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In this paper, a novel reversible blind dual-color image watermarking algorithm is proposed by using singular value decomposition (SVD), Hachimoji Deoxyribonucleic Acid (HDNA) biogenetic encryption, coupled map lattice-based Tent–Sine system (TSS-CML) and mathematical Julia set. For watermark embedding, the watermark image is firstly encrypted using 8-bases HDNA sequences, TSS-CML and Julia set image. Then the encrypted HDNA watermark is obtained. Next, decompose the host image into equal non-overlapping blocks and utilize SVD on the randomly selected blocks. Further, embed the HDNA watermark through modifying the relation between the elements in the first column of the matrix U or V. The watermarked image can be eventually attained by carrying out the inverse SVD on all selected blocks. Also a reliable extraction algorithm is designed to recover the watermark from the possibly attacked watermarked images without resorting to the original image. Experimental and analysis results demonstrate that the proposed watermarking scheme has not only an excellent imperceptibility but a strong robustness against the common image processing attacks, geometric attacks and some composite attacks. In addition, the running time taken for hiding and exacting is about 1 s, which is suitable for real-time network transmission and application. In conclusion, the proposed method outperforms the related dual images watermarking algorithms in terms of time performance, extraction effect and robustness.

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.

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.

The newly introduced family of complex chaotic maps has different behavior concerning the well-known complex chaotic maps. These complex maps have excellent properties such as invariant measures and fractal features. The present study introduces a new algorithm that, as a standard format, can be used in cyberspace. To achieve a low automatic correlation, S-box is constructed based on a complex map. Then, an image encryption algorithm based on a designated S-box is proposed. The complex and ergodic nature of the introduced hierarchy proposed the large and safe Keyspace for encryption. Theoretical and experimental results have indicated that the proposed algorithm is high sensitivity to key, and robust to attacks.

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.

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $$\mathbb C^k$$ , $$k \ge 3.$$ First, we study the relation between two polynomial shift-like maps in $$\mathbb C^k$$ , $$k \ge 3$$ , that share the same forward and backward Julia sets. Secondly, we study the relation between any pair of skew products of Henon maps in $$\mathbb C^3$$ having the same (forward and backward) Julia sets. Also in the same spirit, we further establish a similar relation between skew products of Henon maps fibered over a compact metric space, sharing the same Julia sets.

Newton’s method is a well-known iterative method to find roots of a function. The related Newton maps have primarily been studied for polynomials, but recently extended to rational and transcendental functions. We describe how a rational function r influences the degree and fixed points of its Newton map R. We then analyze the Julia sets of the Newton maps of Mobius transformations. In doing so, we verify a conjecture of Corte and expand on that result. We also consider Newton maps of rational functions of the form $$\displaystyle \frac{(z-r_1)(z-r_2)}{z-p}$$ . We prove that these Newton maps are all conjugate to $$z^2$$ , allowing us to completely describe their Julia sets.

We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point belonging to the Julia set of a rational map.

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

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincare plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

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.

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a real parameter $c$ such that the computational complexity of computing $J_c$ with $n$ bits of precision is higher than $T(n)$. This is the first known class of real parameters with a non poly-time computable Julia set., 9 pages, 1 figure. To be published in the journal Found. Comp. Math. (FoCM). arXiv admin note: text overlap with arXiv:1703.04660

It is known that Julia sets, as a rule, have a fractal structure. In this paper, we give examples of smooth Julia sets, among them: a circle, a segment, an infinite interval, a straight line, and the complex plane. It is shown that the functions studied in the paper are chaotic on their Julia sets. The results obtained by analytical research are visualized using computer programs. The algorithms for constructing the Julia sets being considered are indicated.

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.

The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.

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.

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.

By investigating which level of universality composition operators $C_f$ can have, where the symbol $f$ is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of $f$, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.

In a classical work of the 1950s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee–Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee–Yang–Fisher (LYF) zeros. In this paper, we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $$R$$ in two variables (the Migdal–Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the LYF zeros are equidistributed with respect to a dynamical (1, 1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal–Kadanoff renormalization transformation of various other hierarchical lattices.

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m, Comment: 18 pages, 3 figures

This paper reports the investigation of a fractional Lotka–Volterra model from the fractal viewpoint. A Julia set of a discrete version of this model is introduced and its state feedback control is realized. Coupled terms are designed to realize the synchronization of two Julia sets with different parameters. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

Schroder’s iterative formula of the second kind (S2 formula) for finding zeros of a function f(z) is a generalization of Newton’s formula to an arbitrary order m of convergence. For iterative formulae, convergence regions of initial values to zeros in the complex plane z are essential. From numerical experiments, it is suggested that as order m of the S2 formula grows, the complicated fractal structure of the boundary of convergence regions gradually diminishes. We propose a method of estimating the convergence regions with the circles of Apollonius to verify this result for polynomials f(z) with simple zeros. We indeed show that as m grows, each region surrounded by the circles of Apollonius monotonically enlarges to the Voronoi cell of a zero of f(z). Numerical examples illustrate convergence regions for several values of m and some polynomials.

In this paper, entire solutions f of a class of nonlinear difference equations are studied. By considering the order and deficiency of the coefficients in the equations, we investigate the properties of the radial distribution of the Julia set of f, and estimate the lower bound of the measure of the set defined by the common limiting directions of Julia sets of shifts of f.

