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In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T (z) = z n + m z + r , where n ≥ 2 and m , r ∈ C , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set. [ABSTRACT FROM AUTHOR]

A neutrosophic fuzzy set that generalizes the classical set is represented by a closed interval [0,1]. Let us generalize the fuzzy set to the Neutrosophic set, which is defined as three membership functions between interval]-0,1+ [. This paper examines the bird collision problem within a more grounded Neutrosophic framework. The collision between the bird and the aircraft is represented in the Neutrosophic domain as a stationary point because it occurs at a distance too great for any other auditory signal to discern. Numerous methods exist for solving the airplane problem, including a Julia set in complex numbers. Bird collisions resulting from aircraft crashes add complication. An aviation signal is utilized to avoid an accident in a bird strike or bird collision. [ABSTRACT FROM AUTHOR]

This paper presents a detailed study of the dynamics of transcendental entire maps which are symmetric with respect to rays of the form Λ (a , θ) = { t e i θ + a | t ≥ 0 } , with a ∈ C and θ ∈ [ 0 , 2 π) . The symmetric families considered in the paper contain many well-known entire maps. Let f be such a symmetric entire map. Then (i) there exists a point z 0 = t 0 e i θ + a , (t 0 ≥ 0) , such that for every point z on the ray Λ 0 (a , θ) = { t e i θ + a | t ≥ t 0 } , and any point z n in its forward orbit, any connected compact set containing z and z n must intersect the Julia set of f; (ii) The map f either has a multiply-connected wandering domain, or its Julia set contains the ray Λ 0 (a , θ) ; (iii) If t 0 > 0 , then the Fatou set of f has an invariant immediate attractive basin, or an invariant Leau domain containing a specific disc { z : | z - a | < t 0 } , with z 0 being a repelling fixed point and f ′ (z 0) > 1 , or a rationally indifferent fixed point and f ′ (z 0) = 1 , respectively; As applications, new dynamical properties of parametric families of the most well-known maps, including exponential maps of the forms a e z n + b e - z n , hyperbolic maps of the forms λ sinh z , and cosh z + λ , trigonometric maps of the forms λ sin z , and λ cos z , etc, are obtained. Moreover, a new map with wandering domains is constructed. [ABSTRACT FROM AUTHOR]

We establish a version of the Pommerenke–Levin–Yoccoz inequality for the modulus of a polynomial-like (PL) restriction of a polynomial and give two applications. First we show that if the modulus of a PL restriction of a polynomial is bounded from below then this restricts the combinatorics of the polynomial. The second application concerns parameter slices of cubic polynomials given by the non-repelling multiplier of a fixed point. Namely, the intersection of the so-called Main Cubioid and the multiplier slice lies in the closure of the principal hyperbolic domain, with possible exception of queer components. [ABSTRACT FROM AUTHOR]

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any $ s\in (0,+\infty ] $ s ∈ (0 , + ∞ ] , there is a function taken from this family with the order of growth s. Baker proved that the Hausdorff dimension of a transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth. [ABSTRACT FROM AUTHOR]

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families. [ABSTRACT FROM AUTHOR]

When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals. [ABSTRACT FROM AUTHOR]

A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any s ∈ ( 0 , + ∞ ] , there is a function taken from this family with the order of growth s. Baker proved that the Hausdorff dimension of a transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bishop's construction, the family of functions here have arbitrarily positive or even infinite order of growth. [ABSTRACT FROM AUTHOR]

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. [ABSTRACT FROM AUTHOR]

For the class of meromorphic functions outside a compact countable set of essential singularities, we study the dynamics of some functions in for which the unit disc and its complement are invariant. These functions are products and compositions of the function E 1 = exp (z − 1 z + 1) with Blaschke products, that include the family E n = exp z n − 1 z n + 1 , which are the main topic of the article. Slices of the space of parameters of E n are given with a discussion of their main features. We construct a Poincaré extension of E n to the hyperbolic three-dimensional space and to the 3-sphere. Also we study their dynamics. [ABSTRACT FROM AUTHOR]

The majority of fractals' dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of "escape" is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine T c (z) = a s i n (z r) + b z + c and complex exponential T c (z) = a e z r + b z + c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters 'a' and 'b', and the parameters involved in the series expansion of the sine and exponential functions. [ABSTRACT FROM AUTHOR]

