Search

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.

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.

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.

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 show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

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.

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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.

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 ) .

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 study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K., 1 Figure

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

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

Let f be a post-critically finite endomorphism (PCF map for short) on $${\mathbb {P}}^2$$ , let $$J_1$$ denote the Julia set and let $$J_2$$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $$J_1\setminus J_2$$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $$x\in J_2$$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing x. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of PC(f) are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case $$J_1\setminus J_2$$ is contained in the union of the basins of critical component cycles, and for every $$x\in J_2$$ there is no Fatou disk containing x. As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on $${\mathbb {P}}^2$$ with no sporadic super-saddle cycles. 2. We give a new proof of de Thelin’s laminarity of the Green currents in $$J_1\setminus J_2$$ for PCF maps on $${\mathbb {P}}^2$$ . 3. We show that for PCF maps on $${\mathbb {P}}^2$$ an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on $${\mathbb {P}}^2$$ which are expanding on $$J_2$$ or satisfy Axiom A.

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.

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1., Comment: Abstract slightly reformulated

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 .

For a sequence $(\lambda_n)$ of positive real numbers we consider the exponential functions $f_{\lambda_n} (z) = \lambda_n e^z$ and the compositions $F_n = f_{\lambda_n} \circ f_{\lambda_{n-1}} \circ ... \circ f_{\lambda_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(\lambda_n)$. Among other results, we prove the Julia set for a random sequence $\{\lambda_n \}$, chosen uniformly from a neighbourhood of $\frac{1}{e}$, is the whole plane with probability $1$. We also prove the Julia set for $\frac{1}{e} + \frac{1}{n^p}$ is the whole plane for $p < \frac{1}{2}$, and give an example of a sequence $\{\lambda_n \} $ for which the iterates of $0$ converge to infinity starting from any index, but the Fatou set is non-empty.

Let P be a non-linear polynomial, K_P the filled Julia set of P, f a renormalization of P and K_f the filled Julia set of f. We show, loosely speaking, that there is a finite-to-one function \lambda from the set of P-external rays having limit points in K_f onto the set of f-external rays to K_f such that R and \lambda(R) share the same limit set. In particular, if a point of the Julia set J_f=\partial K_f of a renormalization is accessible from C\setminus K_f then it is accessible through an external ray of P (the inverse is obvious). Another interesting corollary is that: a component of K_P\setminus K_f can meet K_f only at a single (pre-)periodic point. We study also a correspondence induced by \lambda on arguments of rays. These results are generalizations to all polynomials (covering notably the case of connected Julia set K_P) of some results of Levin-Przytycki, Blokh-Childers-Levin-Oversteegen-Schleicher and Petersen-Zakeri where the case is considered when K_P is disconnected and K_f is a periodic component of K_P., Comment: Minor corrections. To appear in Bull. Polish Acad. Sci. Math

Fractal geometry is a structure that is constructed of an element geometry (points, lines, areas, and space) and these elements are experiencing faults equation is not continuous, monotonous go up or down the course, the graph circular, blending and converging to the center, and size scale in each substructure same. This is similar to the principle of cell division, mitosis is the process of cell division that splits into two cells, and each cell has the same chromosomes as their parent. Mitosis is usually followed by cytokinesis, the division of the cytoplasm to two identical daughter cells. The stages of mitotic division consists of prophase, metafese, anaphase, telophase and interphase. Of the five stages only obtained three stages which can be searched equation through the Julia set algorithm, namely prophase, telophase, and interphase. The mathematical equation for prophase and interphase are the same, namely , the difference is the position x. At prophase position x is -0,9 ? x ? 0,1 while at the interphase stage position x is -0,9 ? x ? 0,9. The mathematical equation for telophase stage is .

The family of exponential maps ƒα(z)=ez + α is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(ƒα). When α ∈ (−∞,−1), and more generally when α belongs to the Fatou set F(ƒα), it is known that J(ƒα) can be written as a union of hairs and endpoints of these hairs. In 1990, Mayer proved for α ∈ (−∞,−1) that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where α ∈ F(ƒα), and showed that it holds even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a ‘spider’s web’; in particular we give a new topological characterisation of spiders’ webs that maybe of independent interest. We also show how our results can be applied to Fatou’s function, z ↦ z + 1 + e−z.

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 this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely \begin{document}$ P_{c} (z) = z^{2} + c $\end{document} with \begin{document}$ |c| and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely \begin{document}$ P_{(a_{1}, \, a_{0})} (z) = z^{3} + a_{1} z + a_{0} $\end{document} with \begin{document}$ |a_{i}| , restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.

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

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 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

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

Permutable rational functions have identical Julia set. However, is the converse right? We have found a class of rational functions with degree two, they have identical Julia set but are not permutable. Moreover, they have hyperbolic periodic points only.

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

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

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 $\mathbb{R}^+$.

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.

Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in $\mathbb{C}$. We show that there are many transcendental self-maps of $\mathbb{C}^*$ for which the Julia set is such a spider's web, and we construct a transcendental self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this structure and hence is connected. By way of contrast with transcendental entire functions, we conjecture that there is no transcendental self-map of $\mathbb{C}^*$ for which the fast escaping set $A(f)$ is such a spider's web.

We construct a family of transcendental entire functions whose Julia sets have packing dimension in $(1,2)$. These are the first examples where the computed packing dimension is not $1$ or $2$. Our construction will allow us further show that the set of packing dimensions attained is dense in the interval $(1,2)$, and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension., Comment: 37 pages, 5 figures

We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic, i.e., for which the critical point $0$ is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric $\rho(z) |dz|$, where $\rho(z) = 1+\frac{1}{\textrm{dist}(z,P(f))^{1-1/d}}$, and where $P(f)$ is the postcritical set of $f$. We also show that this metric is H\"older equivalent to the usual Euclidean metric., Comment: 14 pages, 1 figure; simplified and streamlined the proof of Theorem 12, several other small changes and corrections of minor errors

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 $$\lambda E_n $$ .