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We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions $\Sigma$. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial $z\mapsto\overline{z}^d$ is the Schwarz reflection map arising from the corresponding map in $\Sigma$. We characterize the image of this embedding in $\Sigma$ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial., Comment: Figure 1 added to illustrate the main result, and some other minor changes to the introduction

We prove fixed point results for branched covering maps f of the plane. For complex polynomials P with Julia set $$J_{P}$$ these imply that periodic cutpoints of some invariant subcontinua of $$J_{P}$$ are also cutpoints of $$J_{P}$$ . We deduce that, under certain assumptions on invariant subcontinua Q of $$J_{P}$$ , every Riemann ray to Q landing at a periodic repelling/parabolic point $$x\in Q$$ is isotopic to a Riemann ray to $$J_{P}$$ relative to Q.

For entire or meromorphic function f, a value θ ∈ [0, 2π) is called a Julia limiting direction if there is an unbounded sequence {zn} in the Julia set satisfying $$\mathop {\lim }\limits_{n \to \infty } \;\arg {z_n} = \theta $$ . Our main result is on the entire solution f of P(z, f) + F(z)fs = 0, where P(z, f) is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F, with the integer s being no more than the minimum degree of all differential monomials in P(z, f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.

For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $$\mathbb {S}^n$$ , for $$n\ge 2$$ . Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in $$\mathbb {S}^n$$ , then it is the Julia set of a hyperbolic UQR map $$f:\mathbb {S}^N \rightarrow \mathbb {S}^N$$ where $$N=n$$ if $$n=2$$ and $$N=n+1$$ otherwise.

For any integers $$d\ge 3$$ and $$n\ge 1$$ , we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its Julia set except single points and Jordan curves.

For any $ \delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=\delta$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known., Comment: 53 pages, 25 figures

Diophantine approximation in dynamical systems concerns the Diophantine properties of the orbits. In classic Diophantine approximation, the powerful mass transference principle established by Beresnevich and Velani provides a general principle to the dimension for a limsup set. In this paper, we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system. More precisely, let (X, T) be a topological dynamical system where X is a compact metric space and $$T:X\rightarrow X$$ is an expanding continuous transformation. Given $$y_o\in X$$ , we consider the following limsup set $$\mathcal {W}(T,f)$$ , driven by the dynamical system (X, T), $$\begin{aligned} \Big \{x\in X: x\in B(z, e^{-S_n(f+\log |T'|)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {with infinitely many}}\ n\in \mathbb {N}\Big \}, \end{aligned}$$ where $$\log |T'|$$ is a function reflecting the local conformality of the transformation T, f is a non-negative continuous function over X, and $$S_n (f+ \log |T'|) (z)$$ denotes the ergodic sum $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$$ . By proposing a dynamical ubiquity property assumed on the system (X, T), we obtain that the dimensions of X and $$\mathcal {W}(T,f)$$ are both related to the Bowen-Manning-McCluskey formulae, namely the solution to the pressure functions $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log |T'|+f))=0, \text {respectively}. \end{aligned}$$ We call this phenomenon a dynamical dimension transference principle, because of its partial analogy with the mass transference principle. This general principle unifies and extends some known results which were considered only separatedly before. These include the b-adic expansions, expanding rational maps over Julia sets, inhomogeneous. Diophantine approximation on the triadic Cantor set and finite conformal iterated function systems.

It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map f, if f has an attracting fixed point p and all critical points are attracted to p under the iteration of f, then the Julia set J(f) is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for J(f) to be a Sierpinski curve when it is connected.

Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2π) is called a Julia limiting direction of f if there is an unbounded sequence {zn} in the Julia set J(f) satisfying limn→∞ arg zn = θ (mod 2π). We denote by L(f) the set of all Julia limiting directions of f. Our main result is that, for any non-empty compact set E ⊆ [0, 2π) and ρ ∈ [0, ∞], there are an entire function f of infinite lower order and a transcendental meromorphic function g of order ρ such that L(f) = L(g) = E. In addition, we have also constructed some transcendental entire functions whose lower order is ρ ∈ (1/2, ∞) and whose L(f) coincides with a certain kind of compact set. To prove our results, we have established a criterion for a direction θ to be a Julia limiting direction of a function by utilizing the growth rate of the function in the direction θ. The criterion may be of independent interest.