To investigate the degree d connectedness locus, Thurston [ On the geometry and dynamics of iterated rational maps , Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied \sigma _d-invariant laminations , where \sigma _d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f(z) = z^2 +c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C_sCL together with the induced factor space \mathbb {S}/C_sCL of the unit circle \mathbb {S}. As will be verified in the third paper of the series, \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus , i.e. the space of all cubic symmetric polynomials with connected Julia sets. [ABSTRACT FROM AUTHOR]

Levin, Przytycki and Shen [Invent. Math. 205 (2016), pp. 363–382] proved for a polynomial map f_c(z)=z^d+c, d\geq 2 and c \in \mathbb C, with Julia set J(f) of positive measure that for a.e. z \in J(f) the Lyapunov exponent \chi _s(z)=0. The aim of this paper is to show that the extension to non-entire transcendental meromorphic functions is not possible. [ABSTRACT FROM AUTHOR]

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type T (z) = z n + p z + r , where p , r ∈ C , and n ≥ 2 . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method. [ABSTRACT FROM AUTHOR]

In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and study the consequences of underlying parameters on the variation of dynamics, colour, time of generation, and shape of generated fractals. The black points in the obtained fractals are the "non-chaotic" points and the dynamical behaviour in the black area is easily predictable. The coloured points are the points that "escape", that is, they tend to infinity under one of iterative methods based on a four-step fixed-point iteration scheme extended with s-convexity. The different colours tell us how quickly a point escapes. The order of escaping of coloured points is red, orange, yellow, green, blue, and violet, that is, the red point is the fastest to escape while the violet point is the slowest to escape. Mostly, these generated fractals have symmetry. The Julia set, where we find all of the chaotic behaviour for the dynamical system, marks the boundary between these two categories of behaviour points. The Mandelbrot set, which was originally observed in 1980 by Benoit Mandelbrot and is particularly important in dynamics, is the collection of all feasible Julia sets. It perfectly sums up the Julia sets. [ABSTRACT FROM AUTHOR]

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the main cubioid in this parameter space. The main cubioid is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials z^2+c for c in the filled main cardioid. [ABSTRACT FROM AUTHOR]

In this research, we look at the Julia set patterns that are linked to the entire transcendental function f (z) = a e z n + b z + c , where a , b , c ∈ C and n ≥ 2 , using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function. [ABSTRACT FROM AUTHOR]

Letting f and h denoting rational maps of degree at least 2 and 1 respectively, and denote by f n the nth iterate of f. We give upper and lower bounds on the set of accumulation points of the Julia sets J (h ∘ f n) with respect to the Hausdorff metric on non-empty compact subsets of the sphere. [ABSTRACT FROM AUTHOR]

Quantum tunneling in nonintegrable systems is studied based on the complex semiclassical analysis in the time domain. The theory of complex dynamical systems predicts that complexified stable and unstable manifolds form the skeleton of the dynamics in phase space, and mixing and ergodicity hold even for the mixed system. After surveying some rigorous results derived for the HĂ©non map, we numerically verify that some fundamental properties thus predicted can actually be observed, and then investigate natures of complexified stable and unstable manifolds since they guide the complex orbits controlling the tunneling transition in the mixed phase space. In particular, for a properly designed scattering map, we examine the imaginary action associated with the stable manifolds for the periodic orbits in the real plane and find that the orbits tending to the sticky zone of the KAM region provide the most dominant contribution in the semiclassical sum. It is worth emphasizing that the behavior of such dominant orbits is affected by the presence of chaos in the complex plane, which is an aspect far beyond the scope of the dynamics in the real plane. [ABSTRACT FROM AUTHOR]

Some inherited properties of exceptional Jacobi polynomials are derived and as an application it is shown that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense. [ABSTRACT FROM AUTHOR]

