Filled Julia Set Research Articles

FRACTAL ORACLE NUMBERS

Consider orbits [Formula: see text] of the fractal iterator [Formula: see text], [Formula: see text], that start at initial points [Formula: see text], where [Formula: see text] is the set of all rational complex numbers (their real and imaginary parts are rational) and [Formula: see text] consists of all such [Formula: see text] whose complexity does not exceed some complexity parameter value [Formula: see text] (the complexity of [Formula: see text] is defined as the number of bits that suffice to describe the real and imaginary parts of [Formula: see text] in lowest form). The set [Formula: see text] is a bounded-complexity approximation of the filled Julia set [Formula: see text]. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number [Formula: see text] is the analog of a program [Formula: see text], an iterator [Formula: see text] is analogous to a universal Turing machine [Formula: see text] which executes program [Formula: see text], and an unbounded orbit [Formula: see text] is analogous to an execution of a program [Formula: see text] on [Formula: see text] that halts. We define a real number [Formula: see text] which resembles Chaitin’s [Formula: see text] number, where, instead of being based on all programs [Formula: see text] whose execution on [Formula: see text] halts, it is based on all rational complex numbers [Formula: see text] whose orbits under [Formula: see text] are unbounded. Hence, similar to Chaitin’s [Formula: see text] number, [Formula: see text] acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set [Formula: see text]. We present a procedure that, when given [Formula: see text] and [Formula: see text], it uses [Formula: see text] to generate [Formula: see text]. Several numerical examples of sets that estimate [Formula: see text] are presented.

Fractal viewpoint in supply chain price competition

This paper investigates the dynamical behaviors of the classical dual-channel supply chain from the perspective of fractal theory. First, the definition of filled Julia set in supply chain price competition model is introduced. Second, according to the periodicities of the attractors, the filled Julia set is divided into several sub basins. On this basis, the importance of mixed coexisting attraction basins’ connectivity is further illustrated. Then, the inner characteristics of filled Julia sets under different online preferences are analyzed, which indicates that online preference increase has negative influence on filled Julia sets’ structures. Finally, for the unstable cases caused by high online preference, we adopt three control strategies and analyze their control effects by quantifying the connectivity restorations. The results suggest that the fractal viewpoint is beneficial for a global observation of the prices evolutions in different initial states, and could also evaluate the market status more comprehensively.

On dynamics of asymptotically minimal polynomials

We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we show that the filled Julia sets converge to polynomial convex hull of E in the Klimek topology.

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

Rigidity of Newton dynamics

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set.In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”.Our main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.

The Dynamics of Complex Box Mappings

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);the QC-Criterion and the Spreading Principle from KSS. The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:puzzle pieces shrink to points,(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting.We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