We prove that every quasisymmetric homeomorphism of a standard square Sierpinski carpet $$S_p$$ , $$p\ge 3$$ odd, is an isometry. This strengthens and completes earlier work by the authors (Bonk and Merenkov in Ann Math (2) 177:591–643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of $$S_p$$ across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet $$S_p$$ is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.

We prove the existence of rational maps whose Julia sets are Sierpinski carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpinski carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $$s\in (1,2)$$, we prove the existence of Sierpinski carpet Julia sets having Hausdorff dimension exactly s.

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, $$R_{1}, R_{2}, \ldots , R_{N}$$ , each with degree $$d_{i};\ 1 \le i \le N$$ and with at least one $$R_{i}$$ in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanski (equilibrium) measure associated to some real-valued Holder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

In this article we investigate the dynamics of the family $$F_{\lambda ,c,\mu }(z)= \lambda e^{1/(z^2+c)} + \mu $$ , where $$\lambda ,c\in {\mathbb C}{\setminus }\{0\}$$ and $$\mu \in {\mathbb C}{\setminus }\{\pm i\sqrt{c}\}$$ , with two essential singularities which are not omitted values. Choosing a slice of the space of parameters, we prove that for certain parameters $$\lambda ,c$$ and $$\mu $$ , the Fatou set contains a completely invariant and multiply connected attracting domain, a parabolic domain and a Siegel disc. Moreover, we prove that the triple $$(F_{\lambda ,c,\mu }, U, V)$$ is a polynomial-like mapping of degree two for certain values of the parameters $$\lambda $$ , c, $$\mu $$ , and some domains U and V. Also, some examples of the Fatou and Julia sets for the polynomial-like mapping are given.

In this article we derive an inequality of Łojasiewicz–Siciak type for certain sets arising in the context of the complex dynamics in dimension 1. More precisely, if we denote by dist the Euclidean distance in $$\mathbb {C}$$ , we show that the Green function $$G_K$$ of the filled Julia set K of a polynomial such that $$\mathring{K}\ne \emptyset $$ satisfies the so-called ŁS condition $$\displaystyle G_A\ge c\cdot \hbox {dist}(\cdot , K)^{c'}$$ in a neighborhood of K, for some constants $$c,c'>0$$ . Relatively few examples of compact sets satisfying the ŁS condition are known. Our result highlights an interesting class of compact sets fulfilling this condition. For instance, this is the case for the filled Julia sets of quadratic polynomials of the form $$z\mapsto z^2+a$$ , provided that the parameter a is parabolic, hyperbolic or Siegel. The fact that filled Julia sets satisfy the ŁS condition may seem surprising, since they are in general very irregular and sometimes they have cusps. However, we provide an explicit example of a curve which has a cusp and satisfies the ŁS condition. In order to prove our main result, we define and study the set of obstruction points to the ŁS condition. We also prove, in dimension $$n\ge 1$$ , that for a polynomially convex and L-regular compact set of non-empty interior, these obstruction points are rare, in a sense which will be specified.

We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must be at least 1, but whether this minimum could be attained has remained open until now. Our example also has packing dimension 1, and is the first transcendental Julia set known to have packing dimension strictly less than 2. It is also the first example with a multiply connected wandering domain where the dynamics can be completely described.