For maps of one complex variable, f, given as the sum of a degree n power map and a degree d polynomial, we provide necessary and sufficient conditions that the geometric limit as n approaches infinity of the set of points that remain bounded under iteration by f is the closed unit disk or the unit circle. We also provide a general description, for many cases, of the limiting set.

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.

When a semi-flow is induced by a d-fold branched covering $$ f:M \rightarrow M $$ defined on a Riemannian manifold M, the associated Julia set J(f) is a compact invariant subset of M and, therefore, there exists an induced restriction $$ f | _ {J (f)} :J (f) \rightarrow J (f) $$ . In order to construct an inverse system of regular sub-complexes whose inverse limit is J(f) we use computational techniques to iterate subdivision processes for a regular CW-structure given in M. The invariants of this inverse system can be used to study some topology and shape properties of J(f) . In particular, for the case of an expanding rational map we have constructed a resolution using global multipliers. The advantage of this resolution is that we can develop many algorithms that give an explicit description of the complexes of this resolution and implemented versions of this procedure can be used to give nice visualizations of the Julia set or to compute its shape invariants. If J(f) does not contain critical points of f, the restriction $$f | _ {J (f)} $$ inherits a d-fold overlay structure which is the limit of d-fold coverings and the classification of this overlay structure can be given in terms of representations of the fundamental pro-groupoid of J(f) in the symmetric group $$\varSigma _d$$ .

The iteration of the map $$z \mapsto \lambda + \tan z$$ on the complex plane is investigated for each complex number $$\lambda $$ . It is shown that its Fatou set always contains a completely invariant component and the Julia set is either totally disconnected or connected. A necessary and sufficient condition for totally disconnected Julia set is proved. When the Julia set is connected, it is seen that the Fatou set can consist of additional components, which can be completely invariant attracting domain, a Siegel disk, a two periodic attracting domain or parabolic domain. Examples of each case are given. Further, a necessary and sufficient condition is provided for the Julia set to be a Jordan curve passing through infinity. It is proved that the set of parameters corresponding to Jordan curve Julia sets is connected, whereas the set of those corresponding to totally disconnected Julia sets is disconnected.

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

We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time., Comment: arXiv admin note: text overlap with arXiv:1404.1236

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ2×2 and Julia set of one dimensional complex dynamic system.

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

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

The Fig. 1 was drawn by Shigehiro Ushiki using his software called HenonExplorer. This complicated object is the Julia set of a complex Henon map $$f_{c, b}(x, y)=(x^2+c-by, x)$$ defined on $$\mathbb {C}^2$$ together with its stable and unstable manifolds, hence it is a fractal set in the real 4-dimensional space! The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of $$\mathbb {C}^2$$ including complex Henon maps with an emphasis on the combinatorial and topological aspects of their Julia sets.

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.

Mandelbrot and Julia sets are examples of fractal patterns generated in the complex plane. In the literature we can find many generalizations of those sets. One of such generalizations is the use of switching process. In this paper we introduce some switching processes to another type of complex fractals, namely polynomiographs. Polynomiograph is an image presenting the visualization of the complex polynomial's root finding process. The proposed switching processes will be divided into four groups, i.e., switching of: the root finding methods, the iterations, the polynomials and the convergence tests. All the proposed switching processes change the dynamics of the root finding process and allowed us to obtain new and diverse fractal patterns.

We study distributions of the harmonic and Lebesgue measures on the boundary of the connectedness locus $$\mathcal{M}_{d}$$ for uni-critical polynomials $$z^{d}+c$$ and dynamical properties which are typical with respect to the harmonic measure. One result is a conformal similarity between $$\mathcal{M}_{d}$$ and the corresponding Julia set for almost every point of $$\mathcal{M}_{d}$$ with respect to the harmonic measure. It is also shown that the harmonic measure is supported on Lebesgue density points of the complement of $$\mathcal{M}_{d}$$ which are not accessible from outside within John angles and at which the boundary of $$\mathcal{M}_{d}$$ ‘spirals’ infinitely often in both directions. In the case of quadratic polynomials, we prove that every parameter c from the boundary of the Mandelbrot set $$\mathcal{M}$$ that is recurrent under iterates of $$z^{2}+c$$ but not infinitely tunable is a Lebesgue weak density point of the complement of $$\mathcal{M}$$ . A direct consequence of our theory is the Yoccoz local connectivity theorem. A new technical ingredient in the paper is the method of “amplification” which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding Julia set. The pointwise conformality of the similarity map is derived through inducing techniques and a classical TWB-theory.

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

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 $$\lambda E_n $$ .

We study the dynamics of the family of rational maps of the form $$\begin{aligned} f_{d,{\uplambda }}(z)={\uplambda }\left( z+\frac{1}{z^{d-1}}\right) , \quad d \ge 3, \quad {\uplambda }\in \mathbb C{\setminus } \{0\}. \end{aligned}$$ Among other things, we show that the parameter planes for these maps contain infinitely many copies of the Mandelbrot set as well as infinitely many “blowup points”, i.e., parameters for which the critical orbits map to $$\infty $$ , so the Julia set is the entire sphere. Our efforts are aided by the useful observation that for fixed $$d \ge 3$$ , this family is conformally conjugate on the entire Riemann sphere to the family of relaxed Newton maps for $$p_d(z) = z^d-1$$ . The conjugacy allows us to move from one family to the other in order to find simpler proofs of our results, as well as establishing a dictionary of results from one family to the other.

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