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The Arakelov-Zhang pairing ⟨ ψ , ϕ ⟩ \langle \psi ,\phi \rangle is a measure of the “dynamical distance” between two rational maps ψ \psi and ϕ \phi defined over a number field K K . It is defined in terms of local integrals on Berkovich space at each completion of K K . We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height h ϕ h_\phi and the standard Weil height h h , and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.

For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal{H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal{H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in\mathcal{H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.

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.

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set., Comment: 22 pages, 2 figures, writing polished, results unchanged. To appear in Transactions of the American Mathematical Society

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

In this paper, we investigate the dynamics of the following family of rational maps \begin{document}$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $\end{document} with one parameter \begin{document}$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $\end{document} , where \begin{document}$ n\geq 2 $\end{document} . This family of rational maps is viewed as a singular perturbation of the bi-critical map \begin{document}$ P_{-n}(z) = z^{-n} $\end{document} if \begin{document}$ \lambda \neq 0 $\end{document} is small. It is proved that the Julia set \begin{document}$ J(f_\lambda) $\end{document} is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerate Sierpinski carpet provided the free critical orbits of \begin{document}$ f_\lambda $\end{document} are attracted by the super-attracting cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Furthermore, we prove that there exists suitable \begin{document}$ \lambda $\end{document} such that \begin{document}$ J(f_\lambda) $\end{document} is a Cantor set of circles but the dynamics of \begin{document}$ f_{\lambda} $\end{document} on \begin{document}$ J(f_{\lambda}) $\end{document} is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of \begin{document}$ J(f_{\lambda}) $\end{document} is also proved if the free critical orbits are not attracted by the cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Finally we give an estimate of the Hausdorff dimension of the Julia set of \begin{document}$ f_\lambda $\end{document} in some special cases.

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 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.

We consider the family of rational maps given by $$F_\lambda (z)=z^n+\lambda /z^d$$ where $$n, d \in \mathbb {N}$$ with $$1/n+1/d 1$$ is given by $$\tau _k^{n,d} = dn^{k-2}(n-1)-n^{k-1} + 1$$ .

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d, Comment: 21 pages, 2 figures

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.

We study the escaping set of functions in the class \begin{document} $\mathcal{B}^*$ \end{document} , that is, transcendental self-maps of \begin{document} $\mathbb{C}^*$ \end{document} for which the set of singular values is contained in a compact annulus of \begin{document} $\mathbb{C}^*$ \end{document} that separates zero from infinity. For functions in the class \begin{document} $\mathcal{B}^*$ \end{document} , escaping points lie in their Julia set. If \begin{document} $f$ \end{document} is a composition of finite order transcendental self-maps of \begin{document} $\mathbb{C}^*$ \end{document} (and hence, in the class \begin{document} $\mathcal{B}^*$ \end{document} ), then we show that every escaping point of \begin{document} $f$ \end{document} can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence \begin{document} $e∈\{0,∞\}^{\mathbb{N}_0}$ \end{document} , we show that the escaping set of \begin{document} $f$ \end{document} contains a Cantor bouquet of curves that accumulate to the set \begin{document} $\{0,∞\}$ \end{document} according to \begin{document} $e$ \end{document} under iteration by \begin{document} $f$ \end{document} .

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush., V2: Author accepted manuscript. To appear in Discrete Contin. Dyn. Syst

For a sequence \begin{document}$ (a_n) $\end{document} of complex numbers we consider the cubic parabolic polynomials \begin{document}$ f_n(z) = z^3+a_n z^2+z $\end{document} and the sequence \begin{document}$ (F_n) $\end{document} of iterates \begin{document}$ F_n = f_n\circ\dots\circ f_1 $\end{document}. The Fatou set \begin{document}$ \mathcal{F}_0 $\end{document} is the set of all \begin{document}$ z\in\hat{\mathbb{C}} $\end{document} such that the sequence \begin{document}$ (F_n) $\end{document} is normal. The complement of the Fatou set is called the Julia set and denoted by \begin{document}$ \mathcal{J}_0 $\end{document}. The aim of this paper is to study some properties of \begin{document}$ \mathcal{J}_0 $\end{document}. As a particular case, when the sequence \begin{document}$ (a_n) $\end{document} is constant, \begin{document}$ a_n = a $\end{document}, then the iteration \begin{document}$ F_n $\end{document} becomes the classical iteration \begin{document}$ f^n $\end{document} where \begin{document}$ f(z) = z^3+a z^2+z $\end{document}. The connectedness locus, \begin{document}$ M $\end{document}, is the set of all \begin{document}$ a\in\mathbb{C} $\end{document} such that the Julia set is connected. In this paper we investigate some symmetric properties of \begin{document}$ M $\end{document} as well.

The Julia set of the exponential family $E_{\kappa}:z\mapsto\kappa e^z$, $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0, Comment: 33 pages, 3 figures, final version, to appear in Ergodic theory and Dynamical Systems

We prove P. Alexandersson’s conjecture that for every complex polynomial p p of degree d ≥ 2 d \geq 2 the convex hull H p H_p of the Julia set J p J_p of p p satisfies p − 1 ( H p ) ⊂ H p p^{-1}(H_p) \subset H_p . We further prove that the equality p − 1 ( H p ) = H p p^{-1}(H_p) = H_p is achieved if and only if p p is affinely conjugated to the Chebyshev polynomial T d T_d of degree d d , to − T d -T_d , or to a monomial c z d c z^d with | c | = 1 |c|=1 .

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 present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

In [ 6 ], regularly ramified rational maps are constructed and Julia sets of these maps in some one-parameter families are explored through computer-generated pictures. It is observed that they have classifications similar to the Julia sets of maps in the families \begin{document}$ f_n^{c}(z) = z^n+\frac{c}{z^n}$\end{document} , where \begin{document}$ n≥ 2$\end{document} and \begin{document}$ c$\end{document} is a complex number. A new type of Julia set is also presented, which has not appeared in the literature. We call such a Julia set an exploded McMullen necklace. We prove in this paper: if a map \begin{document}$ f$\end{document} in the one-parameter families given in [ 6 ] has a superattracting fixed point of order greater than 2, then its Julia set \begin{document}$ J(f)$\end{document} is either connected, a Cantor set, or a McMullen necklace (either exploded or not); if such a map \begin{document}$ f$\end{document} has a superattracting fixed point of order equal to 2, then \begin{document}$ J(f)$\end{document} is either connected or a Cantor set.

We prove that for any finite collection of quadratic Julia sets, a polynomial and a transcendental entire function exist whose Julia sets include copies of the given quadratic Julia sets. In order to prove the result, we construct quasiregular maps with required dynamics and employ the quasiconformal surgery to obtain the desired functions.

We consider the family of rational maps given by Fλ(z)=zn+λ/zd where n,d∈N with 1/n+1/d

Much recent work on the iterates of a transcendental entire function f f has been motivated by Eremenko’s conjecture that all the components of the escaping set I ( f ) I(f) are unbounded. We prove several general results about the topological structure of I ( f ) I(f) including the fact that if I ( f ) I(f) is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of the fast escaping set. We give analogous results for the intersection of I ( f ) I(f) with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.

Let $d(\varepsilon)$ and $\mathcal D(\delta)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_\delta(z)=(1+\delta)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(\delta)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $\delta$ parametrization) which is outside the Mandelbrot set and is related to the parabolic implosion phenomenon. We prove that for directions in the closed left half-plane the derivative of $d$ is negative. Computer calculations show that it is negative except a cone (with opening angle approximately $150^\circ$) around $\mathbb{R}^+$.

In this work we define a stochastic adding machine associated to the Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$. We obtain a Markov chain whose states are the set of nonnegative integers. We study probabilistic properties of this chain, such as transience and recurrence. We also prove that the spectrum associated to this Markov chain is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$.

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

Let ƒ be Fatou's function, that is, ƒ(z)= z+1+e-z. We prove that the escaping set of ƒ has the structure of a 'spider's web', and we show that this result implies that the non-escaping endpoints of the Julia set of ƒ together with infinity form a totally disconnected set. We also present a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web, and we point out that the same property holds for some families of functions.

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.

The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size $4$, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form $f_c(z)= z^2+ c$ with $|c|< \frac{1}{4}$, we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being $3$ or $4$ for the diffusive coupling. For $c=0$ and $-2$, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to $5$. Global synchronization for the individual map of the form $g_c(z)= z^3+ cz$ is also reported.