We consider $f:\hat I\to \R$ being a $C^3$ (or $C^2$ with bounded distortion) real-valued multimodal map with non-flat critical points, defined on $\hat I$ being the union of closed intervals, and its restriction to the maximal forward invariant subset $K\subset I$. We assume that $f|_K$ is topologically transitive. We call this setting the (generalized multimodal) real case. We consider also $f:\C\to \C$ a rational map on the Riemann sphere and its restriction to $K=J(f)$ being Julia set (the complex case). We consider topological pressure $P_{\spanning}(t)$ for the potential function $\varphi_t=-t\log |f'|$ for $t>0$ and iteration of $f$ defined in a standard way using $(n,\e)$-spanning sets. Despite of $\phi_t=\infty$ at critical points of $f$, this definition makes sense (unlike the standard definition using $(n,\e)$-separated sets) and we prove that $P_{\spanning}(t)$ is equal to other pressure quantities, called for this potential {\it geometric pressure}, in the real case under mild additional assumptions, and in the complex case provided there is at most one critical point with forward trajectory accumulating in $J(f)$. $P_{\spanning}(t)$ is proved to be finite for general rational maps, but it may occur infinite in the real case. We also prove that geometric tree pressure in the real case is the same for trees rooted at all `safe' points, in particular at all points except the set of Hausdorff dimension 0., Comment: 22 pages, 1 figure

For any polynomial diffeomorphism f of \({\mathbb C}^2\) with positive entropy, the Julia set of f is never \(C^1\) smooth as a manifold-with-boundary.

We consider a family of rational functions which is given by $$\begin{aligned} f_{\lambda }(z)=\frac{z^n(z^{2n}-\lambda ^{n+1})}{z^{2n}-\lambda ^{3n-1}}, \end{aligned}$$ where $$n\ge 2$$ and $$\lambda \in {\mathbb {C}}^*-\{\lambda :\lambda ^{2n-2}=1\}$$ . When $$\lambda \ne 0$$ is small, $$f_{\lambda }$$ can be seen as a perturbation of the unicritical polynomial $$z\mapsto z^n$$ . It was known that in this case the Julia set $$J(f_\lambda )$$ of $$f_\lambda $$ is a Cantor set of circles on which the dynamics of $$f_\lambda $$ is not topologically conjugate to that of any McMullen maps. In this paper, we prove that this is the unique case such that $$J(f_\lambda )$$ is disconnected.

The dynamics of functions $$f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0$$ is studied showing that there exists $$\lambda ^* > 0$$ such that the Julia set of $$f_\lambda $$ is disconnected for $$0< \lambda < \lambda ^*$$ whereas it is the whole Riemann sphere for $$\lambda > \lambda ^*$$ . Further, for $$0< \lambda < \lambda ^*$$ , the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of $$\infty $$ is shown to be disconnected for $$0 \lambda ^*$$ . For complex $$\lambda $$ , it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions $$E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}$$ , which we call exponential-like, are studied as a generalization of $$f(z)=\frac{\mathrm{e}^{z}}{z+1}$$ which is nothing but $$E_1(z)$$ . This name is justified by showing that $$E_n$$ has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over $$\infty $$ and both are direct. Non-existence of Herman rings are proved for $$\lambda E_n $$ .

In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem., Comment: 35 pages. To appear in Comput. Methods Funct. Theory. V2: Authors' final accepted manuscript. Revisions and clarifications have been made throughout from V1. This includes improvements in the proof of Proposition 6.11 and Theorem 8.1, as well as corrections in Remarks 7.1 and 7.3 concerning differing definitions of escaping endpoints in greater generality

In this paper we prove the existence of infinitely many accessible Mandelbrot sets in the parameter plane for the family of maps $$z^n + \lambda / z^n$$ when $$n > 1$$ . These are Mandelbrot sets for which the cusp of the main cardioid touches the outer boundary of the connectedness locus. We show that there is a unique such Mandelbrot set at the landing point of each external ray that is periodic under $$\theta \mapsto n \theta $$ .

Let f be a transcendental entire function with order ρ < ½ and let σ be a sufficiently large constant. We prove that if there exists r 0 > 1 such that, for all r > r 0 and any small ɛ > 0, $$M(r^\sigma ,f) \geqslant M(r,f)^{\sigma + \varepsilon } ,$$ then every component of the Fatou set F(f) is bounded.

