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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]

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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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 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]

We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of the escaping sets if the function has no logarithmic singularities over ∞, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or ∞. [ABSTRACT FROM AUTHOR]

Spaces of sequences which are valued in a topological space E are considered in order to study universality properties of generalized backward shifts associated to certain selfmappings of E . After providing a condition which guarantees the mixing property of backward Φ-shifts in terms of the dynamical properties of the generating selfmappings, applications to universality theory within the framework of complex dynamics are furnished. [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]

In this note, we give fixed point results in fractal generation (Julia sets and Mandelbrot sets) by using Noor iteration scheme with s-convexity. Researchers have already presented fixed point results in Mann and Ishikawa orbits that are examples of one-step and two-step feedback processes respectively. In this paper we present fixed point results in Noor orbit, which is a three-step iterative procedure. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of the functions $$ f_{\mu }(z)=z\exp (z+\mu ) $$ with the real parameter is studied. We say that a real parameter $$ \mu $$ belongs to the set $$ B_n $$ for a positive integer n if $$ f_\mu $$ has an attracting cycle of n-order. We prove that the Fatou set $$ F(f_\mu ) $$ is a completely invariant attracting basin for every parameter $$ \mu <0 $$ . Further, regarding the set $$ B_n $$ for $$ n>1 $$ , we prove the following results: (1) There exists $$ \mu _{*}\ne +\infty $$ such that $$ B_2 = (2,\mu _{*}) $$ . (2) For every positive integer $$ n>2 $$ , the set $$ B_n $$ is non-empty. (3) For every prime number $$ p>3 $$ , the set $$ B_p $$ has at least two components. [ABSTRACT FROM AUTHOR]

Consider the standard family of complex Hénon maps H ( x , y ) = ( p ( x ) − a y , x ) , where p is a quadratic polynomial and a is a complex parameter. Let U + be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set U + is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of ( C − D ‾ ) × C by a discrete group of automorphisms Γ isomorphic to Z [ 1 / 2 ] / Z . On the other hand, the boundary J + of U + is a complicated fractal object on which the Hénon map behaves chaotically. We show how to extend the group action to S 1 × C , in order to represent the set J + as a quotient of S 1 × C / Γ by an equivalence relation. We analyze this extension for Hénon maps that are perturbations of hyperbolic polynomials with connected Julia set or polynomials with a parabolic fixed point. [ABSTRACT FROM AUTHOR]

Let P be a polynomial of degree d with Julia set J P . Let N ˜ be the number of non-repelling cycles of P . By the famous Fatou–Shishikura inequality N ˜ ≤ d − 1 . The goal of the paper is to improve this bound. The new count includes wandering collections of non-(pre)critical branch continua , i.e., collections of continua or points Q i ⊂ J P all of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval ( Q i ) ≥ 3 external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense. A weak version of the inequality reads N ˜ + N irr + χ + ∑ i ( eval ( Q i ) − 2 ) ≤ d − 1 where N irr counts repelling cycles with no periodic rays landing at points in the cycle, { Q i } form a wandering collection B C of non-(pre)critical branch continua, χ = 1 if B C is non-empty, and χ = 0 otherwise. [ABSTRACT FROM AUTHOR]

Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behavior in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J ( f ) = J ( g ) provided that neither f nor g has a simply connected wandering domain in the fast escaping set. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the radial distribution of Julia sets of Newton's method of entire solutions of some complex linear differential equations. Under certain conditions, we find the lower bound of it and also obtain some related results. [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]

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]

We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of Boyd's theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpiński carpets that are Julia sets of hyperbolic rational maps. [ABSTRACT FROM AUTHOR]

If x and y belong to a metric space X , we call ( x , y ) a DC1 scrambled pair for f : X → X if the following conditions hold: 1) for all t > 0 , lim sup n → ∞ 1 n | { 0 ≤ i < n : d ( f i ( x ) , f i ( y ) ) < t } | = 1 , and 2) for some t > 0 , lim inf n → ∞ 1 n | { 0 ≤ i < n : d ( f i ( x ) , f i ( y ) ) < t } | = 0 . If D ⊂ X is an uncountable set such that every x , y ∈ D form a DC1 scrambled pair for f , we say f exhibits distributional chaos of type 1. If there exists t > 0 such that condition 2) holds for any distinct points x , y ∈ D , then the chaos is said to be uniform . A dendrite is a locally connected, uniquely arcwise connected, compact metric space. In this paper we show that a certain family of quadratic Julia sets (one that contains all the quadratic Julia sets which are dendrites and many others which contain circles) has uniform DC1 chaos. [ABSTRACT FROM AUTHOR]

The forced Brusselator model is investigated from the fractal viewpoint. A Julia set of the discrete version of the Brusselator model is introduced and control of the Julia set is presented by using feedback control. In order to discuss the relations of two different Julia sets, a coupled item is designed to realize the synchronization of two Julia sets with different parameters, which provides a method to discuss the relation and the changing of two different Julia sets, one Julia set can be changed to be the other. Numerical simulations are used to verify the effectiveness of these methods. [ABSTRACT FROM AUTHOR]

