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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

This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia sets are used as examples.

Using the computer program creating Julia sets for two-dimensional maps we have made computer animation showing how Julia sets for the Peckham map alters when the parameter of the map is changing. The Peckham map is a one-parameter map which includes the complex map z=z^2+c, and is nonanalytical for other values of the parameter. Computer animation of Julia fractal sets allows seeing how patterns typical for complex maps gradually destroy.

We construct a canonical Green current $T_f$ for every quasi-algebraically stable meromorphic self-map $f$ of $\mathbb{P}^k$ such that its first dynamical degree $\lambda_1(f)$ is a simple root of its characteristic polynomial and that $\lambda_1(f)>1.$ We establish a functional equation for $T_f$ and show that the support of $T_f$ is contained in the Julia set, which is thus non empty.

We consider the family of complex maps given by fλ,a (z) = z n + λ/(z − a)d where n, d ≥ 1 are integers, and a and λ are complex parameters such that a = 1 and is sufficiently small. We focus on the topological characteristics of the Julia and Fatou sets of fλ,a.

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} .

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

In this paper, the dynamics of a non-autonomous tabu learning two-neuron model is investigated. The model is obtained by building a tabu learning two-neuron (TLTN) model with a composite hyperbolic tangent function consisting of three hyperbolic tangent functions with different offsets. The possibility to adjust the compound activation function is exploited to report the sensitivity of non-trivial equilibrium points with respect to the parameters. Analysis tools like bifurcation diagram, Lyapunov exponents, phase portraits, and basin of attraction are used to explore various windows in which the neuron model under the consideration displays the uncovered phenomenon of the coexistence of up to six disconnected stable states for the same set of system parameters in a TLTN. In addition to the multistability, nonlinear phenomena such as period-doubling bifurcation, hysteretic dynamics, and parallel bifurcation branches are found when the control parameter is tuned. The analog circuit is built in PSPICE environment, and simulations are performed to validate the obtained results as well as the correctness of the numerical methods. Finally, an encryption/decryption algorithm is designed based on a modified Julia set and confusion–diffusion operations with the sequences of the proposed TLTN model. The security performances of the built cryptosystem are analyzed in terms of computational time (CT = 1.82), encryption throughput (ET = 151.82 MBps), number of cycles (NC = 15.80), NPCR = 99.6256, UACI = 33.6512, χ2-values = 243.7786, global entropy = 7.9992, and local entropy = 7.9083. Note that the presented values are the optimal results. These results demonstrate that the algorithm is highly secured compared to some fastest neuron chaos-based cryptosystems and is suitable for a sensitive field like IoMT security.

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.

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.

We study the balanced resistance forms on the Julia sets of Misiurewicz-Sierpinski maps, which are self-similar resistance forms with equal weights. In particular, we use a theorem of Sabot to prove the existence and uniqueness of balanced forms on these Julia sets. We also provide an explorative study on the resistance forms on the Julia sets of rational maps with periodic critical points., 31 pages, 14 figures

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.

We consider the symmetries of Julia sets of polynomial skew products on C^2, which are birationally conjugate to rotational products. Our main results give the classification of the polynomial skew products whose Julia sets have infinitely many symmetries., Comment: 12 pages

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 fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a fractional rational system in the complex plane. Then, the fractal dynamics of this model is discussed from the fractal viewpoint. Julia set of the fractional Potts model is given, and control items of this fractional model are designed to control the Julia set. To associate two different Julia sets of the fractional model with different parameters and fractional orders, nonlinear coupling items are taken to make one Julia set change to another. The simulations are provided to illustrate the efficacy of these methods.

Suppose that $f$ is a transcendental entire function, $V \subsetneq \mathbb{C}$ is a simply connected domain, and $U$ is a connected component of $f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to see that $g$ is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of $f$ in $V$ lie away from the boundary, there is a strong relationship between singularities of $g$ and accesses to infinity in $U$. In the case where $U$ is a forward-invariant Fatou component of $f$, this leads to a very significant generalisation of earlier results on the number of singularities of the map $g$. If $U$ is a forward-invariant Fatou component of $f$ there are currently very few examples where the relationship between the pair $(f, U)$ and the function $g$ have been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this way, and we show the following: For every finite Blaschke product $g$ whose Julia set coincides with the unit circle, there exists a transcendental entire function $f$ with an invariant Fatou component such that $g$ is associated to $f$ in the above sense. Furthermore, there exists a single transcendental entire function $f$ with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated to the restriction of $f$ to a wandering domain., Comment: 32 pages, 6 figures. V4: Author accepted manuscript. To appear in Arnold Mathematical Journal (special volume dedicated to Prof. Mikhail Lyubich). A number of figures added from V1; general revision throughout; minor corrections of the proofs in Sections 8 and 9

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 construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.