We study a sequence of one-parameter matrices, $$A_{k}\left( c\right) $$ , with $$k\in \mathbb {N}$$ and parameter $$c\in \mathbb {C}$$ , which is obtained recursively. For the given recursion, the characteristic polynomial of $$A_{k}\left( c\right) $$ is given by $$-f_{c}^{k}\left( x\right) $$ , where $$f_{c}\left( x\right) =c-x^{2}$$ and $$f_{c}^{k}\left( x\right) =f_{c}\circ \cdots \circ f_{c}\left( x\right) $$ . Therefore, the spectrum of each matrix $$A_{k}\left( c\right) $$ corresponds to the pre-images of $$0$$ under iteration of $$f_{c}$$ . We obtain the block structure and a recursion for the eigenvectors of $$A_k(c),\ k=0,1,\ldots $$ . The structure of the eigenspaces of $$A_k(c)$$ change dramatically with the parameter $$c$$ , in particular their dimension as real linear spaces. For $$c\in \mathbb {R}$$ , we extensively use symbolic dynamics for unimodal maps of an interval to study this problem. We show that the logarithm of the growth rate of the real eigenspaces dimensions is equal to the topological entropy of $$f_c$$ . Moreover, for $$c\in \mathbb {C}$$ , the relation between the sequence $$A_{k}\left( c\right) $$ and the iteration of $$f_{c}$$ gives us an interesting interpretation of the spectrum of $$A_{k}\left( c\right) $$ as the Julia set of $$f_{c}$$ .

Let d(c) denote the Hausdorff dimension of the Julia set J c of the polynomial f c (z) = z 2 + c. The function c ↦ d(c) is real-analytic on the interval (−5/4,−3/4), which is included in the 1/2 bulb of the Mandelbrot set. The number c = −3/4 is the parameter at which, going from the right, the attracting fixed point bifurcates to an orbit of period two. Recently [13], we studied d′(c) when c ↦ −3/4 from the right. Here, under numerically verified assumption d(−3/4) < 4/3, we will show that there exists K −3 4/− > 0 such that d′(c)(−3/4 − c)−3d(−3/4)/2+2 → −K −3 4/− , when c tends to −3/4 from the left. In particular we obtain d′(c) ↦ −∞. This case is much harder than that considered in [13].

We introduce a recurrence function in order to analyze the dynamics of semigroups of complex polynomials. We show that under a regularity hypothesis, the recurrence function is continuous in the complex plane. This is a new notion even for the case of a semigroup with just one generator.

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.

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show that for any escape parameter t, the boundary of the basin at infinity A t is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpinski curve.

We study the projection \({\pi : \mathcal{M}_d \rightarrow \mathcal{B}_d}\) which sends an affine conjugacy class of polynomial \({f : \mathbb{C} \rightarrow \mathbb{C}}\) to the holomorphic conjugacy class of the restriction of f to its basin of infinity. When \({\mathcal{B}_d}\) is equipped with a dynamically natural Gromov–Hausdorff topology, the map π becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of π are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which model the local behavior of a polynomial.

For an infinitely renormalizable quadratic map \({f_c: z\mapsto z^2+c}\) with the sequence of renormalization periods {km} and rotation numbers {tm = pm/qm}, we prove that if \({\limsup k_m^{-1} \log |p_m| >0 }\), then the Mandelbrot set is locally connected at c. We prove also that if \({\limsup |t_{m+1}|^{1/q_m}

Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B m (z), m≥2. Let θ be a simple root of p(z), V(θ) its Voronoi cell, and A m (θ) its basin of attraction with respect to B m (z). We prove that given any closed subset C of V(θ), including any homothetic copy of V(θ), there exists m 0 such that for all m≥m 0, C is also a subset of A m (θ). This implies that when all roots of p(z) are simple, the basins of attraction of B m (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B m (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions.

It is proved that the rational maps in the family {z → z m +λ/z d : λ ∈ ℂ\{0}} for integers m, d ≥ 2 have no Herman rings.