We study a nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map that has been introduced by Bamón et al. [2006]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimages. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. For parameters away from the complex quadratic family we define a generalized notion of the Julia set as the basin boundary of infinity. We are interested in how the Julia set changes when saddle points along with their stable and unstable sets appear as the perturbation is switched on. Advanced numerical techniques enable us to study the interactions of the Julia set with the critical set and the (un)stable sets of saddle points. We find the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set; these bifurcations lead to three complicated types of Julia sets that are given by the closure of stable sets of saddle points of the map, namely, a Cantor bouquet and what we call a Cantor tangle and a Cantor cheese. We are able to illustrate how bifurcations of the nonanalytic map connect to those of the complex quadratic family by computing two-parameter bifurcation diagrams that reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family. [ABSTRACT FROM AUTHOR]

An efficient and secure environment is necessary for data transmission and storage, especially for large‐column multimedia data. In this study, a novel compression–encryption scheme is presented using a fractal dictionary and Julia set. For the compression in this scheme, fractal dictionary encoding not only reduces time consumption, but also gives good quality image reconstruction. For the encryption in the scheme, the key has large key space and high sensitivity, even to tiny perturbation. Besides, the stream cipher encryption and the diffusion process adopted in this study help spread perturbation in the plaintext, achieving high plain sensitivity and giving an effective resistance to chosen‐plaintext attacks. [ABSTRACT FROM AUTHOR]

Experiment with discrete time dynamical systems, may represent an important educational resource in the investigation of the properties of dynamical systems and their potential applications to disciplines such as economics. As an illustration of this possibility teaching, this paper provides a brief introduction to the dynamics of discretetime dynamic systems using examples assisted by the symbolic language Mathematica. Such systems are essentially iterated maps. In the first part, we construct orbits of points under iteration of real and complex functions. If x is a real number or a complex number, then the orbit of x under f is the sequence {x, f (x), f (f (x)), ...}. These sequences may be convergent or sequences that tend to infinity. In particular, to test this behavior in complex sequences will require the concept of derivative of a complex function. In a second part, we use the concepts reviewed in the first to build Julia sets, these sets are obtained by assigning colors to a rectangular grid points according to the behavior of their orbits under the studied complex function, the colors are assigned according the classification of the points. The pattern obtained, the Julia set is a fractal. However, the image obtained is always an approximation. [ABSTRACT FROM AUTHOR]

Our main theorem establishes that the Fatou set of the functions fλ,μ(z) = λez + μ/z contains a two-cycle of Baker domains {F∞, F0} if Re(λ) < 0 and |Im(λ)| < 1/2|Re(λ)|. We show that under every point in F∞ tends to infinity and in F0 tends to zero. Moreover, if |Im(λ)| < 1/2|Re(λ)| - 4, the set F∞ contains infinitely many critical points of fλ,μ and F0 contains infinitely many critical values; also, infinitely many critical values of are contained in both F∞ and F0. Finally, the images of the Baker domains are displayed for some parameters. [ABSTRACT FROM AUTHOR]

In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces (X, d), the existence of at least one limit function on a compact subset A of X implies the existence of plenty of them on many supersets of A. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to studying the limit directions of Julia sets of solutions of , where is an integer and are entire functions of finite lower order. With some additional conditions on coefficients, we know that every non-trivial solution of such equations has infinite lower order, and prove that the limit directions of Julia sets of must have a definite range of measure. [Copyright &y& Elsevier]

Thurston defined invariant laminations, i.e. collections of chords of the unit circle S (called leaves) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a polynomial, a lamination has to be generated by an equivalence relation with specific properties on S; then it is called a q-lamination. Since not all laminations are q-laminations, then from the point of view of studying polynomials the most interesting are those which are limits of q-laminations. In this paper we introduce an alternative definition of an invariant lamination, which involves only conditions on the leaves (and avoids gap invariance). The new class of laminations is slightly smaller than that defined by Thurston and is closed. We use this notion to elucidate the connection between invariant laminations and invariant equivalence relations on S. [ABSTRACT FROM AUTHOR]

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods. [ABSTRACT FROM AUTHOR]

In this paper, we define (p , q) -cosine and sine sigmoid polynomials. Based on this, the properties of each polynomial, and the structure and assumptions of its roots, can be identified. Properties can also be determined by the changes in p and q. [ABSTRACT FROM AUTHOR]

The dynamical and fractal behaviors of the complex perturbed rational maps are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied. [ABSTRACT FROM AUTHOR]

Abstract: We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a -parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than , then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than . We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each with , the family of small perturbations of the semigroup generated by satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive. [Copyright &y& Elsevier]

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]

In this paper, we consider the family of rational maps where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids. [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]