Filled Julia Set Research Articles

Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

Tricomplex Distance Estimation for Filled-In Julia Sets and Multibrot Sets

In this article, we present a distance estimation formula that can be used to ray trace 3D slices of the filled-in Julia sets and the Multibrot sets generated by the tricomplex polynomials of the form [Formula: see text] where [Formula: see text] is any integer greater than [Formula: see text].

Boundaries of Filled Julia Sets in Generalized Jungck Mann Orbit

Boundaries of Filled Julia Sets in Generalized Jungck Mann Orbit

ŁS condition for filled Julia sets in $$\mathbb {C}$$ C

In this article, we derive an inequality of {\L}ojasiewicz-Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by $dist$ the euclidian distance in $\mathbb{C}$, we show that the Green function $G_K$ of the filled Julia set $K$ of a polynomial such that $\mathring{K}\neq \emptyset$ satisfies the so-called {\L}S condition $\displaystyle G_A\geq c\cdot dist(\cdot, K)^{c'}$ in a neighborhood of $K$, for some constants $c,c'>0$. Relatively few examples of compact sets satisfying the {\L}S condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. The fact that filled Julia sets satisfy the {\L}S condition may seem surprising, since they are in general very irregular. In order to prove our main result, we define and study the set of obstruction points to the {\L}S condition. We also prove, in dimension $n\geq 1$, that for a polynomially convex and L-regular compact set of non empty interior, these obstruction points are rare, in a sense which will be specified.

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed in [27] and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in C. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.

Julia Sets of Orthogonal Polynomials

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K.

Non-Archimedean Hénon maps, attractors, and horseshoes

We study the dynamics of the Henon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Henon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Henon maps over $${\mathbb {Q}}_3$$ , this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over $${\mathbb {Q}}_5$$ and $${\mathbb {Q}}_7$$ as well. On a different region of the parameter space, we show that the Henon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols.

Julia sets for Fibonacci endomorphisms of ℝ2

ABSTRACTWe study the dynamics of the family f c(x, y) = (xy + c, x) of endomorphisms of , where c is a real parameter. We investigate several topological properties of the forward and backward filled Julia sets. Then, through the study of adapted dynamical filtrations of the plane, we prove that for the interval of parameters given by 0 < c <, these two filled Julia sets can be described explicitly as finite unions of invariant manifolds.

Polynomial semiconjugacies, decompositions of iterations, and invariant curves

We study the functional equation A◦ X = X ◦ B, where A, B, and X are polynomials over C. Using results of [12] about polynomials sharing preimages of compact sets, we show that for given B its solutions may be described in terms of the filled-in Julia set of B. On this base, we prove a number of results describing a general structure of solutions. The results obtained imply in particular the result of Medvedev and Scanlon [4] about invariant curves of maps F : C 2 → C 2 of the form (x,y) → (f(x),f(y)),

Fekete polynomials and shapes of Julia sets

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

Probabilistic approximation of partly filled-in composite Julia sets

We study properties of the metric space of pluriregular sets and of contractions on that space induced by finite families of proper polynomial mappings of several complex variables. In particular, ...

Rigorous bounds for polynomial Julia sets

We present an algorithm for computing images of polynomial Julia setsthat are reliable in the sensethat they carry mathematical guarantees againstsampling artifacts and rounding errors in floating-point arithmetic.We combine cell mappingbased on interval arithmeticwith label propagation in graphsto avoid function iteration and rounding errors.As a result,our algorithm avoids point sampling and canreliably classify entire rectanglesin the complex plane as being on either side of the Julia set.The union of the rectangles that cannot be so classified is guaranteedto contain the Julia set.Our algorithm computes a refinable quadtree decomposition of the complex planeadapted to the Julia set which can be used for rendering and forapproximating geometric propertiessuch asthe area of the filled Julia set andthe fractal dimension of the Julia set.

Complete invariance and compactness of the filled Julia sets of polynomial maps in mathbb {C}^N

We gather known results on escape radii and add some facts and examples concerning the subject of complete invariance and compactness of filled Julia sets.

Convergence properties of the Gronwall area formula for quadratic Julia sets

Using parabolic enrichment, it is shown that Gronwall area formula for the filled Julia set along the boundary of the main cardioid of the Mandelbrot set cannot be well approximated by replacing it by a finite sum. A revised version of this article will appear in the Journal of the London Mathematical Society.

A Case of the Dynamical André–Oort Conjecture

We prove a special case of the Dynamical André–Oort Conjecture formulated by Baker and DeMarco [2]. For any integer |$d\ge 2$|⁠, we show that for a rational plane curve |$C$| parameterized by |$(t, h(t))$| for some nonconstant polynomial |$h\in \mathbb {C}[z]$|⁠, if there exist infinitely many points |$(a,b)\in C(\mathbb {C})$| such that both |$z^d+a$| and |$z^d+b$| are postcritically finite maps, then |$h(z)=\xi z$| for a |$(d-1)$|st root of unity |$\xi $|⁠. As a key step in our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial |$g\in \mathbb {C}[z]$|⁠.Communicated by Prof. Umberto Zannier

Continuity of core entropy of quadratic polynomials

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends the entropy theory for real unimodal maps to complex polynomials: the real segment is replaced by an invariant tree, known as the Hubbard tree, which lies inside the filled Julia set. We prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle in parameter space, answering a question of Thurston.

Emerging Julia Sets

SummaryFunctions from the complex plane to itself are difficult to visualize; we consider the real and imaginary projections. In this paper, we explore the connections between the graphs of the real and imaginary parts of various complex functions and their corresponding filled Julia sets. We begin by examining the family of complex quadratic functions.We then expand our results to a broader collection of rational maps, including functions whose Julia sets form a Cantor set of simple closed curves, checkerboards, and a perturbed rat.

The Real Dynamics of Bieberbach’s Example

Bieberbach constructed, in 1933, domains in $${\mathbb {C}}^2$$ which were biholomorphic to $${\mathbb {C}}^2$$ but not dense. The existence of such domains was unexpected. The special domains Bieberbach considered are basins of attraction of a cubic Henon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis. In this paper, the boundaries of the real sections of Bieberbach’s domains will be calculated explicitly as the stable manifolds of the saddle points. The real filled Julia sets and the real Julia sets of Bieberbach’s map will also be calculated explicitly and illustrated with computer generated graphics. Basic differences between real and the complex dynamics will be shown.

Parabolic-like mappings

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

Visualizing Research of Construction of Planar Dynamic Systems with Truncated Fourier Series

The paper constructs the planar dynamic systems with truncated Fourier series. We propose a method to generate planar tiling images of the iterative mapping with truncated Fourier series. It constructed the planar tessellations of planar dynamic systems with cyclic windows. The iterative mapping is constructed. Then we generate cyclic windows of the dynamic plane with these mappings. Filled-in Julia sets are created by generating the coordinates of any cyclic windows. The images in different windows are continuous. We can choose any cyclic windows as the basic computing region and stretch it into a square, which is transferred to the plane to compose the new form chaos attractor and filled-in Julia sets images in the Euclidian plane. Experimental results show the effectiveness of the proposed approach.