Search

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]

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]

We study the group TA of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). We prove that TA is generated by a copy of Thompson's group F and a copy of Thompson's group T, hence it is finitely generated. Then we study the commutator subgroup [TA; TA], proving that the abelianization of TA is isomorphic to Z and that [TA; TA] is simple, finitely generated and acts 2-transitively on the so-called components of the airplane limit space. Moreover, we show that TA is contained in T and contains a natural copy of the basilica rearrangement group TB studied by Belk and Forrest (2015). [ABSTRACT FROM AUTHOR]

Suppose $ X $ is a compact connected metric space and $ f\colon X \to X $ is a metric coarse expanding conformal map in the sense of Haïssinsky-Pilgrim. We show that if $ X $ contains a homeomorphic copy of the letter 'Y', then the Hausdorff dimension of $ X $ is greater than one. As an application, we show that for a semi-hyperbolic rational map $ f $ its Julia set $ \mathcal{J}_f $ is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if $ \mathcal{J}_f $ is homeomorphic to a circle or a closed interval. [ABSTRACT FROM AUTHOR]

In this paper it is shown that for a transcendental entire map with certain properties on the distribution of either its zeros, or points in its periodic and preperiodic Fatou domains, all of its Fatou components are simply connected, the union of its Julia set and { ∞ } is connected and the Julia set is uniformly perfect. Moreover, it is shown that any transcendental entire map that interpolates the 3n+1 problem has only simply connected Fatou components and its Julia set is uniformly perfect. [ABSTRACT FROM AUTHOR]

In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between $ T (r, f) $ and $ \log M (r, f) $ when the modulus of variable $ |z| = r $ is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.

We show the structure of the parameter space for a family of rational maps containing Blaschke products. Through numerical simulations using the orbit of a single critical point, we reveal the existence of infinitely many Mandelbrot-like sets along the unit circle, as well as eight-like structures in other regions of parameter space. We pose some open questions related to the parameter space of these functions.

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]

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component could be bounded even if the orbit of the component contains a sequence of annuli whose moduli tend to infinity, and this cannot happen when the maximal modulus of the meromorphic function is uniformly large enough. In this way we extend certain related results for entire functions to meromorphic functions with infinitely many poles. [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]

We will give a complete description of the dynamics of the rational map $N_{F_{M_c}}(z)=\frac{3z^4-2z^3+c}{4z^3-3z^2+c}$ where c is a complex parameter. These are rational maps $N_{F_{M_c}}$ arising from Newton's method. The polynomial of Newton iteration function is obtained from singularly perturbed of the Multibrot set polynomial.

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 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.

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 Tc(z)=asin(zr)+bz+c and complex exponential Tc(z)=aezr+bz+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.

We study the Chebyshev–Halley methods applied to the family of polynomials f n , c (z) = z n + c , for n ≥ 2 and c ∈ C ∗ . We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for n ≥ 2 , the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton's method to f n , - 1 . [ABSTRACT FROM AUTHOR]

Nous montrons que la mesure d’équilibre d’un polynôme quadratique peut être extraite de la mesure harmonique de l’ensemble de Mandelbrot par un procédé du type renormalisation. [ABSTRACT FROM AUTHOR]

We study the dynamics of a meromorphic perturbation of the family λsinz\lambda \sin z by adding a pole at zero and a parameter μ\mu , that is, fλ,μ(z)=λsinz+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 fλ,μ{f}_{\lambda ,\mu } and prove that the imaginary axis is invariant under fn{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 fλ,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 f1,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 fλ,μ{f}_{\lambda ,\mu } and the Fatou set of fλ,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 fλ,μ{f}_{\lambda ,\mu }, where the Fatou set contains two types of domains, for λ,μ\lambda ,\mu given.

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]

The limit sets of zeros of the partition function for $\lambda $ -state Potts models on diamond-like hierarchical lattice are the Julia sets of functions in a family of rational functions. In this paper, the consensus problem of Julia sets generated by $\lambda $ -state Potts models on diamond-like hierarchical lattice is studied. Two types of the consensus problem of Julia sets are considered, one is with a leader and the other is with no leaders. Based on these two types, two different control protocols are proposed respectively to make systems achieve consensus of Julia sets. The simulations confirm the efficacy of control protocols.

Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here $ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $ where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.

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]

The purpose of this paper is to generate fractals of sine and cosine functions for a complex polynomial z k + c via Picard–Mann iterations. To generate Mandelbrot and Julia sets, escape criteria are established and proven for sin z k + c and cos z k + c for Picard–Mann iterations. Various input parameters for different k are used to compare images between sine and cosine. generation time and average number of iterations of the generated Mandelbrot and Julia sets are presented. [ABSTRACT FROM AUTHOR]

We present some properties the Mandelbrot set of Quartic rational maps. Every Quartic rational functions is conjugate toz4+c or λ(z³+1/z+b). We study the Mandelbrot set 𝑀λ the set of parramrters b for which the Julia set of λ(z³+1/z+b) is connected. [ABSTRACT FROM AUTHOR]

We show that for each d ∈ (0 , 2 ] there exists a meromorphic function f such that the inverse function of f has three singularities and the Julia set of f has Hausdorff dimension d. [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]