Julia Set Research Articles

On the spatial Julia set generated by fractional Lotka-Volterra system with noise

Abstract This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system’s Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

Squeezing functions and Cantor sets

We construct large Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct large Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.

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.

Rearrangement groups of fractals

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson’s groups $F$, $T$, and $V$. Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite $\mathrm {CAT}(0)$ cubical complex, and we use this action to prove that certain rearrangement groups are of type $F_{\infty }$.

Attractor sets and Julia sets in low dimensions

If X X is the attractor set of a conformal IFS (iterated function system) in dimension two or three, we prove that there exists a quasiregular semigroup G G with a Julia set equal to X X . We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.

Construction of stable and globally convergent schemes for the matrix sign function

In this paper, we propose a dynamics-based approach to develop efficient matrix iterations for computing the matrix sign function. Considering an iterative method with unknown weight functions, our approach constructs a double sequence of weight functions so that for a quadratic polynomial with distinct roots, the Julia set of the method associated with each term of the sequence is the perpendicular bisector of the line segment from one root to another. Based on the sequence-generated methods, we introduce a double sequence of matrix iterations for computing the matrix sign function, and perform precise convergence and stability analyses of these iterations. We show that our matrix iterations contain the well-known globally convergent principal Padé family and its reciprocal as special cases. Based on the result of the numerical experiments, we found several of our methods that do not belong to the principal Padé family and its reciprocal, which are more efficient in computing the matrix sign function.

From Cantor to semi-hyperbolic parameters along external rays

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let c ^ \hat {c} be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each z = z ( c ) z = z(c) in the Julia set, the derivative d z ( c ) / d c dz(c)/dc is uniformly O ( 1 / | c − c ^ | ) O(1/\sqrt {|c-\hat {c}|}) when c c belongs to a parameter ray that lands on c ^ \hat {c} . We also characterize the degeneration of the dynamics along the parameter ray.

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].

Fast Growth Entire Functions Whose Escaping Set Has Hausdorff Dimension Two

The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.

Eremenko points and the structure of the escaping set

Much recent work on the iterates of a transcendental entire function f f has been motivated by Eremenko’s conjecture that all the components of the escaping set I ( f ) I(f) are unbounded. We prove several general results about the topological structure of I ( f ) I(f) including the fact that if I ( f ) I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of the fast escaping set. We give analogous results for the intersection of I ( f ) I(f) with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.

TOPOLOGICAL PROPERTIES OF FRACTAL JULIA SETS RELATED TO THE SIGNS OF REAL AND REACTIVE ELECTRIC POWERS

This paper focuses on applying fractal Julia sets to observe the topological properties related to the signs of the real and reactive electric powers. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Julia sets. The study considers three cases: the first study considers the change of real power when the reactive power is fixed; the second study deals with the change of the reactive power when the real power is fixed; and finally, the third study contemplates that both real and reactive powers change. Furthermore, the fractal diagrams of the power in the four quadrants of the complex plane are studied to identify the topological properties for each sign. A qualitative analysis of the diagrams helps identify that complex power loads present some fractal graphic patterns with respect to the signs considered in the different quadrants of the complex planes. The diagrams represented in the complex planes save a relation in the forms and structure with other points studied, concluding that the power is related to other figures in other quadrants. Thus, this result allows a new study of the behavior of power in an electrical circuit by showing a clear relation of the different fractal diagrams obtained by the Julia sets.

A structure theorem for semi-parabolic Hénon maps

Consider the parameter space Pλ⊂C2 of complex Hénon mapsHc,a(x,y)=(x2+c+ay,ax),a≠0, which have a semi-parabolic fixed point with one eigenvalue λ=e2πip/q. In this paper we provide a complete topological description of the Julia set and forward Julia set within a family of semi-parabolic Hénon maps, i.e. those Hénon maps from the curve Pλ which are perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in Pλ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J+. The forward Julia set J+ has a nice local description: inside a certain bidisk it is a trivial fiber bundle over Jp, the Julia set of the polynomial p, with fibers biholomorphic to a disk. The Julia set J is homeomorphic to the quotient of a solenoid by an equivalence relation and J=J⁎, the closure of the saddle periodic points.

Area and Hausdorff dimension of Sierpiński carpet Julia sets

We prove the existence of rational maps whose Julia sets are Sierpinski carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpinski carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $$s\in (1,2)$$, we prove the existence of Sierpinski carpet Julia sets having Hausdorff dimension exactly s.

Connectivity of the Julia sets of singularly perturbed rational maps

We consider a family of rational functions which is given by $$\begin{aligned} f_{\lambda }(z)=\frac{z^n(z^{2n}-\lambda ^{n+1})}{z^{2n}-\lambda ^{3n-1}}, \end{aligned}$$ where $$n\ge 2$$ and $$\lambda \in {\mathbb {C}}^*-\{\lambda :\lambda ^{2n-2}=1\}$$ . When $$\lambda \ne 0$$ is small, $$f_{\lambda }$$ can be seen as a perturbation of the unicritical polynomial $$z\mapsto z^n$$ . It was known that in this case the Julia set $$J(f_\lambda )$$ of $$f_\lambda $$ is a Cantor set of circles on which the dynamics of $$f_\lambda $$ is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that $$J(f_\lambda )$$ is disconnected.

Geometric limits of Julia sets and connectedness locus of the family of polynomials Pc(z) = zn + czk

ABSTRACTLet n and k be positive integers. For , we consider the connectedness locus of the family of polynomials , where c is a complex parameter. We show that the geometric limit of the connectedness locus sets , when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of .When parameter c belongs to the open unit disk, we show that the geometric limit of the Julia sets , when n tends to infinity, exists and is the unit circle. Finally, we have established some properties on the hyperbolic components of this family.

The Topology of the Set of Non-Escaping Endpoints

Abstract There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set of finite endpoints. It was recently shown that, for certain functions in the exponential family, there is a strong dichotomy between the topological properties of the set of endpoints that escape and those of the set of endpoints that do not escape. In this paper, we show that this result holds for large families of functions in the Eremenko–Lyubich class. We also show that this dichotomy holds for a family of functions, outside that class, which includes the much-studied Fatou function defined by $$\begin{equation*}f(z):= z + 1+ e^{-z}.\end{equation*}$$Finally, we show how our results can be used to demonstrate that various sets are spiders’ webs, generalising results such as those in [9].

Stochastic canonical heights

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.

On the dynamics of n-circle inversion

The article deals with singular perturbation of polynomial maps where is a complex parameter and n is the degree, which is a particular case of the family of rational maps known as McMullen maps. Our main result shows that even when the geometric limit of the Julia set converges to the unit circle or the annulus for a.e. Lebesgue , as n tends to infinity, the measure of maximal entropy always converges to the Lebesgue measure supported on the unit circle.Additionally we describe the dynamics on the Julia set and show that is related to a quotient of a shift of n symbols by an equivalence relation. Finally we prove that the Thurston pull-back map associated with a particular four-circle inversion is a ramified Galois covering. From the arithmetical point of view we prove that each n-circle inversion can be defined over its field of moduli.

Universality of Composition Operators and Applications to Holomorphic Dynamics

By investigating which level of universality composition operators Cf can have, where the symbol f is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of f, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.

Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation

In a classical work of the 1950s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee–Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee–Yang–Fisher (LYF) zeros. In this paper, we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $$R$$ in two variables (the Migdal–Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the LYF zeros are equidistributed with respect to a dynamical (1, 1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal–Kadanoff renormalization transformation of various other hierarchical lattices.