In this paper, we analyze the theory of the Julia set (J set) of Newton’s method, construct the Julia sets of Newton’s method of function $F(z)=ze^{z^{w}}$ (w∈ℂ) through iteration method, and analyze the attracting region of the two fixed points 0 and ∞ when w are different values. Consequently, we draw the following conclusions: (1) When the judge conditions for the iterative algorithm are changed to |N(z n )−z n |≤EOF, the properties of the figures in our experiments are contrary to the conclusions in (Wegner and Peterson, Fractal Creations, pp. 168–231, 1991); (2) The attracting regions of the fixed points 0 and ∞ for w=2n (n=0,±2,±4,…) are symmetrical about x-axis and y-axis; select the main argument to be in [−π,π), for arbitrary w=α (α∈ℂ), the attracting regions of the fixed points 0 and ∞ are symmetrical about the x-axis; (3) The attracting regions of the two fixed points 0 and ∞ of J set for w=±η have rotational symmetry of η times; (4) If w=−4.7, k=0.8, then the attracting regions of different magnifications display a startling similarity, J set holds infinite self-similar structures; (5) When w is a complex number, because the selection of main argument θ z in the negative x-axis is not continuous, the fault and rupture of the attracting regions of the two fixed points 0 and ∞ appear only in the negative x-axis.

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form fc(z) = z2 + c, having an associated periodic kneading sequence of the form \({\overline{\alpha\ast}}\) which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial fc, with similar dynamical properties.

The independence polynomial of a graph G is the polynomial ∑ikxk, where ik denote the number of independent sets of cardinality k in G. In this paper, we obtain the relationships between the independence polynomial of path Pn and cycle Cn with Jacobsthal polynomial. We find all roots of Jacobsthal polynomial. As a consequence, the roots of independence polynomial of the family {Pn} and {Cn} are real and dense in \((-\infty,-\frac{1}{4}]\). Also we investigate the independence fractals or independence attractors of paths, cycles, wheels and certain trees.

We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.

A complex exponential map is said to be Misiurewicz if the forward trajectory of the asymptotic value 0 lies in the Julia set and is bounded. We prove that the set of Misiurewicz parameters in the exponential family \({\lambda\exp(z),\lambda\in\mathbb{C}{\setminus}\{0\}}\), has the Lebesgue measure zero.

Julia sets are considered one of the most attractive fractals and have wide range of applications in science and engineering. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck (Physica D 125(3---4):171---182, 1999) to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter. Argyris et al. (Chaos Solitons Fractals 11(13):2067---2073, 2000) have studied the effect of noise on Julia sets and concluded that Julia sets are stable for noises of low strength, and a small increment in the strength of noise may cause considerable deterioration in the configuration of the Julia sets. It is well-known that the method of function iterates plays a crucial role in discrete dynamics utilizing the techniques of fractal theory. However, recently Rani and Kumar (J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):261---277, 2004) introduced superior iterations as a generalization of function iterations in the study of Julia sets and studied superior Julia sets. This technique is further utilized to study effectively new Mandelbrot sets and related properties (see, for instance, Negi and Rani, Chaos Solitons Fractals 36(2):237---245, 2008; 36(4):1089---1096, 2008, Rani and Kumar, J. Korea Soc. Math. Edu. Ser. D: Res. Math. Edu. 8(4):279---291, 2004). The intent of this paper is to study certain effects of noise on superior Julia sets. We find that the superior Julia sets are drastically more stable for higher strength of noises than the classical Julia sets. Finally, we make a humble attempt to discuss some applications of superior orbit in discrete dynamics and of superior Julia sets in particle dynamics.

One of the favorite display shelves in every recreational programmer’s cabinet of curiosities belongs to the Julia sets. Julia sets, named after the French mathematician Gaston Julia, have a variety of equivalent mathematical definitions; they are also remarkably easy graphical entities to generate with a minimal degree of programming. Here’s the basic idea: suppose you have a complex polynomial function f(z). (A good example-to-think-with is the quadratic function f(z) = z 2 ? c, where c is some constant complex number.) If you iterate f(z) a very large number of times, starting from a particular initial value z0, and follow the values of the successive points under iteration, you will find that the sequence of complex number values either becomes arbitrarily large (goes to infinity), or it doesn’t. So some starting z0 points ‘‘go to infinity’’ and some don’t. Plot the values of z0 whose sequences go to infinity in one color (say, white); and those whose sequences don’t in another (say, black). The black points constitute the filled Julia set of the function f, and the boundary between black and white is the Julia set. Given this definition, you might imagine that these sets are of interest only to a dedicated subculture among mathematicians; but the graphical plots of the Julia sets are in fact remarkably appealing, and form an easily-programmed class of fractals for computer programming students. Here’s the general algorithm (written in pseudocode) through which you can program a Julia set for a quadratic polynomial

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that a component of the filled-in Julia set of any polynomial is a point if and only if its forward orbit contains no periodic critical components. It follows immediately that the Julia set of a polynomial is a Cantor set if and only if each critical component of the filled-in Julia set is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992.