On the correspondence of external angles under renormalization

Let P $P$ be a monic polynomial of degree D ⩾ 3 $D \geqslant 3$ whose filled Julia set K P $K_P$ has a non-degenerate periodic component K $K$ of period k ⩾ 1 $k \geqslant 1$ and renormalization degree 2 ⩽ d < D $2 \leqslant d<D$ . Let I = I K $I=I_K$ denote the set of angles θ $\theta$ on the circle T = R / Z ${\mathbb {T}}={\mathbb {R}}/{\mathbb {Z}}$ for which the (smooth or broken) external ray R θ P $R^P_\theta$ for P $P$ accumulates on ∂ K $\partial K$ . We prove the following. I $I$ is a compact set of Hausdorff dimension ⩽ log d / ( k log D ) < 1 $\leqslant \log d/(k \log D)<1$ and there is a continuous surjection Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ which semiconjugates θ ↦ D k θ ( mod Z ) $\theta \mapsto D^k \theta \ (\operatorname{mod} {\mathbb {Z}})$ on I $I$ to θ ↦ d θ ( mod Z ) $\theta \mapsto d \theta \ (\operatorname{mod} {\mathbb {Z}})$ on T ${\mathbb {T}}$ . Moreover, Π $\Pi$ is unique up to postcomposition with a power of the rotation θ ↦ θ + 1 / ( d − 1 ) ( mod Z ) $\theta \mapsto \theta +1/(d-1) \ (\operatorname{mod} {\mathbb {Z}})$ . Any hybrid conjugacy φ $\varphi$ between a renormalization of P ∘ k $P^{\circ k}$ on a neighborhood of K $K$ and a degree d $d$ monic polynomial Q $Q$ induces a semiconjugacy Π : I → T $\Pi : I \rightarrow {\mathbb {T}}$ as above with the property that for every θ ∈ I $\theta \in I$ the external ray R θ P $R^P_\theta$ has the same accumulation set as the curve φ − 1 ( R Π ( θ ) Q ) $\varphi ^{-1}(R^Q_{\Pi (\theta )})$ . In particular, R θ P $R^P_\theta$ lands at z ∈ ∂ K $z \in \partial K$ if and only if R Π ( θ ) Q $R^Q_{\Pi (\theta )}$ lands at φ ( z ) ∈ ∂ K Q $\varphi (z) \in \partial K_Q$ . The projection Π $\Pi$ is uniformly finite-to-one. In fact, the cardinality of each fiber of Π $\Pi$ is ⩽ D − d + 2 $\leqslant D-d+2$ , and the inequality is strict when the component K $K$ has period k = 1 $k=1$ . The upper bound in the above result is sharp. Using a new type of quasiconformal surgery, we construct polynomials P $P$ with a prescribed hybrid class near K = P ( K ) $K=P(K)$ and a prescribed set of D − d + 1 $D-d+1$ consecutive fixed points of θ ↦ D θ ( mod Z ) $\theta \mapsto D \theta \ (\operatorname{mod} {\mathbb {Z}})$ in the same fiber of Π $\Pi$ .

Geometric limit of Julia set of a family of rational functions with odd degree

For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by with . Also, when we show that the geometric limit of the Julia set and filled Julia set of the family exists and is the unit circle.

Visualization of Mandelbrot and Julia Sets of Möbius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

Filled Julia Sets of Chebyshev Polynomials

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set \(K\subset {\mathbb C}\) and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

Lacunary Möbius Fractals on the Unit Disk

Centered polygonal lacunary functions are a type of lacunary function that exhibit behaviors that set them apart from other lacunary functions, this includes rotational symmetry. This work will build off of earlier studies to incorporate the automorphism group of the open unit disk D, which is a subgroup of the Möbius transformations. The behavior, dimension, dynamics, and sensitivity of filled-in Julia sets and Mandelbrot sets to variables will be discussed in detail. Additionally, several visualizations of this three-dimensional parameter space will be presented.

Value distribution of derivatives in polynomial dynamics

AbstractFor every$m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in$\mathbb {C}\setminus \{0\}$under the$m$th order derivatives of the iterates of a polynomials$f\in \mathbb {C}[z]$of degree$d&gt;1$towards the harmonic measure of the filled-in Julia set offwith pole at$\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number fieldkfor a sequence of effective divisors on$\mathbb {P}^1(\overline {k})$having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of$\mathbb {C}^2$has a given eigenvalue.

Computing the Filled Julia Set

The Julia set is one of the most remarkable and exciting fields of dynamic mathematics Because it contains very beautiful graphics.Through this study we will highlight how the filled Julia set is computed. It is worth noting that there is an integrative work between computers and mathematical calculations to get great results to complete Julia Set filling process

Spectral Decimation of the Magnetic Laplacian on the Sierpinski Gasket: Solving the Hofstadter–Sierpinski Butterfly