In this paper we describe a new structure that arises in the parameter plane of the family of maps zn+λ/zd where n≥2 is even but d≥3 is odd. We call these structures Mandelbrot–Sierpinski spokes (or, for short, ‘Mandelpinski spokes’). It is known that there are infinitely many baby Mandelbrot sets in these parameter planes that are part of what is called the Mandelpinski maze for these maps. We show here that there are infinitely many ‘spokes’ emanating from each of these Mandelbrot sets. Each spoke consists of infinitely many alternating Mandelbrot sets and Sierpinski holes that lie along a certain arc that tends away from the given Mandelbrot set in a certain direction.

For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points $1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{1/4-c})$ when real $c\nearrow 1/4$; and is uniformly $O(1/\sqrt{-2-c})$ when real $c\nearrow -2$. These estimates of the derivative imply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches these boundary points. In particular, the Hausdorff distance between $J(q_c)$ with $0\le c, 15 pages, 2 figures

We study the Chebyshev-Halley family of root finding algorithms from the point of view of holomorphic dynamics. Numerical experiments show that the speed of convergence to the roots may be slower when the basins of attraction are not simply connected. In this paper we provide a criterion which guarantees the simple connectivity of the basins of attraction of the roots. We use the criterion for the Chebyshev-Halley methods applied to the degree $n$ polynomials $z^n+c$, obtaining a characterization of the parameters for which all Fatou components are simply connected and, therefore, the Julia set is connected. We also study how increasing $n$ affects the dynamics., 27 pages, 11 figures

In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$., Comment: 16 pages, 1 figue

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urb\'{a}nski \cite{RU}., Comment: 19 pages, 2 figures To appear in Discrete and Continuous Dynamical Systems

If the Julia set of a quartic polynomial with certain conditions is neither connected nor totally disconnected, there exists a homeomorphism between the set of all components of the filled-in Julia set and some subset of the corresponding symbol space. The question is to determine the quartic polynomials exhibiting such a dynamics and describe the topology of the connected components of their filled-in Julia sets. In this paper, we answer the question, namely we show that for any two quadratic Julia sets, there exists a quartic polynomial whose Julia set includes copies of the two quadratic Julia sets.

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.

We investigate the link between the relations of the saddle sets and the continuity of fiber Julia sets for Axiom A polynomial skew products on C2. It turns out that the fiber Julia set Jz continuously depends on z in Kp if and only if there exist no relations between saddle sets over Jp and over Ap.

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 $$\theta \mapsto n \theta $$ .

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.

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 are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph-directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. For example, it fails in general for self-conformal sets, self-affine sets and Julia sets. We also give applications of our results concerning Ahlfors regularity. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and $\delta$-approximate packing pre-measure coincide for sufficiently small $\delta>0$., Comment: 21 pages. This version includes applications concerning the weak separation property and Ahlfors regularity. To appear in Journal of Fractal Geometry

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family $F_{\lambda,n}(z) = z^n + \lambda/z^n$, $n \ge 2$, $\lambda \in \mathbb{C} \setminus \{0\}$. In particular, we show that $\dim_H J(F_{\lambda,n}) = \mathcal O (1/\ln |\lambda|)$ for large $|\lambda|$, and $\dim_H J(F_{\lambda,n}) = 1 + \mathcal O (1/\ln n)$ for large $n$ in the three cases: when $J(F_{\lambda,n})$ is a Cantor set, a Cantor set of quasicircles and a Sierpinski curve.

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 $$f_{c}$$ .

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics., Some new remarks, examples and bibliographic entries

It was known that the Sierpi\'{n}ski carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well known rational maps, such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpi\'{n}ski carpets., Comment: 12 pages, 3 figures

Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of $f$ on the Julia set if and only if $f$ does not have parabolic basins and Siegel disks. If such $g$ exists, then the degree can be chosen such that $\text{deg}(g)\leq 7d-2$. In particular, if $f$ is a polynomial, then $g$ can be chosen such that $\text{deg}(g)\leq 4d+4$. Moreover, some quartic and cubic rational maps whose Julia sets contain buried Jordan curves are also constructed., 37 pages, 10 figures; to appear in Transactions of AMS