In this paper, a novel reversible blind dual-color image watermarking algorithm is proposed by using singular value decomposition (SVD), Hachimoji Deoxyribonucleic Acid (HDNA) biogenetic encryption, coupled map lattice-based Tent–Sine system (TSS-CML) and mathematical Julia set. For watermark embedding, the watermark image is firstly encrypted using 8-bases HDNA sequences, TSS-CML and Julia set image. Then the encrypted HDNA watermark is obtained. Next, decompose the host image into equal non-overlapping blocks and utilize SVD on the randomly selected blocks. Further, embed the HDNA watermark through modifying the relation between the elements in the first column of the matrix U or V. The watermarked image can be eventually attained by carrying out the inverse SVD on all selected blocks. Also a reliable extraction algorithm is designed to recover the watermark from the possibly attacked watermarked images without resorting to the original image. Experimental and analysis results demonstrate that the proposed watermarking scheme has not only an excellent imperceptibility but a strong robustness against the common image processing attacks, geometric attacks and some composite attacks. In addition, the running time taken for hiding and exacting is about 1 s, which is suitable for real-time network transmission and application. In conclusion, the proposed method outperforms the related dual images watermarking algorithms in terms of time performance, extraction effect and robustness. [ABSTRACT FROM AUTHOR]

We prove that there is a sequence of purely imaginary parameter values λ n converging to 0 such that the Julia set for z ↦ λ n (z + 1 / z) is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate. [ABSTRACT FROM AUTHOR]

We study the dynamics of a meromorphic perturbation of the family λ sin z \lambda \sin z by adding a pole at zero and a parameter μ \mu , that is, f λ , μ (z) = λ sin z + μ / z {f}_{\lambda ,\mu }\left(z)=\lambda \sin z+\mu \hspace{-0.08em}\text{/}\hspace{-0.08em}z , where λ , μ ∈ C ⧹ { 0 } \lambda ,\mu \in {\mathbb{C}}\hspace{-0.16em}\setminus \hspace{-0.16em}\left\{0\right\}. We study some geometrical properties of f λ , μ {f}_{\lambda ,\mu } and prove that the imaginary axis is invariant under f n {f}^{n} and belongs to the Julia set when ∣ λ ∣ ≥ 1 | \lambda | \ge 1. We give a set of parameters (λ , μ) \left(\lambda ,\mu) , such that the Fatou set of f λ , μ {f}_{\lambda ,\mu } has two super-attracting domains. If λ = 1 \lambda =1 and μ ∈ (0 , 2) \mu \in \left(0,2) , the Fatou set of f 1 , μ {f}_{1,\mu } has two attracting domains. Also, we give parameters λ , μ \lambda ,\mu such that ± π / 2 \pm \pi \hspace{-0.08em}\text{/}\hspace{-0.08em}2 are fixed points of f λ , μ {f}_{\lambda ,\mu } and the Fatou set of f λ , μ {f}_{\lambda ,\mu } contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of f λ , μ {f}_{\lambda ,\mu } , where the Fatou set contains two types of domains, for λ , μ \lambda ,\mu given. [ABSTRACT FROM AUTHOR]

Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a criterion for fractionalizing traditional fractal sets, and reports several fractional generalizations of the Julia set and Mandelbrot set of the classic quadratic map. The discrepancies among those extensions are investigated preliminarily. The influence of the map's memory effect on its fractal sets is discussed through visualization techniques and dimension analysis. [ABSTRACT FROM AUTHOR]

Julia set is one of the most important branches in fractal theory, and has achieved much progress on both theoretical analysis and application research. Nevertheless, most of the existing investigations relied on the systems with known parameters. Few works have been done on the inverse problem about how to determine the system parameter based on a given Julia set's shape. This work aims to address this issue by starting with the most classical polynomial map. First, by constructing a proper fitness function measuring the Julia sets' area error, the Julia sets' parameter estimation is formulated as a kind of optimization problem. According to the known conditions of Julia set to be estimated, the optimization problem is classified into two cases. For one case, when both the shape and size of Julia set are given, we only need to estimate the system's own parameter c. For the other case, when the Julia set has only a shape, we redesign the problem into three dimensions by adding a new scaling factor parameter γ and applying the partial coverage principle. Second, a particle swarm optimization (PSO) approach including dynamically adjustment strategy is adopted to solve the proposed optimization problem. At last, we present seven groups of simulations in which both randomly generated Julia sets and those in existing literature (or on the internet) are included. The simulation results verify the effectiveness of the proposed parameter estimation scheme. [ABSTRACT FROM AUTHOR]

For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by R c (z) = c (z d − (d − 1)) / z with c ∈ R ∖ { 0 }. Also, when c ∈ C ∖ { 0 } we show that the geometric limit of the Julia set and filled Julia set of the family R c exists and is the unit circle. [ABSTRACT FROM AUTHOR]

