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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]

We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time. [ABSTRACT FROM AUTHOR]

Let Σ ( R ) be the group of symmetries of the Julia set of a rational map R and K f , n be the König's method for polynomial f of order n ( ≥ 2 ) . For any given integer n ≥ 2 , we prove that if f is in normal form, then Σ ( f ) is a subgroup of Σ ( K f , n ) . We also obtain a necessary and sufficient condition for the Julia set of K f , n to be a line. [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]

We relate periodic and recurrent points in dendritic Julia sets. This generalizes well-known results for interval dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of the map z ↦ exp( z)/ z on the punctured plane ℂ* = ℂ\{0}. We show that for almost every point z ∈ ℂ*, the ω-limit set of z is equal to {0,∞}. In particular, the map is not recurrent. [ABSTRACT FROM AUTHOR]

Abstract: Considering a family of rational maps relating to renormalization transformations, we prove that each Fatou component of is a Hölder domain for odd integers and some parameters , but is not a Collet–Eckmann map even if there is only one critical point in its Julia set. Furthermore, we show that there exists a buried open interval in the Julia set , but the two endpoints of this interval belong to the boundary of some Fatou component of . [Copyright &y& Elsevier]

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees d ≥ 2. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure. [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]

Abstract: We study the dynamics on the Julia set for holomorphic endomorphisms of . The Julia set is the support of the so-called Green current T, so it admits a natural filtration , where for we put . We show that for a generic point of there are at least “Fatou directions” in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions. [Copyright &y& Elsevier]

Abstract: We prove some results concerning the possible configurations of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period and give a sufficient condition for a configuration to be realizable. [Copyright &y& Elsevier]

Abstract: In this paper, we study certain dynamical properties of the relaxed Newton’s method for entire functions. We present a procedure for constructing entire functions so that superattracting cycles of any prescribed period occur when the relaxed Newton’s method is applied. Based on the above conclusion, we give a model of any prescribed type cycles for four disjoint subclasses of meromorphic functions, including polynomials, transcendental entire functions, rational functions but not polynomials, and transcendental meromorphic functions but not entire functions. [Copyright &y& Elsevier]

Abstract: Let f be a transcendental entire function of finite order in the Eremenko–Lyubich class (or a finite composition of such maps), and suppose that f is hyperbolic and has a unique Fatou component. We show that the Julia set of f is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic with connected Fatou set, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function , a natural compactification of the dynamical plane by adding a “circle of addresses” at infinity. [Copyright &y& Elsevier]

Abstract: This paper is devoted to studying the dynamical properties of solutions of , where is an integer, and is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of provided that and is a solution base of such equations. [Copyright &y& Elsevier]

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]

Abstract: In the present work we expand our previous work in by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations. [ABSTRACT FROM AUTHOR]

With the help of the Schwarzian derivative, critical points and asymptotic values, we present a sufficient condition for some classes of transcendental entire functions to have their Julia set to be the complex plane. Further, it is shown that the composition with a function which preserves these properties (but does not necessarily have the plane as a Julia set) again gives a function with empty Fatou set. By the two theorems we obtain new classes of simple entire functions whose Julia set is the complex plane. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the theory of Julia sets of Schröder iteration functions is introduced, the Julia sets of the Schröder functions of a one-parameter family polynomials with high degree are constructed through iteration method, and their structures are analyzed. Consequently, the following results are found in the study: (1) the Julia sets of the Schröder iteration functions of a one-parameter family polynomials with high degree contain the structure of classical Mandelbrot-like set; (2) the orbits of the critical points may escape from the zero points of the corresponding polynomial to converge to the k-cycle attractive basin or the extra fixed points; (3) if critical points on parameter plane are selected to construct Julia sets on dynamics plane, then attractive k-cycle basin will emerge, while it will not emerge if no critical points are selected; (4) the extra fixed points may be repulsive, litmusless or attractive, but the former takes the major role and (5) the Julia sets of the Schröder iteration functions have symmetry. [Copyright &y& Elsevier]

The theory of Fatou and Julia has been extended in [ 3 ] for the class of functions that are meromorphic outside a sufficiently small, non-empty totally disconnected compact set of essential singularities. For a certain subclass of these functions we study (a) accessibility of boundary points of the Fatou components and (b) the number of completely invariant domains. [ABSTRACT FROM AUTHOR]