In this paper we study branched coverings of metrized, simplicial trees F : T → T which arise from polynomial maps f : C → C with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space PTD compactifying the moduli space of polynomials of degree D; that F records the asymptotic behavior of the multipliers of f ; and that any meromorphic family of polynomials over ∆∗ can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial enumeration of the trees that arise, and show that PT3 is itself a tree. Resume Dans ce travail, nous etudions des revetements ramifies d’arbres metriques simpliciaux F : T → T qui sont obtenus a partir d’applications polynomiales f : C → C possedant un ensemble de Julia non connexe. Nous montrons que la collection de tous ces arbres, a un facteur d’echelle pres, forme un espace contractile PTD qui compactifier l’espace des modules des polynomes de degre D. Nous montrons aussi que F enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur ∆∗ peut etre completee par un unique arbre comme sa fibre centrale. Dans le cas cubique, nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que PT3 est lui-meme un arbre.

In this paper we give a solution of the following problem: under what conditions on infinite compact sets $K_1,K_2\subset \C$ and polynomials $f_1,$ $f_2$ the preimages $f_1^{-1}\{K_1\}$ and $f_2^{-1}\{K_2\}$ coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets., Comment: 21 pages, last version accepted by GAFA

Let f be an entire transcendental map of finite order, such that all the singularities of f −1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from $${\hat {\mathbb C}}$$ . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps.

In this note we prove that the so-called Sierpi\’nski holes in the parameter plane 0 < ¦λ¦ < ∞ of the McMullen family Fλ(z) = z m + λ/z l, m ≥ 2 and l ≥ 1 fixed, are simply connected, and determine the total number of these domains.

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide.

Let ƒ(z = e2πiθz(1 + z/d)d, θ ∈ ℝ \ ℚ be a polynomial. If θ is an irrational number of bounded type, it is easy to see that ƒ(z has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of ƒ(z satisfies Dim(J(ƒ)) < 2.

We consider the dynamics of a transcendental meromorphic function f(z) with only finitely many poles and prove that if f has only finitely many weakly repelling fixed points, then there is no multiply-connected wandering domain in its Fatou set.

We establish several new properties of the escaping setI(f)={z∶f n (z)→∞ andf n(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)⊂J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.

In this paper we consider families of rational maps of degree 2n on the Riemann sphereF λ: $$F_\lambda :\bar {\mathbb{C}} \to \bar {\mathbb{C}}$$ given by $$F_\lambda (z) = z^n + \frac{\lambda }{{z^n }}$$ where $$\lambda \in \mathbb{C} - \{ 0\} $$ andn>-2. One of our goals in this paper is to describe a type of structure that we call aCantor necklace that occurs in both the dynamical and the parameter plance forF λ. Roughly speaking, such a set is homeomorphic to a set constructed as follows. Start with the Cantor middle thirds set embedded on thex-axis in the plane. Then replace each removed open interval with an open circular disk whose diameter is the same as the length of the removed interval. A Cantor necklace is a set that is homeomorphic to the resulting union of the Cantor set and the adjoined open disks. The second goal of this paper is to use the Cantor necklaces in the parameter plane to prove the existence of several new types of Sierpinski curve Julia sets that arise in these families of rational maps. Unlike most examples of this type of Julia set, the maps on these Julia sets are structurally unstable. That is, small changes in the parameter λ give rise to Julia sets on which the dynamical behavior is quite different. In addition, we also describe a new type of related Julia set which we call a hybrid Sierpinski curve.