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We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order \ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding \ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when \ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point, To appear in Ergodic Theory and Dynamical Systems

The article describes an algorithm for generating fractals using polar coordinates. The classic Julia and Mandelbrot polynomial iteration applied to a complex number is replaced by an iteration with separate functions for distance and angle. A polynomial function is used for an angle and a power function for a distance. Varying the functions parameters allows to create a wide range of attractive pictures. Distance values ​​are used for coloring fractal images.

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity., Comment: 19 pages. V2: Small number of minor corrections made from V1

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.

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.

We study the Poincare series of rational maps. By investigating the property of conical Julia set and dissipative measure, we prove that the Poincare critical exponents are equal to the hyperbolic dimensions for a large class of rational maps.

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

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.

Since the introduction of complex fractals by Mandelbrot they gained much attention by the researchers. One of the most studied complex fractals are Mandelbrot and Julia sets. In the literature one can find many generalizations of those sets. One of such generalizations is the use of the results from fixed point theory. In this paper we introduce in the generation process of Mandelbrot and Julia sets a combination of the S -iteration, known from the fixed point theory, and the s -convex combination. We derive the escape criteria needed in the generation process of those fractals and present some graphical examples.

Let f: $$\hat {\Bbb C} \to \hat {\Bbb C}$$ be an arbitrary rational map of degree larger than 1. Denote by J(f) its Julia set. Let φ: J(f) → ℝ be a Holder continuous function such that P(φ) > sup(φ). It is known that there exists a unique equilibrium measure $${\mu _\varphi }$$ for this potential. We introduce a special inducing scheme with fine recurrence properties. This construction allows us to prove four main results. Firstly, dimension rigidity, i.e., we characterize all maps and potentials for which $$HD({\mu _\varphi }) = HD(J(f))$$ . As its consequence we obtain that $$HD({\mu _\varphi }) = 2$$ if and only if both the function φ: J(f) → ℝ is cohomologous to a constant in the class of continuous functions on J(f), and the rational function f: $$\hat {\Bbb C} \to \hat {\Bbb C}$$ is a critically finite rational map with a parabolic orbifold. Secondly, real analyticity of topological pressure P(tφ) as a function of t. Third, some bold stochastic laws, namely, exponential decay of correlations, and, as its consequence, the Central Limit Theorem and the Law of Iterated Logarithm for Holder continuous observables. Also, the Law of Iterated Logarithm for all linear combinations of Holder continuous observables and the function log |f′|. Finally, its geometric consequences that allow us to compare equilibrium states with the appropriate generalized Hausdorff measures in the spirit of [PUZ].

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, the dynamical behavior of different optimal iterative schemes for solving nonlinear equations with increasing order, is studied. The tendency of the complexity of the Julia set is analyzed and referred to the fractal dimension. In fact, this fractal dimension can be shown to be a powerful tool to compare iterative schemes that estimate the solution of a nonlinear equation. Based on the box-counting algorithm, several iterative derivative-free methods of different convergence orders are compared.

We give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f. Among other applications, we construct a function f with bounded singular set, whose Fatou set contains a wandering domain.

We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated arbitrarily closely in the Hausdorff topology by the Julia set of a post critically finite polynomial with two finite postcritical points.

We give an asymptotic formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps fλ(z) = zd + λ/zd, where d ≥ 3 and λ is small.

International audience; In this short note we give an updated account of the recent results on Fatou components for polynomial skew-products in complex dimension two in a neighbourhood of an invariant fiber, dividing our discussion according to the different possible kinds of invariant fibers.

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. Generalizamos as noções de estabilidade estrutural e hiperbolicidade para a família de correspondências holomorfas Hc(z) = zr + c; onde r > 1 é racional e zr = exp r log z: Descobrimos que Hc é estruturalmente estável em todos os parâmetros hiperbólicos satisfazendo a condição de fuga. Tipicamente Hc possui infinitos pontos periódicos atratores, fato totalmente inesperado, uma vez que este número é sempre finito para aplicações racionais. O conjunto de tais pontos dá origem ao chamado conjunto de Julia dual, que é um conjunto de Cantor proveniente de um Conformal Iterated Function System. Tanto o conjunto de Julia e quanto seu dual são projeções de movimentos holomorfos de sistemas definidos em subconjuntos compactos denotados por Xc e Wc; respectivamente de um espaço de Banach. Para todo c próximo de zero: (1) mostramos que Jc é reunião de arcos quase-conformes próximos do círculo unitário; (2) o conjunto Xc é um movimento holomorfo do solenóide X0; (3) utilizando o formalismo dos estados de Gibbs, exibimos um limitante superior para a dimensão de Hausdorff de Jc. Consequentemente, Jc possui medida de Lebesgue nula.