It is of great significance to study the three-dimensional financial system model based on the discrete fractional-order theory. In this paper, the Julia set of the three-dimensional discrete fractional-order financial model is identified to show its fractal characteristics. The sizes of the Julia sets need to be changed in some situations, so it is necessary to achieve control of the Julia sets. In combination with the characteristics of the model, two different controllers based on the fixed point are designed, and the control of the three-dimensional Julia sets is realized by adding the controllers into the model in different ways. Finally, the simulation graphs show that the controllers can effectively control the fractal behaviors. [ABSTRACT FROM AUTHOR]

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function hl(z) = z2 +l. Their generalization was based on the composition of hl with the Möbius transformation m(z) = 1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of m(hl(z)) (at z = 0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation. [ABSTRACT FROM AUTHOR]

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function h(z) = z2 + l. 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. [ABSTRACT FROM AUTHOR]

This work focuses on a kind of fractals Parrondo's paradoxial phenomenon "deiconnected+diconnected=connected" in an alternated superior complex system zn+1 = β b(zn² + ci) + (1-β) zn, i = 1, 2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. [ABSTRACT FROM AUTHOR]

This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]

We consider a family of quartic polynomials generated by the composition of two quadratic polynomials. The elements of this family have two complex parameters, however they have at most two dynamic behaviors, since every map in this family have two critical points with the same forward orbits. In this paper, we study this quartic family in the complex parameter space, and we describe the dynamical plane for some special parameters. Moreover, we analyze the parameter space for these quartic polynomials with a super attracting fixed point. We describe the connectedness locus for this family, and we prove the locally connectedness of the boundary of hyperbolic components in the parameter space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical examples are also included to verify the conclusions. [ABSTRACT FROM AUTHOR]

Let n and k be positive integers. For n > k ≥ 2 , we consider the connectedness locus M n , k of the family of polynomials P c (z) = z n + c z k , where c is a complex parameter. We show that the geometric limit of the connectedness locus sets M n , k , when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of M n , k .When parameter c belongs to the open unit disk, we show that the geometric limit of the Julia sets J c , when n tends to infinity, exists and is the unit circle. Finally, we have established some properties on the hyperbolic components of this family. [ABSTRACT FROM AUTHOR]

The objective of this paper is to generate fractals for complex valued polynomials using Jungck-SP orbit. We present an algorithm to generate fractals. With the help of this algorithm, we generate such beautiful graphics which may be useful for graphic designers. In this paper, we extend and improve the corresponding results of Kang et al. (2015) and Kumari et al. (2017) existing in the literature. One can further generalize our results and derive a new escape criterion to generate some more beautiful fractals. [ABSTRACT FROM AUTHOR]

It is of crucial significance to study the infectious disease phenomenon by using the SIRS model and thoughts of Julia set. In this paper, Julia set of the discrete version of the SIRS model is established to analyze the fractal dynamics of this model. Then, controller is designed to change the Julia set. Furthermore, the box-counting dimensions of the controlled Julia sets by selecting different appropriate parameters are computed to show the complexity of the model. Finally, a nonlinear coupling method is introduced to synchronize the Julia sets with different parameters of the same system. Simulation results show the efficacy of these methods. [ABSTRACT FROM AUTHOR]

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic. By results of Kiwi, any dendritic polynomial is semiconjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus. [ABSTRACT FROM AUTHOR]

Consider the parameter space P λ ⊂ C 2 of complex Hénon maps H c , a (x , y) = (x 2 + c + a y , a x) , a ≠ 0 , which have a semi-parabolic fixed point with one eigenvalue λ = e 2 π i p / 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 P λ 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 P λ 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 J p , 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study on the local and semi-local convergence and dynamic of an iterative optimal two-step Steffensen's method. We study local convergence by using Lipschitz conditions and continuity of differentiable function F. We also obtain semi-local convergence by using majorizing sequences. By this approach, we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions. Moreover, we analyze the stability of the method via Julia sets and parameter planes that is a novelty for Steffensen-type methods. We extend this method for obtaining the sign function of a matrix. Numerical examples are given to confirm our theoretical results. [ABSTRACT FROM AUTHOR]