubic polynomials Part I: The global topology of parameter space

1988 ◽  
Vol 160 (0) ◽  
pp. 143-206 ◽  
Author(s):  
Bodil Branner ◽  
John H. Hubbard
Keyword(s):  
Parameter Space ◽  
Global Topology ◽  
Cubic Polynomials

scholarly journals Reduction of dynatomic curves

2018 ◽  
Vol 39 (10) ◽  
pp. 2717-2768 ◽  
Author(s):  
JOHN R. DOYLE ◽  
HOLLY KRIEGER ◽  
ANDREW OBUS ◽  
RACHEL PRIES ◽  
SIMON RUBINSTEIN-SALZEDO ◽  
...  
Keyword(s):  
Complex Dynamics ◽  
Uniform Boundedness ◽  
Mandelbrot Set ◽  
Modular Curves ◽  
Good Reduction ◽  
One Dimensional ◽  
Formula One ◽  
Polynomial Maps ◽  
Ramification Theory ◽  
The Family

In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$, where $m\geq 2$. We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$. The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$, a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$. In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$: one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$, yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $Y_{1}(n)$ for several primes dividing $D_{n}$ when $n=7,8,11$, and $f_{c}(x)=x^{2}+c$. The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

scholarly journals Slices of the parameter space of cubic polynomials

2021 ◽  
Author(s):  
Alexander Blokh ◽  
Lex Oversteegen ◽  
Vladlen Timorin
Keyword(s):  
Parameter Space ◽  
Cubic Polynomials

Connectivity of the Mandelbrot set for the family of renormalization transformations

2010 ◽  
Vol 53 (3) ◽  
pp. 849-862 ◽  
Author(s):  
XiaoGuang Wang ◽  
WeiYuan Qiu ◽  
YongCheng Yin ◽  
JianYong Qiao ◽  
JunYang Gao
Keyword(s):  
Mandelbrot Set ◽  
The Family

VISUALIZATION OF CHAOTIC DYNAMICAL SYSTEMS BASED ON MANDELBROT SET METHODOLOGY

Fractals ◽  
2008 ◽  
Vol 16 (01) ◽  
pp. 89-97 ◽  
Author(s):  
JIN CHENG ◽  
JIANRONG TAN ◽  
CHUNBIAO GAN
Keyword(s):  
Parameter Space ◽  
Chaotic Systems ◽  
Mandelbrot Set ◽  
Control Parameters ◽  
Limited Information ◽  
Visualization Technique ◽  
Lyapunov Spectra ◽  
The Given ◽  
Local Nature

Although there are lots of methods to analyze chaotic systems, they tend to be of local nature and reveal only limited information about a certain group of control parameters. This paper explores a visualization technique on the basis of Mandelbrot set (M-set) methodology to give an overall view of a chaotic system's dynamical performance in the parameter space. Firstly, the Lyapunov spectra with regard to different points in the parameter space are calculated, according to which the color of these points is defined. Then points in the given parameter space are mapped to the computer screen and a colorful image named Lyapunov distribution map (LDM) is generated, which conveys a wealth of information about a system's dynamic behavior across variations in its control parameters. The results of visualizing two typical chaotic systems proved the feasibility and validity of this technique. The study suggests that desired dynamical performance of a system can be conveniently achieved by selecting a point from some color region in its LDM and adjusting the control parameters according to the point value. Furthermore, there is no need for any prior knowledge of the system under evaluation and the complicated mathematical analysis before the formulation of appropriate rules for a new system can also be avoided since uniform rules are employed in the classification of parameter points for any system and the LDM is generated via the same algorithm.

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