The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights $\{\omega_{xy}\}_{xy\in E}$, satisfying the condition that $\omega_{xy}=\overline{\omega_{yx}}$ for every directed edge $xy$. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph ($SG$) where the magnetic fluxes equal $\alpha$ through the upright triangles, and $\beta$ through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map $\mathcal{U}$ involving non-rational functions, which takes into account $\alpha$, $\beta$, and the spectral parameter $\lambda$. In doing so we provide a quantitative answer to a question of Bellissard [Renormalization Group Analysis and Quasicrystals (1992)] on the relationship between the dynamical spectrum and the actual magnetic spectrum. Our main theorems lead to two applications. In the case $\alpha=\beta$, we demonstrate the approximation of the magnetic spectrum by the filled Julia set of $\mathcal{U}$, the Sierpinski gasket counterpart to Hofstadter's butterfly. Meanwhile, in the case $\alpha,\beta\in \{0,\frac{1}{2}\}$, we can compute the determinant of the magnetic Laplacian determinant and the corresponding asymptotic complexity.

Spectral Properties of Laplacians on Snowflake Domains and Filled Julia Sets

We present eigenvalue data and pictures of eigenfunctions of the classic and quadratic snowflake fractal and of quadratic filled Julia sets. Furthermore, we approximate the area and box-counting dimension of selected Julia sets to compare the eigenvalue counting function with the Weyl term.

Rays to renormalizations

Let $K_P$ be the filled Julia set of a polynomial $P$, and $K_f$ the filled Julia set of a renormalization $f$ of&nbsp;$P$. We show, loosely speaking, that there is a finite-to-one function $\lambda $ from the set of $P$-external rays having limit points

Filled Julia set of some class of Hénon-like maps

ABSTRACTIn this work we consider a class of endomorphisms of defined by , where is a real number and we prove that when , the forward filled Julia set of f is the union of stable manifolds of fixed and periodic points of f. We also prove that the backward filled Julia set of f is the union of unstable manifolds of the saddle fixed and periodic points of f.

Area-capacity-modulus inequality and area of the filled Julia set

In this paper, we prove the general area-modulus inequality$\\operatorname{area}(K)\\leq\\operatorname{area}(D){\\rm~e}^{-4\\pi\\operatorname{mod}(A)},$where $K$ is a full regular compact set in the plane, $D$ is a bounded simply connected domain containing $K$ and $A=D\\backslash~K$.Here the equality holds if and only if $A$ is a round annulus.Then we establish the area-capacity-modulus inequality$\\operatorname{area}(K)\\leq\\pi\\operatorname{cap}(K)^2{\\rm~e}^{2L-4\\pi\\operatorname{mod}(\\Lambda)}$for full regular compact set $K$ in the plane, where $L$ is the maximal critical value of the Greenfunction induced by $K$ and $\\Lambda$ is the tree (in set theory) induced by $K$.Here the equality holds if and only if $K$ is a closed disk or $\\operatorname{mod}(\\Lambda)=\\infty$.Using this inequality, we obtain an upper bound for the area of the filledJulia set $K(f)$ for the monic complex polynomial $f$ of degree $d$. That is,$\\operatorname{area}(K(f))\\leq\\pi~{\\rm~e}^{-\\frac{2qL}{d-q-1}},$where $L=\\max\\{G_f(c)\\mid~f&apos;(c)=0\\}$ and $q$ is the critical index for $f$ onthe set $G_f^{-1}(L)\\backslash~K(f)$.

Parabolic implosion for endomorphisms of $\mathbb C^2$

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.

Conformal equivalence of measures and dynamics of orthogonal polynomials

ABSTRACTWe introduce a notion of asymptotically orthonormal polynomials for a Borel measure μ with compact non-polar support in . Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials, see also Christiansen et al. [Julia sets of orthogonal polynomials, Potential Anal. 50(3) (2019), pp. 401–413]. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study the dynamics of sequences of asymptotically orthonormal polynomials for the pullback measure under affine mappings φ. We prove that the sequences of Julia sets and filled Julia sets of affine deformations of sequences of asymptotically orthonormal polynomials for μ also have the same convergence properties as the Julia sets and filled Julia sets of the orthonormal polynomials. This leads to theorems on the convergence properties of affine deformations of the family of iterates of any fixed monic centred polynomial and, in the case the polynomial is hyperbolic, on the corresponding family of affine parameter spaces.