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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 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]

We consider hyperbolic random complex dynamical systems on the Riemann sphere with separating condition and multiple minimal sets. We investigate the Hölder regularity of the function T of the probability of tending to one minimal set, the partial derivatives of T with respect to the probability parameters, which can be regarded as complex analogues of the Takagi function, and the higher partial derivatives C of T . Our main result gives a dynamical description of the pointwise Hölder exponents of T and C , which allows us to determine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum α − is strictly less than 1, which allows us to show that the averaged system acts chaotically on the Banach space C α of α -Hölder continuous functions for every α ∈ ( α − , 1 ) , though the averaged system behaves very mildly (e.g. we have spectral gaps) on C β for small β > 0 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some fixed point results in the generation of Julia and Mandelbrot sets by using Jungck Mann and Jungck Ishikawa iterations with s-convexity. [ABSTRACT FROM AUTHOR]

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances. [ABSTRACT FROM AUTHOR]

Dynamics of transcendental entire maps with coefficients in an algebraically closed and complete non-Archimedean field $$K$$ is studied. It is shown, among other things, that periodic repelling points are dense in Berkovich Julia set, the forward union of any open set which intersects the Julia set is the whole Berkovich affine line, and a multi-connected Fatou component is wandering, in which all points go to infinity under iteration. [ABSTRACT FROM AUTHOR]

Escape time algorithm is a classical algorithm to calculate the Julia sets, but it has the disadvantage of dull color and cannot record the iterative process of the points. In this paper, we present the equipotential point algorithm to calculate the Julia sets by recording the strike frequency of the points in the iterative process. We calculate and analyze the Julia sets in the complex plane by using this algorithm. Finally, we discuss the iteration trajectory of a single point. [ABSTRACT FROM AUTHOR]

We show that, for many parameters aϵC, the set I(fa) of points that converge to infinity under iteration of the exponential map fa(z) = ez + a is connected. This includes all parameters for which the singular value a escapes to infinity under iteration of fa. [ABSTRACT FROM AUTHOR]

Given a complex polynomial p( z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B( z), m≥2. Let θ be a simple root of p( z), V( θ) its Voronoi cell, and A( θ) its basin of attraction with respect to B( z). We prove that given any closed subset C of V( θ), including any homothetic copy of V( θ), there exists m such that for all m≥ m, C is also a subset of A( θ). This implies that when all roots of p( z) are simple, the basins of attraction of B( z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B( z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings of the invariant set, where each consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given. [Copyright &y& Elsevier]

In this paper, we first establish a rational iteration method which can be used as a root-finding algorithm for almost every polynomial. It has no nonrepelling extraneous fixed point in the complex plane and is generally convergent for both quadratic and cubic polynomials. Then some properties of this algorithm are given. By the aid of computer, we produce pictures of the Julia sets for the iterations of some polynomials. Numerical results show that it is a root-finding method with convergence order the same as Halley's method. [ABSTRACT FROM AUTHOR]

Stable versions of Newton's iteration for computing the principal matrix pth root A1/P of an n x n matrix A are provided. In the case in which X0 is the identity matrix, it is proved that the method converges for any matrix A having eigenvalues with modulus less than 1 and with positive real parts. Based on these results we provide a general algorithm for computing the principal pth root of any matrix A having no nonpositive real eigenvalues. The algorithm has quadratic convergence, is stable in a neighborhood of the solution, and has a cost of O(n3 log p) operations per step. Numerical experiments and comparisons are performed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze the Julia set of Newton method for multiple roots, construct the Julia sets of standard, relaxed and multiple roots. Through the experimental mathematics method we drew the following conclusion: (1) the Julia sets of above methods for f(z)= z α (z β −1) has β time rotation symmetry and its center is origin; (2) the multiple roots domain of attraction of these kinds of Julia sets are sensitive to α; (3) there is not simple root domain of attraction in relax method, because z*, the root of f(z), is neutral or repelling fixed point of F(z); (4) ∞ is not the fixed point of F(z), so multiple root’s Julia set consists of a great number of attraction domain of multiple and simple roots; (5) the experimental errors and truncation of coefficients cause the minimal effect to multiple root, the more to relax, the maximal to standard Newton’ method. [Copyright &y& Elsevier]

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum. [ABSTRACT FROM AUTHOR]

A frame is an overcomplete family of vectors in a Hilbert space in which the orthogonality condition is relaxed. The Julia set is the chaotic regime of a rational function. In this note, we label frames of an abstract Hilbert space by elements of the Julia set of a rational function. [Copyright &y& Elsevier]

We consider the family of entire transcendental maps given by F[sub λ,m](z)=λz[sup m]exp(z) where m≥2. All functions F[sub λ,m] have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle. [ABSTRACT FROM AUTHOR]

Schröder iteration functions, a generalization of the Newton­Raphson method to determine roots of equations, are generally rational functions which possess some critical points free to converge to attracting cycles. These free critical points, however, satisfy some higher-degree polynomial equations. We present a new algorithmic construction to compute in general all of the Schröder functions' terms as well as to maximize the computational efficiency of these functions associated with a one-parameter family of cubic polynomials. Finally, we examine the Julia sets of the Schröder functions constructed to converge to the nth roots of unity, these roots' basins of attraction, and the orbits of all free critical points of these functions for order higher than four, as applied to the one-parameter family of cubic polynomials mentioned above. [ABSTRACT FROM AUTHOR]