Bieberbach constructed, in 1933, domains in $${\mathbb {C}}^2$$ which were biholomorphic to $${\mathbb {C}}^2$$ but not dense. The existence of such domains was unexpected. The special domains Bieberbach considered are basins of attraction of a cubic Henon map. This classical method of construction is one of the first applications of dynamical systems to complex analysis. In this paper, the boundaries of the real sections of Bieberbach’s domains will be calculated explicitly as the stable manifolds of the saddle points. The real filled Julia sets and the real Julia sets of Bieberbach’s map will also be calculated explicitly and illustrated with computer generated graphics. Basic differences between real and the complex dynamics will be shown.

The intent of this paper is to study the fractal patterns of one dimensional complex logistic map by finding the optimum values of the control parameter using Ishikawa iterative scheme. The logistic map is shown to have bounded and stable behaviour for larger values of the control parameter. This is well depicted via time series analysis and interesting fractal patterns as well are presented

We study analytic maps of the sphere that project to well-defined maps on the nonorientable real surface R P 2 \mathbb {RP}^2 . We parametrize all maps with two critical points on the Riemann sphere C ∞ \mathbb {C}_\infty , and study the moduli space associated to these maps. These maps are also called quasi-real maps and are characterized by being conformally conjugate to a complex conjugate version of themselves. We study dynamics and Julia sets on R P 2 \mathbb {RP}^2 of a subset of these maps coming from bicritical analytic maps of the sphere.

In this paper we mainly investigate the radial distribution of Julia set of derivatives of entire solutions to some complex linear differential equations. Under certain conditions, we find the lower bound of it, and prove the radial distribution of Julia sets of the entire solutions and their derivatives are similar, which improve some recent results.

In the theory of dynamical systems, the notion of ω \omega -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 ) (X,d) , the existence of at least one limit function on a compact subset A A of X X implies the existence of plenty of them on many supersets of A 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.

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.

We study the qualitative theory of first order di fferential equations consisting of the iteration of complex quadratic rational functions and we focus on the configuration, namely location and stability, of simple equilibrium points which correspond to periodic points of the quadratic rational functions. Our main tools are properties of Julia sets of the quadratic rational functions and the Euler-Jacobi formula.

We prove, that the Julia set of a rational function $f$ is computable in polynomial time, assuming that the postcritical set of $f$ does not contain any critical points or parabolic periodic orbits.

In this paper, we consider a rational map f of degree at least two acting on Riemman sphere Open image in new window that is expanding away from critical points. Assuming that all critical points of f in the Julia set J(f) are reluctantly recurrent, we prove that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension, and the Lebesgue measure of Julia set is zero when the Julia set J(f) ≠ Open image in new window.

This article studies the iterative behaviour of a quasiregular mapping that is an analogue of a sine function. We prove that the periodic points of S form a dense subset of . We also show that the Julia set of this map is in the sense that the forward orbit under S of any non-empty open set is the whole space . The map S was constructed by Bergweiler and Eremenko (Ann. Acad. Sci. Fenn. Math. 36 (2011), pp. 165–175) who proved that the escaping set is also dense in .

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.

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, we show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin at infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier in [9] using grand orbits. In addition, we have combinatorial rigidity in the sense that if a finite set of external rays separates the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component.

Consider the family $f_{\lambda, \gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}$ where $\lambda$ and $\gamma$ are non-zero complex numbers. It contains the sine family $\lambda \sin z$ and is a natural extension of the sine family. We give a direct proof of that the escaping set $I_{\lambda, \gamma}$ of $f_{\lambda, \gamma}$ supports no $f_{\lambda,\gamma}$-invariant line fields.

Any planar shape $P\subset \mathbb{C}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^3$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q = S \setminus P$ is the associated {\em cap}. We study the cap construction when the curvature is harmonic measure on the boundary of $(\hat{\mathbb{C}}\setminus P, \infty)$. Of particular interest is the case when $P$ is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy., We make significant changes to the structure of the article, reordering sections and adjusting definitions. We also added details to clarify arguments

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.

We prove the form of dynamical rigidity of transcendental meromorphic functions which asserts that if two tame transcendental meromorphic functions restricted to their Julia sets are topologically conjugate via a locally bi-Lipschitz homeomorphism, then they, treated as functions defined on the entire complex plane , are topologically conjugate via an affine map, i.e. a map from to of the form z???az?+?b. As an intermediate step we show that no tame transcendental meromorphic function is essentially affine.

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

The iterates of a uniformly quasiregular map acting on a Riemannian manifold are quasiregular with a uniform bound on the dilatation. There is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We construct the first examples of uniformly quasiregular mappings that have a 2-torus as the Julia set. The spaces supporting this type of mappings include the Hopf link complement and its lens space quotients.

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

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

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.