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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.

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.

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 ) .

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.

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 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.

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.

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 .

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.

We show the continuity of the Julia set and its Hausdorff dimension about a family of rational maps concerning 2-dimensional diamond hierarchical Potts models about anti-ferromagnetic coupling in statistical mechanics.

Let E denote the class of all transcendental entire functions f ( z ) = ∑ n = 0 ∞ a n z n for z ∈ C and a n ⩾ 0 for all n ⩾ 0 such that f ( x ) > 0 for x 0 and the set of all (finite) singular values of f forms a bounded subset of R . For each f ∈ E , one parameter family S = { f λ ≡ λ f : λ > 0 } is considered. In this paper, we mainly study the dynamics of functions in the one parameter family S . If f ( 0 ) ≠ 0 , we show that there exists a positive real number λ ∗ (depending on f) such that the bifurcation and the chaotic burst occur in the dynamics of functions in the one parameter family S at the parameter value λ = λ ∗ . If f ( 0 ) = 0 , it is proved that the Julia set of f λ is equal to the complement of the basin of attraction of the super attracting fixed point 0 for all λ > 0 . It is also shown that the Fatou set F ( f λ ) of f λ is connected whenever it is an attracting basin and the immediate basin contains all the finite singular values of f λ . Finally, a number of interesting examples of entire transcendental functions from the class E are discussed.

In the present paper, a class F of critically finite transcendental meromorphic functions having rational Schwarzian derivative is introduced and the dynamics of functions in one parameter family K ≡ { f λ ( z ) = λ f ( z ) : f ∈ F , z ∈ C ˆ and λ > 0 } is investigated. It is found that there exist two parameter values λ ∗ = ϕ ( 0 ) > 0 and λ ∗ ∗ = ϕ ( x ˜ ) > 0 , where ϕ ( x ) = x f ( x ) and x ˜ is the real root of ϕ ′ ( x ) = 0 , such that the Fatou sets of f λ ( z ) for λ = λ ∗ and λ = λ ∗ ∗ contain parabolic domains. A computationally useful characterization of the Julia set of the function f λ ( z ) as the complement of the basin of attraction of an attracting real fixed point of f λ ( z ) is established and applied for the generation of the images of the Julia sets of f λ ( z ) . Further, it is observed that the Julia set of f λ ∈ K explodes to whole complex plane for λ > λ ∗ ∗ . Finally, our results found in the present paper are compared with the recent results on dynamics of one parameter families λ tan z , λ ∈ C ˆ ∖ { 0 } [R.L. Devaney, L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. Ecole Norm. Sup. 22 (4) (1989) 55–79; L. Keen, J. Kotus, Dynamics of the family λ tan ( z ) , Conform. Geom. Dynam. 1 (1997) 28–57; G.M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. 49 (1994) 281–295] and λ e z − 1 z , λ > 0 [G.P. Kapoor, M. Guru Prem Prasad, Dynamics of ( e z − 1 ) z : The Julia set and bifurcation, Ergodic Theory Dynam. Systems 18 (1998) 1363–1383].

Bergweiler proved that for any given integer k ⩾ 2 , every polynomial P of degree d ⩾ 2 has at least one repelling periodic cycle of period k unless ( k , d ) ∈ { ( 2 , 2 ) , ( 2 , 3 ) , ( 2 , 4 ) , ( 3 , 2 ) } . Here we classified these exceptional polynomials. We also showed that the Julia sets of these exceptional polynomials are connected.

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.

Pantograph equations are characterized by the presence of a linear functional argument. These equations arise in several applications and often the argument has a repelling fixed point at the origin. Recently, Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157–170] studied a related class of functional differential equations with nonlinear functional arguments and showed that, generically, solutions to such equations have a natural boundary. Their approach uses some well-known properties of the Julia set and relies heavily on the nonlinearity of the functional argument. The method is not directly applicable to pantograph type equations though some of the techniques can be exploited. In this paper we show that solutions to pantograph equations generally have natural boundaries. We focus on a special set of solutions that have the imaginary axis as a natural boundary.

This paper discusses the limit functions of a random iteration system formed by finitely many rational functions. Applying these results we prove that a hyperbolic iteration system has no wandering domain and that its limit functions are constant. Finally the continuity on its Julia set is considered.

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation. The pantograph equation contains a linear functional argument. In this paper we generalize this functional argument to include nonlinear polynomials. In contrast to the entire solutions generated by the pantograph equation for the retarded case, we show that in the nonlinear case natural boundaries occur. These boundaries are linked to the Julia set of the polynomial functional argument.

In this paper we investigate conditions for conservativity and exactness ofd-to-1 endomorphisms. We prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized Lebesgue measure on the sphere, or have an attracting property inside the Julia set. We provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property.

The complexification of the standard family of circle maps θ↦θ+α+βsin(θ)mod(2π), whose parameter space contains the well-known Arnold tongues, is given byFαβ(ω)=ωeiαe(β/2)(ω−1/ω), a holomorphic map of C* with essential singularities at 0 and ∞. For real values of the parameters, we study the dynamical plane of the familyFαβ. Near the essential singularities we prove the existence ofhairsin the Julia set, an invariant set of curves organized by some symbolic dynamics, whose points (that are not endpoints) tend exponentially fast to 0 or ∞ under iteration. For β1 we prove that, under certain conditions, all hairs end up attached to the unit circle as we increase the parameter.

In this paper we show that the Julia set g (λ sin z) is the whole plane C for certain real λ with ¦λ¦ > 1 by considering some connections between the chaotic set of λ sin x with xϵ[−λ,λ]⊂ R and the Julia set of λ sin with zϵ C . Denote by Fa1,a2,a3,a4 the function a1 sin(a2z + a3) + a4. We also show that J (FFa1,a2,a3,a4= C for certain complex parameters a1, a2, a3, and a4 by considering the dynamical behaviour of the singularities of Fa1,a2,a3,a4−1 under Fa1,a2,a3,a4. The idea used in the proof of Theorem 1 can also be applicable to more general functions.

A useful formula is given for the coefficients of the conformal mapping from the unit disk onto the complement of the reciprocal of the Mandelbrot set. As a consequence, a number of coefficients are determined. Finally, a conjecture is made that would imply the local connectedness of the boundary of the Mandelbrot set.

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E = E ( f ) and the cluster set of f at any a ∈ E with respect to E c = C ˆ \ E is equal to C ˆ . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f ○ g and g ○ f . Denote by F ( f ) and J ( f ) the Fatou and Julia sets of f. Let U be a component of F ( f ○ g ) and V be a component of F ( g ○ f ) which contains g ( U ) . We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.

We characterize the Julia sets of certain exponential functions. We show that the Julia sets J ( F λ n ) of F λ n ( z ) = λ n e z n where λ n > 0 is the whole plane C , provided that lim k → ∞ F k λ n (0) = ∞. In particular, this is true when λ n are real numbers such that λ n > 1 ne 1/ n . On the other hand, if 0 λ n 1 ne 1/ n , then J ( F λ n ) is nowhere dense in C and is the complement of the basin of attraction of the unique real attractive fixed point of F λ n . We then prove similar results for the functions[formula] where λ i ∈ C − {0}, 1 ≤ i ≤ n + 1, a j > 1, 1 ≤ j ≤ n , and m , n ≥ 1.

We show the existence of chaotic (in the sense of Devaney) polynomials on Banach spaces of q-summable sequences. Such polynomials P consist of composition of the backward shift with a certain fixed polynomial p of one complex variable on each coordinate. In general we also prove that P is chaotic in the sense of Auslander and Yorke if and only if 0 belongs to the Julia set of p.