An omitted value of a transcendental meromorphic function $f$ is called a Baker omitted value, in short \textit{bov} if there is a disk $D$ centered at the bov such that each component of the boundary of $f^{-1}(D)$ is bounded. Assuming that the bov is in the Fatou set of $f$, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If $U$ is the Fatou component containing the bov then it is proved that a Fatou component $U'$ is infinitely connected if and only if it lands on $U$, i.e. $f^{k}(U') \subset U$ for some $k \geq 1$. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect $U$ is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a $2$-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a $2$-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components., Comment: 21 pages, 7 figures

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to a repelling fixed point., 21 pages, 13 figures

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 .

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question., Comment: 27 pages, 14 figures

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 .

Degree-$d$-invariant laminations of the disk model the dynamical action of a degree-$d$ polynomial; such a lamination defines an equivalence relation on $S^1$ that corresponds to dynamical rays of an associated polynomial landing at the same multi-accessible points in the Julia set. Primitive majors are certain subsets of degree-$d$-invariant laminations consisting of critical leaves and gaps. The space $\textrm{PM}(d)$ of primitive degree-$d$ majors is a spine for the set of monic degree-$d$ polynomials with distinct roots and serves as a parameterization of a subset of the boundary of the connectedness locus for degree-$d$ polynomials. The core entropy of a postcritically finite polynomial is the topological entropy of the action of the polynomial on the associated Hubbard tree. Core entropy may be computed directly, bypassing the Hubbard tree, using a combinatorial analogue of the Hubbard tree within the context of degree-$d$-invariant laminations.

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.

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge. We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions., 33 pages

The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z \to (1+ \mu) \exp (iz)$, $g : z \to \big(1+ \mu |z - z^*|\big) \exp (iz)$, $h : z \to \big(1+ \mu (z - z^* )\big) \exp (iz)$, with $z,\mu \in \mathbb{C}$, $z^* : \exp (iz^*) = z^*$. For a quasilinear map g having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of "fork" type and "saddle"-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of "valleys" on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of "period doubling" is noted (in one case along the path $39\to 3$, in the other along the path $17\to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. In this thesis, we review existing results on rational maps sharing a Julia set, and offer results of our own on the symmetry group of such maps. Desde a década de oitenta, um enorme progresso foi feito no problema de determinar quais funções têm o mesmo conjunto de Julia. O caso polinomial foi completamente respondido em 1995, e mostrou-se que as simetrias do conjunto de Julia têm um papel central nessa questão. O caso racional permanece aberto, mas já se sabe que ele é muito mais complexo do que o polinomial. Nesta dissertação, nós revisamos resultados existentes sobre aplicações racionais com o mesmo conjunto de Julia e apresentamos nossos próprios resultados sobre o grupo de simetrias de tais aplicações.

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of exponential functions., 18 pages; slight improvement of lower bound in Theorem 1.1; some overall revision

We prove Zhang’s dynamical Manin–Mumford conjecture and dynamical Bogomolov conjecture for dominant endomorphisms$\unicode[STIX]{x1D6F7}$of$(\mathbb{P}^{1})^{n}$. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.

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.

In memory of Sibe Mardesic, our friend. Sibe Mardesic has enriched algebraic topology developing shape and strong shape theories with important constructions and theorems. This paper relates computational topology to shape theory. We have developed some algorithms and implementations that under some conditions give a shape resolution of some Julia sets. When a semi-flow is induced by a rational map g of degree d defined on the Riemann sphere, one has the associated Julia set J ( g ) . The main objective of this paper is to give a computational procedure to study the shape of the compact metric space J ( g ) . Our main contribution is to provide an inverse system of cubic complexes approaching J ( g ) by using implemented algorithms based in the notion of spherical multiplier. This inverse system of cubical complexes is used to: (i) obtain nice global visualizations of the fractal structure of the Julia set J ( g ) ; (ii) determine the shape of the compact metric space J ( g ) . These techniques also give the possibility of applying overlay theory (introduced by R. Fox and developed among others by S. Mardesic) to study the symmetry properties of the fractal geometry of the Julia set J ( g ) .

Iterated monodromy groups of postcritically finite rational maps form a rich class of self‐similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non‐polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpinski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non‐renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth.