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Recently a hybrid secure image encryption scheme based on Julia set and three-dimensional Lorenz chaotic system was suggested. The offered scheme was based on the multiplicative operation carried out by Julia set fractals and bitwise XOR which was implemented by the trajectories obtained from the Lorenz chaotic system. This paper presents the cryptanalysis of the scheme through a chosen-plaintext attack by taking three different chosen images. In this work, a practical chosen-plaintext attack is performed which reveals that the cryptosystem depending on only multiplication and bitwise XOR operation can be effortlessly attacked. The results of the analysis performed on the retrieved image reflect the effectiveness of the chosen-plaintext attack which fruitfully produces the original image.

This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system’s Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

In this paper, we propose a dynamics-based approach to develop efficient matrix iterations for computing the matrix sign function. Considering an iterative method with unknown weight functions, our approach constructs a double sequence of weight functions so that for a quadratic polynomial with distinct roots, the Julia set of the method associated with each term of the sequence is the perpendicular bisector of the line segment from one root to another. Based on the sequence-generated methods, we introduce a double sequence of matrix iterations for computing the matrix sign function, and perform precise convergence and stability analyses of these iterations. We show that our matrix iterations contain the well-known globally convergent principal Pade family and its reciprocal as special cases. Based on the result of the numerical experiments, we found several of our methods that do not belong to the principal Pade family and its reciprocal, which are more efficient in computing the matrix sign function.

This paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. The fractal characteristic of the Julia set is discussed after introducing the discrete version of this model. Then, control of the model’s Julia set is studied via two different control strategies. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

The aim iof ithis ipaper iis ito discuss ithe connectedness of Julia sets of quadratic polynomials. This paper includes the iiteration of irational ifunction iand ithe theorems iwhich icharacterize the iJulia iset as iconnected or itotally idisconnected. The repelling fixed points of f are discussed. If f(z)=z2-2, J(f) iis ia isubset of the real interval i[-2,2]. J(f) iis iconnected and iit iis the interval i[-2,2].

This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in M ( 2 , C ) in terms of the corresponding polynomials in C . A normality criterion given by Aladro–Krantz ( [1] ) plays a crucial role in proving this result. Further, we construct Green functions and Bottcher-type functions for these matrix-valued polynomials.

This paper introduces an effective image encryption method based on fractals for colored images. Fractals are now widely used in image security due to its randomness property and infinite boundaries. Since images are traversed over the public channel in day-to-day life confidentiality, integrity and authenticity must be pledged for prosperous transmission. This method capsulizes a larger key space and complex behavior to accomplish a productive confusion and diffusion techniques in a fruitful way. Julia set and Mandelbrot set are constructed and active confusion is exerted till the image pixels are altogether disordered and ensuing diffusion with XOR operation. Experimental results show the ability of the proposed system with respect to various statistical measures.

In this paper, the two-parameter family F b ≡ { f λ , μ ( z ) = λ b z + μ b − z for z ∈ C : λ ≥ μ > 0 } of transcendental entire functions is considered. By investigating on the existence and nature of the fixed points of f λ , μ , the dynamics of functions f λ , μ ∈ F b is studied. It is shown that the Julia set of f λ , μ is nowhere dense and not locally connected whenever parameters λ and μ satisfy 1 + 4 λ μ ( ln ⁡ b ) 2 ≤ ln ⁡ ( 1 + 1 + 4 λ μ ( ln ⁡ b ) 2 2 λ ln ⁡ b ) and the Julia set is the whole of extended complex plane for 1 + 4 λ μ ( ln ⁡ b ) 2 > ln ⁡ ( 1 + 1 + 4 λ μ ( ln ⁡ b ) 2 2 λ ln ⁡ b ) .

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.

The first example of a meromorphic function with non-invariant Baker domains was published by Baker et al. (1991) [2]. In this article we answer some questions concerning connectivity of Baker domains of that meromorphic function and location of the iterates of critical values. We analyze dynamics on the Fatou set and connectivity of the Julia set. We extend some of the achieved results to the family f λ , μ ( z ) = λ exp ( z ) + μ z of transcendental meromorphic functions, where λ , μ ∈ C ∖ { 0 } . We also prove that some ‘non-parabolic’ bifurcations appear many times in this family.

Considering a family of rational maps Tnλ concerning renormalization transformation, we study the relations between quasi-discs and the Fatou components of Tnλ. Moreover, we give a perfect description whether the boundary of each Fatou component is a quasi-circle or not for parameters n ∈ N and λ ∈ R .

We show that the dynamics of sufficiently dissipative semi-Siegel complex Henon maps with golden-mean rotation number is not J-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense G δ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.

In memory of Sibe Mardesic, our friend. Sibe Mardesic has enriched algebraic topology developing shape and strong shape theories with important constructions and theorems. This paper relates computational topology to shape theory. We have developed some algorithms and implementations that under some conditions give a shape resolution of some Julia sets. When a semi-flow is induced by a rational map g of degree d defined on the Riemann sphere, one has the associated Julia set J ( g ) . The main objective of this paper is to give a computational procedure to study the shape of the compact metric space J ( g ) . Our main contribution is to provide an inverse system of cubic complexes approaching J ( g ) by using implemented algorithms based in the notion of spherical multiplier. This inverse system of cubical complexes is used to: (i) obtain nice global visualizations of the fractal structure of the Julia set J ( g ) ; (ii) determine the shape of the compact metric space J ( g ) . These techniques also give the possibility of applying overlay theory (introduced by R. Fox and developed among others by S. Mardesic) to study the symmetry properties of the fractal geometry of the Julia set J ( g ) .

A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27] ) shows that if f is a complex polynomial of degree d , then there is a finite set S d depending only on d such that, given any root α of f , there exists at least one point in S d converging under iterations of N f to α . Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton's method. We show that for all order σ ≥ 2 , there exists a complex polynomial f such that the Julia set of Konig's method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of Konig's method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric.

We consider an analogy among Markov shifts, complex dynamical systems and self-similar maps. Their dynamics are given by 0–1 matrices A , rational functions R and self-similar maps γ on a compact metric space K , respectively. If the 0–1 matrix A is irreducible and not a permutation, then the Cuntz–Krieger algebra O A is simple and purely infinite. Similarly, if the rational function R is restricted to the Julia set J R and the self-similar map γ satisfies the open set condition respectively, then the associated C ⁎ -algebras O R ( J R ) and O γ ( K ) are simple and purely infinite. Let Σ A be the associated infinite path space for the 0–1 matrix A , then C ( Σ A ) is known to be a maximal abelian subalgebra of O A . In this paper we shall show that C ( J R ) is a maximal abelian subalgebra of O R ( J R ) and C ( K ) is a maximal abelian subalgebra of O γ ( K ) .

We give heat kernel estimates on Julia sets J(fc) for quadratic polynomials fc(z)= z2 + c for c in the main cardioid or the ± 1 k -bulbs where k ≥2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.

This paper extends the study of noise-perturbed planar Julia sets to the spatial case by studying the Julia set of a complex Lorenz system. We give the definition of the Julia set of a complex Lorenz system and visualize its spatial fractal structure. Then, the symmetry property of the 3-D slice of the CLS Julia set is proved. Finally, the influence of additive and multiplicative noises on the CLS Julia set are analyzed with respect to structural damages and changes in symmetry separately.

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.

We investigate the H\"older 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 H\"older exponents of $T$ and $C$, which allows us to determine the spectrum of pointwise H\"older exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum $\alpha_{-}$ is strictly less than $1$, which allows us to show that the averaged system acts chaotically on the Banach space $C^{\alpha }$ of $\alpha $- H\"older continuous functions for every $\alpha \in (\alpha_{-},1)$, though the averaged system behaves very mildly (e.g. we have spectral gaps) on $C^{\beta }$ for small $\beta >0.$, Comment: Published in Adv. Math. 313 (2017) 839--874. See also http://www.math.shimane-u.ac.jp/~jaerisch/ and http://www.math.h.kyoto-u.ac.jp/~sumi/index.html

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.

In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure., Comment: 12 pages, 7 figures. Submitted

We describe three different exotic topological objects that arise as Julia sets for complex maps, namely, Cantor bouquets, indecomposable continua, and Sierpinski curves.

In this paper we identify a new type of structure that lies in the parameter plane of the family of maps z n + λ / z d where n ≥ 2 is even but d ≥ 3 is odd. We call this structure a Mandelbrot–Sierpinski maze. Basically, the maze consists at the first level of an infinite string of alternating Mandelbrot sets and Sierpinski holes that lie along an arc in the parameter plane for this family. At the next level, there are infinitely many smaller Mandelbrot sets and Sierpinski holes that alternate on the arc between each Mandelbrot set and Sierpinski hole on the previous level, and then finitely many other Mandelbrot sets and Sierpinski holes that extend away from the given Mandelbrot set in a pair of different directions. And then this structure repeats inductively to produce the “Mandelpinski” maze.

Made available in DSpace on 2018-11-26T17:10:42Z (GMT). No. of bitstreams: 0 Previous issue date: 2016-11-01 Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) We define stochastic adding machines based on Cantor Systems of numeration. Our aim here is to compute the parts of spectra of the transition operators associated with these stochastic adding machines in different classical Banach spaces. We show that these spectra are connected to fibered Julia sets. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved. Univ Estadual Paulista, Dept Matemat, Inst Biociencias Letras & Ciencias Exatas, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Univ Fed Rio de Janeiro, Dept Metodos Estat, Inst Matemat, Caixa Postal 68530, BR-21945970 Rio De Janeiro, Brazil Univ Estadual Paulista, Dept Matemat, Inst Biociencias Letras & Ciencias Exatas, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil CNPq: 307776/2015-8 CNPq: 304593/2012-5 CNPq: 482519/2012-6 FAPESP: 2013/24541-0

We prove that if ξ is a quasisymmetric homeomorphism between Sierpinski carpets that are Julia sets of postcritically-finite rational maps, then ξ is the restriction of a Mobius transformation. This implies that the group of quasisymmetric homeomorphisms of a Sierpinski carpet Julia set of a postcritically-finite rational map is finite.

In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.

Consider the standard family of complex Henon 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 Henon 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 Henon maps that are perturbations of hyperbolic polynomials with connected Julia set or polynomials with a parabolic fixed point.

In order to investigate the dynamics of a quantum weightless neuron node we feed its output back as input. Due to the fact that controlled operators used in the neuron circuit usually generate entanglement, we propose a mathematical method to extract the output at time t and build from that output the input at time t + 1 . As a result the time evolution is a real-valued nonlinear map with one real parameter. The dynamics orbits are plotted showing acute sensitivity to initial conditions clearly exhibiting nonlinearity by just looking at amplitude graphs. The fractal geometry and regions of convergence are discussed by their Julia Set images and a new measure for model comparison is put forward.

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.

Let Σ ( R ) be the group of symmetries of the Julia set of a rational map R and K f , n be the Konig'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.

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.

In this article, we introduce the adapted inverse iteration method to generate bicomplex Julia sets associated to the polynomial map $w^2+c$. The result is based on a full characterization of bicomplex Julia sets as the boundary of a particular bicomplex cartesian set and the study of the fixed points of $w^2+c$. The inverse iteration method is used in particular to generate and display in the usual 3-dimensional space bicomplex Julia sets that are dendrites., Comment: 16 pages, 4 figures

The Cobra attractor have attracted very recently and the model has been investigated using classical differential operators with integer and non-integer order. The model, in the case of fractional differential operator, is able to replicate indeed the Cobra for some values of fractional order. On the other hand, Julia set has been used for many purposes, in this paper; we develop a procedure that combines some chaotic attractors with the Julia set mapping to obtain multi-roll attractors. Using our algorithm, we obtained for the first time a lung of human being.

This paper investigates Julia sets for a class of complex uncertain discrete systems involving Mittag-Leffler functions. Julia sets of some classical maps are generalized, and the influence of parameter α on the characteristics of these fractal sets is discussed. An adaptive control strategy is proposed to synchronize Julia sets of two systems with different parameters and identify their unknown parameters. An example is presented to further verify the correctness and effectiveness of main theoretical results.

Let d ( c ) denote the Hausdorff dimension of the Julia set J c of the polynomial f c ( z ) = z 2 + c . Let c 0 ∈ R be such that f c 0 has a parabolic cycle with two petals: we investigate in this paper how the bifurcation that occurs as c ∈ R crosses c 0 reflects on the variations of d ( c ) .

We study the combinatorial types of periodic orbits of the standard covering endomorphisms m k ( x ) = k x ( mod Z ) of the circle for integers k ≥ 2 and the frequency with which they occur. For any q-cycle σ in the permutation group S q , we give a full description of the set of period q orbits of m k that realize σ and in particular count how many such orbits there are. The description is based on an invariant called the “fixed point distribution” and is achieved by reducing the realization problem to finding the stationary state of an associated Markov chain. Our work generalizes earlier results on the special case where σ is a rotation cycle. It is motivated by the problem of understanding the combinatorial structure of the Julia sets of polynomial maps of degree ≥3 and their associated parameter spaces.

In this paper, we study the dynamical behavior of the family of complex rational maps which is given by f λ ( z ) = z n ( z 2 n − λ n + 1 ) z 2 n − λ 3 n − 1 , where n ≥ 2 and λ ∈ C ⁎ − { λ : λ 2 n − 2 = 1 } . This family of rational maps can be seen as a perturbation of the unicritical polynomial z ↦ z n if λ is small. We prove that the Julia set J ( f λ ) of f λ is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerated Sierpinski carpet provided one of the free critical points of f λ is escaping to the origin or to the infinity. In particular, we prove that there exists suitable λ such that the Julia set J ( f λ ) is a Cantor set of circles, but f λ is not topologically conjugate to any McMullen map on their corresponding Julia sets.

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.

This paper is a sequel to Part I [13] and Part II [14] , [15] . In the current article we relate several combinatorial descriptions of the Julia sets for hyperbolic polynomial diffeomorphisms of C 2 : quotients of solenoids [3] , automata [22] and Hubbard trees [14] , [15] . The notion of iterated monodromy groups are defined for such diffeomorphisms and are used to construct automata from Hubbard trees.

In recent literature, researchers have generated Julia sets and Mandelbrot sets in Mann and Ishikawa orbits that are examples of two-step and three-step feedback processes respectively. This paper presents further generalization of Julia and Mandelbrot sets for complex-valued polynomials such as quadratic, cubic and higher degree polynomials using a Noor orbit, which is a four-step iterative procedure. The graphical images of Julia and Mandelbrot sets have been visualized and certain patterns in Mandelbrot sets have been recognized. It is fascinating to see that a few Mandelbrot sets are akin to a butterfly or a coupled urn or a coupled trident.

This paper is devoted to studying the limit directions of Julia sets of solutions of f ( n ) + A n − 1 ( z ) f ( n − 1 ) + ⋯ + A 0 ( z ) f = 0 , where n ( ≥ 2 ) is an integer and A j ( z ) ( j = 0 , 1 , … , n − 1 ) are entire functions of finite lower order. With some additional conditions on coefficients, we know that every non-trivial solution f ( z ) of such equations has infinite lower order, and prove that the limit directions of Julia sets of f ( z ) must have a definite range of measure.

In this paper, we study the dimensions associated with the Cantor Julia set of a rational map whose Fatou set is an attracting domain. We prove that if the Julia set of such a map contains no persistently recurrent critical points, then the conformal dimension and the Hausdorff dimension of the Julia set are equal.

In this paper, control of Julia set of the complex standard family z"n"+"1=z"[email protected][email protected](z"n) is studied by adding the dissipative feedback control item to the system in two different ways. One is added to the complex system directly and the other is added to the imaginary part of the complex system. For this control method, both the system and the fixed point can be controlled at the same time even the unstable periodic orbits or the fixed points are unknown. Synchronization of two different Julia sets of this system with different system parameters is discussed by two methods, which makes one Julia set change to be another one.

Considering a family of rational maps T n λ relating to renormalization transformations, we prove that each Fatou component of T n λ is a Holder domain for odd integers n ≥ 5 and some parameters λ ∈ ( 1 , 2 ) , but T n λ 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 J ( T n λ ) , but the two endpoints of this interval belong to the boundary of some Fatou component of T n λ .

We study the parameter space structure of degree d ≥ 3 one complex variable polynomials as dynamical systems acting on C . We introduce and study straightening maps. These maps are a natural higher degree generalization of the ones introduced by Douady and Hubbard to prove the existence of small copies of the Mandelbrot set inside itself. We establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems. Also, we characterize straightening maps with compact domains. Moreover, we give two classes of bijective straightening maps. The first produces an infinite collection of embedded copies of the ( d − 1 ) -fold product of the Mandelbrot set in the connectedness locus of degree d ≥ 3 . The second produces an infinite collection of full families of quadratic connected filled Julia sets in the cubic connectedness locus, such that each filled Julia set is quasiconformally embedded.

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.

This paper is devoted to studying the dynamical properties of solutions of f ( n ) + A ( z ) f = 0 , where n ( ⩾ 2 ) is an integer, and A ( z ) is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of E ( z ) provided that E = f 1 f 2 ⋯ f n and { f 1 , f 2 , … , f n } is a solution base of such equations.

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to a question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases., Complex dynamics, 51 pages, 13 figures

The limit set of zeros of partition function of the Potts model on diamond-like hierarchical lattices is studied. It is shown that the limit set is the Julia set of a family rational maps, it is shown in a mathematically exact way that the Julia set tends to a geometrical circle and its Hausdorff dimension tends to 1 when the parameter ∣λ∣ → +∞, which gives a true answer that Bambi Hu and Bin Lin proposed in 1989, furthermore, in this paper, it give a perfect description about this relations. Also the continuity of level diameter of Aλ(1) of this physical model about λ is discussed.

In this paper, we achieve the control and synchronization of Julia sets in coupled map lattice using gradient control and optimal control respectively. The control of the Julia sets is accomplished by controlling the stable space of the fixed plane. Moreover, the synchronization of two different Julia sets is also accomplished by their trajectories synchronization. To verify the feasibility of these control methods, we consider the Julia sets, whose lattice length is two, as examples to achieve their control and synchronization using different methods respectively. The numerical simulations are also shown to illustrate the effectiveness of these control methods.