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Iteration of rational functions. Complex analytic dynamical systems
... Optimality principles exist widely in natural and artificial processes, including neuromechanics [1,2], biochemistry [3][4][5], autonomous systems [6][7][8], economics [9], electricity grid [10][11][12], telecommunication [13], logistics [14], human-resource allocation [15], and others [16,17]. These processes exhibit causally determinism, adhering to the classical perspectives. ...
... This system and problem class is common in the real world. Examples include neuromechanics (muscle-force distribution [1], motor-unit recruitment [2]), metabolic networks (flux balance analysis [3], multidimensional optimality [4]), beamforming in localization and communication systems [13], autonomous systems (omnidirectional vehicle [6], multirobot network [7]), electricity grid (optimal power flow [10][11][12]), and economics (Markowitz portfolio optimization [9]). Due to its causally deterministic nature, OTC is not just a new method, it is, most importantly, a system interpretation of optimality principles in physical systems (e.g., neuromechanics, metabolic networks). ...
... Subsequently, we apply feedback linearization on (13) to establish an output tracking controller. This results in the controller ...
Achieving optimality in controlling physical systems is a profound challenge across diverse scientific and engineering fields, spanning neuromechanics, biochemistry, autonomous systems, economics, and beyond. Traditional solutions, relying on time-consuming offline iterative algorithms, often yield limited insights into fundamental natural processes. In this work, we introduce a novel, causally deterministic approach, presenting the closed-form optimal tracking controller (OTC) that inherently solves pseudoconvex optimization problems in various fields. Through rigorous analysis and comprehensive numerical examples, we demonstrate OTC's capability of achieving both high accuracy and rapid response, even when facing high-dimensional and high-dynamical real-world problems. Notably, our OTC outperforms state-of-the-art methods by, e.g., solving a 1304-dimensional neuromechanics problem 1311 times faster or with 113 times higher accuracy. Most importantly, OTC embodies a causally deterministic system interpretation of optimality principles, providing a new and fundamental perspective of optimization in natural and artificial processes. We anticipate our work to be an important step towards establishing a general causally deterministic optimization theory for a broader spectrum of system and problem classes, promising advances in understanding optimality principles in complex systems.
... If a critical point is not a zero of f , then that critical point is called free critical point. For more informations, one can see [4,10,6]. The Fatou set consists of Fatou components, which are maximal connected open subsets of the Fatou set. ...
... is a function such that the inverse image of every compact set in V is compact in U then f is called proper map. Connectivity of a set A is the number of maximally connected open subsets of C \ A. Now, we are giving Riemann-Hurwitz formula [p-85, [4]]. Lemma 1.1 (Riemann-Hurwitz formula). ...
... Hence, A ∞ is infinitely connected. We know that if R is a rational map, then J(R) is connected if and only if every Fatou component is simply connected [Theorem 5.1.6,[4]]. As A ∞ is infinitely connected, the Julia set of St M (z) is disconnected. ...
We study the dynamics of Stirling's iterative root-finding method for rational and polynomial functions. It is seen that the Scaling theorem is not satisfied by Stirling's iterative root-finding method. We prove that for a rational function R(z) with simple zeroes, the zeroes are the superattracting fixed points of and all the extraneous fixed points of are rationally indifferent. For a polynomial p(z) with simple zeroes, we show that the Julia set of is connected. Also, the symmetry of the dynamical plane and free critical orbits of Stirling's iterative method for quadratic unicritical polynomials are discussed. The dynamics of this root-finding method applied to M\"{o}bius map is investigated here. We have shown that the possible number of Herman rings of this method for M\"{o}bius map is at most 2.
... Moreover, any cycle of periodic Fatou components of a rational map has at least a critical point, i.e. a point z ∈ C such that f (z) = 0, related to it. For a more detailed introduction to the dynamics of rational maps we refer to [4] and [13]. ...
... The connectivity of a domain D ⊂ C is given by the number of connected components of its boundary ∂D. It is well known that periodic Fatou components have connectivity 1, 2 or ∞ (see [4]). However, preperiodic Fatou components may have finite connectivity greater that 2. Beardon [4] introduced an example suggested by Shishikura of a family of rational maps with Fatou components of finite connectivity greater than 2. Baker, Kotus and Lü [3] proved that, given any n ∈ N, there exists rational and meromorphic transcendental maps with preperiodic Fatou components of connectivity n by means of a quasiconformal surgery procedure. ...
... It is well known that periodic Fatou components have connectivity 1, 2 or ∞ (see [4]). However, preperiodic Fatou components may have finite connectivity greater that 2. Beardon [4] introduced an example suggested by Shishikura of a family of rational maps with Fatou components of finite connectivity greater than 2. Baker, Kotus and Lü [3] proved that, given any n ∈ N, there exists rational and meromorphic transcendental maps with preperiodic Fatou components of connectivity n by means of a quasiconformal surgery procedure. Later on, Qiao and Gao [15], and Stiemer [19] provided explicit examples of families of rational maps with such dynamical properties. ...
We study the family of singular perturbations of Blaschke products . We analyse how the connectivity of the Fatou components varies as we move continuously the parameter . We prove that all possible escaping configurations of the critical point take place within the parameter space. In particular, we prove that there are maps which have Fatou components of arbitrarily large finite connectivity within their dynamical planes.
... In this paper we answer to the preceding questions, showing that the five above properties (i.e. (D), Gateaux differentiability of P τ , large deviation principle on M(Ω), large deviation principle on R n for all n ∈ N, convexity properties of some maps on R n for all n ∈ N) are in fact equivalent once specified how they take place (Theorem 1); in particular, we obtain a plain and simple comparison between the two above mentioned general sufficient conditions to get the large deviation principle for nets (ν α , t α ) as in (1): (D) is equivalent to the Gateaux differentiability of P τ on an infinite dimensional vector space V dense in C(Ω) possibly excepting zero; furthermore, for each Schauder basis (f n ) of C(Ω) and for each sequence (ε n ) of positive real numbers converging to zero, there is a sequence (h n ) in C(Ω) \ {0} with || h n ||≤ ε n so that one can take V = span({f n + h n : n ∈ N}); when such a space V is obtained, a Schauder basis may be used to get another vector space linearly independent from V whose direct sum with V fulfils the same property as V ; iterating this process gives rise to a new criterion for the validity of (D) (Theorem 2). When there is a unique measure of maximal entropy, the above conditions can be greatly simplified (Corollary 1). ...
... As a natural candidate for (ν α , t α ) as in (1), for each f ∈ C(Ω) we introduce a basic net (ν τ f,α , t τ α ) canonically associated to the system (Ω, τ ): Let ℘ denote the product set ]0, +∞[×N l ]0,+∞[ pointwise directed, where ]0, +∞[ (resp. N, N l ) is endowed with the inverse of the natural order on R (resp. ...
... Example 1. Let (Ω, τ ) be the system given by the iteration of a rational map T of degree at least two ( [1]). More precisely, Ω is the Julia set of T endowed with the induced chordal metric, and the action τ is defined by N ∪ {0} ∋ n → τ (n) = (T |Ω ) n ; ...
We consider a non-uniquely ergodic dynamical system given by a -action (or -action) on a non-empty compact metrisable space , for some . Let (D) denote the following property: The graph of the restriction of the entropy map to the set of ergodic states is dense in the graph of . We assume that is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map on some sets dense in the space of real-valued continuous functions on , level-2 large deviation principle, level-1 large deviation principle, convexity properties of some maps on for all . The one involving the Gateaux differentiability of is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: We show that for each non-empty -compact subset of , (D) is equivalent to the existence of an infinite dimensional vector space V dense in such that f+g has a unique equilibrium state for all ; any Schauder basis of whose linear span contains admits an arbitrary small perturbation so that one can take . Taking , the existence of an infinite dimensional vector space dense in constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of measure of maximal entropy.
... Also note that F(R) = F(R k ) for all k ≥ 1. Further details can be found in [3,16]. ...
... For every rational map, there can be at most two exceptional points (Theorem 4.1.2, [3]). This article shows, under some conditions, that a rational map with an exceptional point has rotational symmetry. ...
... More precisely, we have the following result. [3]) If R 1 and R 2 are two rational functions such that ...
By the symmetry of the Julia set of a polynomial, we mean a Euclidean isometry preserving the Julia set. Each such symmetry is, in fact, a rotation about the centroid of the polynomial. This article conducts a survey of the symmetries of polynomial Julia sets. The Euclidean isometries, which preserve the Julia set of rational maps, are then considered. A rotation preserving the Julia set of a rational map is called a rotational symmetry of its Julia set. A sufficient condition is provided for a rational map to have rotational symmetries whenever the rational map has an exceptional point. Two classes of rational maps are provided whose Julia sets have rotational symmetries of finite orders. Using this, it is proved that where is a rotational symmetry of the McMullen map for all m, n with and . Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynomial coincide with that of its Newton’s method and Chebyshev’s method under certain assumptions.
... We say a holomorphic family H of holomorphic functions is normal in some domain D ⊂ C if every sequence in H has a subsequence that locally uniformly converges to a holomorphic function or locally uniformly diverges to ∞ on D. We say H is normal at z ∈ D, if there exists a neighborhood N (z) such that throughout the neighborhood the family H is normal [12]. The dynamics of holomorphic function was originated in early 20th century with the independent work of Pierre Fatou and Gaston Julia [3,12]. Both of them were motivated from Montel's theory of normal family [12]. ...
... Both of them were motivated from Montel's theory of normal family [12]. Due to Fatou and Julia [3,12], the Fatou set and Julia set of holomorphic function h is defined as F (h) = {z ∈ C : {h n : n ∈ N} is normal at z}, J(h) = C − F (h) respectively. ...
... In particular, the bounded orbit set of polynomial function is also known as filled Julia set [3]. In 1987, Eremenko and Lyubich [2] showed the existence of the bungee set BU (h) of transcendental entire function h and later in 2015, Osborne and Sixsmith [9] formally introduced the notion of bungee set and they proved that for polynomial P , BU (P ) = ∅ and for any transcendental entire function h, BU (h) = ∅ and BU (h) ∩ J(h) = ∅, J(h) = ∂BU (h). ...
In this paper, we study the structure and properties of escaping sets of holomorphic semigroups. In particular, we study the relationship between escaping set of holomorphic semigroup and escaping set of each function that lies in that semigroup. We also study about the invariantness of escaping sets. Also, in this paper, we define the term bounded orbit set K(H) and the set K'(H) of holomorphic semigroup H. Then we study their invariantness and their relations with escaping sets. We also construct a particular class of holomorphic semigroups generated by two holomorphic functions such that bounded orbit set of holomorphic semigroup is equal to bounded orbit set of its generators.
... For example T 2 (x) = x 2 − 2 and T 3 (x) = x 3 − 3x are Chebyshev polynomials of degree 2 and 3 respectively. Bearden in [1] states that the Chebyshev polynomials are important classical examples of the chaotic behaviour. The complex dynamic of polynomial families z 2 + c and z 3 + cz with c ∈ C has been extensively studied (see for instance [2,4,11,13]). ...
... Case 2 when −1 < c < 1 2 : we have 1 < 3 − 4c < 7 and 2 < 3 − 2c < 5. Thus, p 2 and p 4 are repelling fixed points. 1 2 < c < 1: by an straightforward calculation we get 1 < 3−2c < 2 and −1 < 3 − 4c < 1. This implies that p 2 and p 4 are saddle fixed points. ...
... where F c is given in (1) and D c = {z ∈ C| |z| < |c| + 1}. The fact that ...
We consider a non-conformal cubic map given by for c real. We study some properties of filled Julia set of this map denoted by and show that if then is connected. We also show that the map has relation not only to the cubic family , but also to an important holomorphic map on complex projective space given by with .
... We are looking for line arrangements of 15 lines with singularities t 2 = 27, t 3 = 6, t 5 = 6. More precisely, we impose that the following elements are non-basis (2.2) NB 1 : (1, 5, 10), (2,4,13), (3,9,13), (5,7,8), (6,11,12), (12,14,15) (these are the triple of lines meeting at the same points), moreover any triple of lines among the following quintuples ...
... On the dynamics of χ(z) = − 1 z 2 and F Λ = 2z 2 −1. For basic notions on dynamic we refer to the books [2] and [12]. ...
... The Julia set of F Λ is the interval [−1, 1] (see [2]). The map χ restricted to the unit ball (identified with R/2πZ) is the map θ → −2θ ∈ R/2πZ. ...
The operator acting on line arrangements is defined by associating to a line arrangement \mathcal{A}, the line arrangement which is the union of the lines containing exactly three points among the double points of \mathcal{A}. We say that six lines not tangent to a conic form an unassuming arrangement if the singularities of their union are only double points, but the dual line arrangement has six triple points, six 5-points and 27 double points. The moduli space of unassuming arrangements is the union of a point and a line. The image by the operator of an unassuming arrangement is again an unassuming arrangement. We study the dynamics of the operator on these arrangements and we obtain that the periodic arrangements are related to the Ceva arrangements of lines.
... The study of this question is one of the most basic attempts in the understanding of the dynamics of iterations of polynomials and rational functions (see e.g. Beardon [1], Milnor [6]), as well as in the study of polynomial root-finding methods, their fractal behavior and visualization of iterative methods (see e.g. Devaney [3], Mandelbrot [5], and [4]). ...
... On the other hand the basin of attraction of each attractive fixed point of a rational map must contain a critical point (see e.g. Theorem 5.32 in [4], or [1]). Thus not all fixed points of p(z) can be attractive. ...
We prove a complex polynomial of degree n has at most attractive fixed points lying on a line. We also consider the general case.
... More precisely, it is known that any periodic Fatou component has connectivity 1, 2 or ∞ (c.f. [2]), while preperiodic Fatou components can have finite connectivity greater than 2. Beardon [2] introduced an explicit family of rational maps suggested by Shishikura with a Fatou component of finite connectivity greater than 2. This family was studied more deeply in [8]. The authors proved that if the parameter is small enough then there are Fatou components of connectivity 3 and 5. ...
... More precisely, it is known that any periodic Fatou component has connectivity 1, 2 or ∞ (c.f. [2]), while preperiodic Fatou components can have finite connectivity greater than 2. Beardon [2] introduced an explicit family of rational maps suggested by Shishikura with a Fatou component of finite connectivity greater than 2. This family was studied more deeply in [8]. The authors proved that if the parameter is small enough then there are Fatou components of connectivity 3 and 5. ...
The goal of this paper is to study the family of singular perturbations of Blaschke products given by . We focus on the study of these rational maps for parameters a in the punctured disk and small. We prove that, under certain conditions, all Fatou components of a singularly perturbed Blaschke product have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.
... where d µ (x) denotes the lower pointwise dimension of µ at x, i.e. and Supp(µ) denotes the topological support of µ. In particular, if µ is Ahlfors regular of dimension δ, 2 then dim H (BA d ∩ Supp(µ)) = δ = dim H (Supp(µ)). 3 We recall that a measure µ is called doubling (or Federer ) if there exists a constant C > 0 such that for all x ∈ R n (equiv. for all x ∈ Supp(µ)) and for all r > 0, we have µ(B(x, 2r)) ≤ Cµ(B(x, r)). ...
... Note that in Case 2, the Julia set contains ∞, but in Case 1, this may or may not be true. For the basic properties of Julia sets of rational functions the reader is advised to consult the books [3,7,38,44,48] and the survey article [55]; for the topological dynamics of transcendental functions the survey [5] is a good place to start. We now give several definitions and notations: Notation 3.1. ...
We prove that if J is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in J. The same is true if J is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of J that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author ('12) by showing that every hyperplane diffuse set supports an absolutely decaying measure.
... In fact, J f is the closure of the set of repelling periodic points (see e.g. [1,5]). ...
... For example, K f and J f are connected if and only if the critical points of f belong to K f (see e.g. [1,5]). ...
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.
... First, let α be periodic under φ, which we recall means φ n (α) = α for some n ≥ 1, and assume for simplicity that α = ∞. It follows from taking β = α in equation (2) ...
... For α = ∞, we have a similar statement, but with g n (z) replacing f n (z) − αg n (z). Given rational functions φ, ψ ∈ K(z) and γ ∈ K, an easy argument on compositions of power series (see [2,Section 2.5]) gives e φ•ψ (γ) = e φ (ψ(γ)) · e ψ (γ), and hence for all m ≥ 1 we have e φ n+m (γ) = e φ n (φ m (γ)) · e φ m (γ). It follows that, up to a non-zero multiplicative constant, we have ...
For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and alpha any point not periodic under phi (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) phi has good reduction and (2) phi acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backwards orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.
... The complex dynamics studies various issues concerning the convergence of the sequence {H n } ∞ n=1 . We will mention some definitions pertaining to our study and readers can find more information on this subject in [2,18,21]. For the Tchebyshev polynomial T (z) = 2z 2 − 1, the following theorem is known (see [2]). 3. The spectral dynamics of the infinite dihedral group D ∞ ...
... We will mention some definitions pertaining to our study and readers can find more information on this subject in [2,18,21]. For the Tchebyshev polynomial T (z) = 2z 2 − 1, the following theorem is known (see [2]). 3. The spectral dynamics of the infinite dihedral group D ∞ ...
... The most apparent generalisation [9,10] involves using the function x p + c instead of x 2 + c. Other functions have also been examined in the literature, including transcendental functions [11], rational functions [12,13]. Additionally, the study of Mandelbrot sets has been extended from the complex number system to bicomplex numbers [14], quaternions [15], octonions [16], and more. ...
Nowadays, many researchers are employing various iterative techniques to analyse the dynamics of fractal patterns. In this paper, we explore the formation of Mandelbrot and Julia sets using the Picard–Thakur iteration process, extended with s-convexity. To achieve this, we establish an escape criterion using a complex polynomial of the form xk+1+c, where k ≥ 1 and x, c ∈ ℂ. Based on our proposed algorithms, we provide graphical illustrations of the Mandelbrot and Julia sets. Additionally, we extend our research to examine the relationship between the sizes of Mandelbrot and Julia sets and the iteration parameters, utilising some well-known methods from the literature.
... The Fatou and Julia set of a rational function divides the Riemann sphere into two disjoint completely invariant subsets. For a detailed discussion about the dynamics of rational functions, one can read [1]. While investigating the dynamics of rational functions, the topic of rational semigroups was introduced in [5]. ...
In this paper, we introduce the concept of Denjoy-Wolff set in rational semigroups. We show that for finitely generated Abelian rational semigroups, the Denjoy-Wolff like set is countable. Some results concerning the Denjoy-Wolff like set and the Julia set are also discussed. Then we consider a special class of rational semigroups and discuss various properties of Denjoy-Wolff like set for this class. We use the concept of Denjoy-Wolff like set to classify the class into 3 sub-classes. We also show that for any semigroup in this class, the semigroup can be partitioned into k partitions where k is the cardinality of the Denjoy-Wolff like set.
... To prove the final statement we will use the fact that every rational map has a repelling periodic point of each sufficiently large period. This follows from the result of [Bea,Theorem 6.2.2,p.102], that every rational map has a periodic point of a given (minimal) period greater or equal than 4, and from the fact that a rational map possesses at most finitely many non-repelling periodic points. For the proof of the final statement, notice that we can take the sequence ( W n ) n≥1 above, in such a way that (m Wn ) n≥1 is an arithmetic progression for which the difference between 2 consecutive terms is equal to the minimal period of p. ...
We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential - HD(J(f)) ln |f'|.
... Recent references [DJ05,DJ06c,DJ06d,DJ06b,Jor06] deal with wavelet constructions on non-linear attractors X, such as arise from systems of branched mappings, e.g., finite affine systems (affine IFSs), or Julia sets [Bea91] generated by branches of the inverses of given rational mappings of one complex variable. These X come with associated equilibrium measures µ. ...
In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.
... 165-187). On the other hand, dendrites often appear as Julia sets in complex dynamics (see [11]). After finishing this version, we learned that Li, Oprocha, Yang, and Zeng had recently solved the conjecture for graph maps [33]. ...
We prove that the M\"obius disjointness conjecture holds for graph maps and for all monotone local dendrite maps. We further show that this also hold for continuous map on certain class of dendrites. Moreover, we see that there is a transitive dendrite map with zero entropy for which M\"obius disjointness holds.
... Moreover, it turns out that the hairs except for their endpoints lie in the set I(f ) := {z : lim k→∞ f k (z) = ∞}, which we call the escaping set of f . For further information on complex dynamics we refer to [Bea91,Ber93,Mi06,St93]. ...
In 1984 Devaney and Krych showed that for the exponential family , where , the Julia set consists of uncountably many pairwise disjoint simple curves tending to , which they called hairs. Viana proved that these hairs are smooth. Bara\'nski as well as Rottenfusser, R\"uckert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana's result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.
... From the theory of complex dynamics it is wellknown that J(f ) = ∂A(ξ), see [Mi06,Corollary 4.12]. For further information on complex dynamics we refer to [Bea91,Ber93,Mi06,St93]. ...
Devaney and Krych showed that for the exponential family , where , the Julia set consists of uncountably many pairwise disjoint simple curves tending to . Viana proved that these curves are smooth. In this article we consider a quasiregular counterpart of the exponential map, the so-called Zorich maps, and generalize Viana's result to these maps.
... Such a result is very helpful in searching for an answer to [6,Question 5.2]. However, if we only consider upper semi-continuous decompositions then there might be two decompositions D 1 , D 2 of an unshielded continuum K ⊂ C which are both Peano continua under quotient topology, such that the only decomposition finer than D 1 In 2013, Blokh-Curry-Oversteegen obtained for any unshielded compactum K ⊂ C the core decomposition D F S K with respect to the property of being finitely Suslinian [3,Theorem 4]. Here a compactum is finitely Suslinian provided that every collection of pairwise disjoint subcontinua whose diameters are bounded away from zero is finite. ...
A Peano continuum means a locally connected continuum. A compact metric space is called a \emph{Peano compactum} if all its components are Peano continua and if for any constant all but finitely many of its components are of diameter less than C. Given a compact set , there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the \emph{core decomposition of K with Peano quotient}. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into . We also construct a concrete continuum such that the core decomposition of K with Peano quotient does not exist. For specific choices of , the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map restricted to its Julia set.
... Since then, many generalizations of the sets proposed by Mandelbrot have been suggested. In [8,9], the authors replaced the quadratic function with + , where ≥ 2. Some other functions were used, such as rational [10], trigonometric [11][12][13], exponential [13,14]. In recent years, results from fixed point theory have been used to generate fractal graphics like tricorn, multicorn, Julia and Mandelbrot sets. ...
In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form x^{k+1} + c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard-Mann iterations. Moreover, we study the dependency between the iterations' parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.
... On the contrary, the Julia set separates domains containing points whose iterates behave in a qualitatively different manner. In the example of Sect The general theory [2,9] yields the following assertions: ...
We propose a second-order method for unconditional minimization of functions f(z) of complex arguments. We call it the mixed Newton method due to the use of the mixed Wirtinger derivative ∂2f∂z¯∂z for computation of the search direction, as opposed to the full Hessian ∂2f∂(z,z¯)2 in the classical Newton method. The method has been developed for specific applications in wireless network communications, but its global convergence properties are shown to be superior on a more general class of functions f, namely sums of squares of absolute values of holomorphic functions. In particular, for such objective functions minima are surrounded by attraction basins, while the iterates are repelled from other types of critical points. We provide formulas for the asymptotic convergence rate and show that in the scalar case the method reduces to the well-known complex Newton method for the search of zeros of holomorphic functions. In this case, it exhibits generically fractal global convergence patterns.
... The stability analysis of the method is performed via complex dynamics. Research about this topic can be found in references [24,25]. In recent years, dynamic studies have been carried out to analyze the stability of iterative methods [26][27][28]. ...
This paper introduces an iterative method with a remarkable level of accuracy, namely fourth-order convergence. The method is specifically tailored to meet the optimality condition under the Kung–Traub conjecture by linear combination. This method, with an efficiency index of approximately 1.5874, employs a blend of localized and semi-localized analysis to improve both efficiency and convergence. This study aims to investigate semi-local convergence, dynamical analysis to assess stability and convergence rate, and the use of the proposed solver for systems of nonlinear equations. The results underscore the potential of the proposed method for several applications in polynomiography and other areas of mathematical research. The improved performance of the proposed optimal method is demonstrated with mathematical models taken from many domains, such as physics, mechanics, chemistry, and combustion, to name a few.
... To study tunneling effects in nonintegrable systems, discrete-time dynamical systems, commonly referred to as maps, are often used as model systems. This is supported by several reasons: among them, is that the behavior of complex paths has been well studied for discrete-time dynamical systems compared to continuous-time (Hamiltonian) dynamical systems [17][18][19]. However, we should keep in mind that it is not immediately clear whether the results derived for discrete-time dynamical systems, especially those based on complex dynamics, are applicable to continuous-time flow systems. ...
Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant difference. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role for yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.
... To study tunneling effects in nonintegrable systems, discrete-time dynamical systems, commonly referred to as maps, are often used as model systems. This is supported by several reasons, among which is that the behavior of complex paths has been well studied for discrete-time dynamical systems compared to continuous-time (Hamiltonian) dynamical systems [27][28][29]. However, we should keep in mind that it is not immediately clear whether the results derived for discrete-time dynamical systems, especially those based on complex dynamics, are applicable to continuous-time flow systems. ...
Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant differences. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role of yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.
... En este caso, la constante de Lyapunov asociada a la cuenca del 3-ciclo es 0.9824, por lo que se trata de un 3-ciclo atractor, con capacidad atractiva muy débil (pues su constante asociada es un valor muy cercano a 1). Al igual que en el apartado anterior, el punto del infinito ∞ es un punto fijo repulsor, la constante asociada a su cuenca es 3 2 , y las cuencas de las raíces de q tienen constante asociada 0, al ser puntos fijos súper-atractores de la función racional inducida por el método de Newton. ...
... , n} such that f (C k ) ∩ C j = ∅ and we have f (l k ) = l σ (k) . (4) Each l k is clopen relatively to L. (5) σ is a n-cycle. ...
We give a simple proof that any dendrite map for which the set of endpoints is closed and countable fulfilled Sarnak Möbius disjointness. We further notice that the Smital-Ruelle property can be extended to the class of dendrites with closed and countable endpoints. Our proof use only Dirichlet theorem.
... En este caso, la constante de Lyapunov asociada a la cuenca del 3-ciclo es 0.9824, por lo que se trata de un 3-ciclo atractor, con capacidad atractiva muy débil (pues su constante asociada es un valor muy cercano a 1). Al igual que en el apartado anterior, el punto del infinito ∞ es un punto fijo repulsor, la constante asociada a su cuenca es 3 2 , y las cuencas de las raíces de q tienen constante asociada 0, al ser puntos fijos súper-atractores de la función racional inducida por el método de Newton. ...
La resolución de ecuaciones polinómicas se puede abordar con métodos numéricos como por ejemplo el método de Newton. Estos métodos asocian una función racional compleja a cada polinomio de modo que cada solución de la ecuación polinómica es un punto fijo de la función racional asociada. En esta exposición analizamos las cuencas de atracción de estos puntos fijos e incluso abordamos el estudio de cuencas de atracción de n-ciclos de dicha función racional.
Cada función racional compleja se puede interpretar de modo natural como un endo-morfismo de la esfera de Riemann. Con el fin de evitar los desbordamientos computacionales que pueden aparecer cuando el denominador de una función racional es proximo a cero, presentamos un novedoso método que asocia a cada función racional compleja un endo-morfismo de la fibración de Hopf.
Para calcular las cuencas de atracción definimos lo que hemos llamado la función de Lyapunov que es una función real discontinua pero que tiene la propiedad de que es constante en cada cuenca de atracción de un n-ciclo (salvo un conjunto de medida de Lebesgue nula). La iteración de este endo-morfismo de la fibración de Hopf y el calculo de la función de Lyapunov nos permite calcular los n-ciclos atractores, sus cuencas y, en particular, los puntos fijos y sus cuencas que determinan las soluciones de una ecuación polinómica.
... These basins graphically demonstrate how wide is the set of points in the complex plane which ultimately converge to the desired root of a given equation. Let us recall some basic concepts [3] to proceed further in this section. ...
It is generally desirable to construct a higher order iterative method for locating roots of multiplicity m that preserves its theoretical convergence in applications, but in reality, this is not the case in general. With this aim, a three-step Traub-Steffensen-type (derivative-free) iterative family is proposed and investigated for obtaining roots of multiplicity m of nonlinear equations. The iterative scheme uses four function evaluations per iteration and possesses eighth order of convergence, hence it is optimal. Special cases of the family are thoroughly investigated for their dynamical properties and numerical performance. Estimation of the convergence domain in the complex plane and comparison of the results with existing methods provide a substantial idea about the convergence of new methods. The findings of analysis are remarkable in the sense that the developed scheme exhibits superiority over the existing counterparts by displaying the wider convergence regions along with the rapid convergence rates. In addition, numerical accuracy is analyzed by locating the multiple zeros of selected nonlinear problems including some of the applied nature like Van der Waals equation and Eigen value problem. The performance analysis also shows additional properties of the family by demonstrating high precision in numerical computations while preserving the theoretical order of convergence.
... The Riemann sphere is decomposed into the disjoint union of the Fatou set and the Julia set, defined by whether the iterated sequence forms a normal family in a neighborhood of the point (see e.g. [1,15] for their definitions and basic properties). ...
A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are disjoint if and only if they are separated by a coiling curve. Furthermore, we prove that a post-critically finite rational map with a coiling curve is renormalizable. Using a similar argument, we give a sufficient condition for a Fatou domain to qualify as a Jordan domain. By tuning polynomails in such a Fatou domain, we provide examples of post-critically finite rational maps with coiling curves.
... The Julia set J f of f is the complement of the Fatou set. Refer to [7,1] for their basic properties. ...
We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.
... If N is not simple, then any point in V rep (and also all its iterated preimages) has an infinite amount of branching in H ∞ . Indeed, for any point ζ ∈ V rep , a direction v ∈ T ζ P 1 with B ζ ( v) ∩ H rep = ∅ is either an attracting but not superattracting fixed point or the repelling fixed point of T ζ N under the identification T ζ P 1 with P 1 C by Proposition 2.3 (1) and Corollary 3.7, and hence has infinitely many iterated preimages under T ζ N , see [3,Theorem 4.1.2]. Now let us consider the tree structure on H ∞ as following. ...
Let L be the completion of the formal Puiseux series over C, and let P1 be the Berkovich space over L. We study the dynamics of Newton maps, defined over L, on P1. As an application, we give a complete description of the rescaling limits for a holomorphic family of complex Newton maps.
... We consider the iteration of holomorphic functions f : C → C. As first shown by Fatou [16,17] and Julia [22], the complex plane is partitioned into an open set of "regular" dynamics, the Fatou set (denoted F (f )), and a closed set of "chaotic" dynamics, the Julia set (denoted J(f )) -see, for instance, [4] or [7] for introductions to the subject. Both the Fatou and Julia sets are completely invariant under f , meaning that a connected component U of the Fatou set -called a Fatou component -is mapped by f n into another Fatou component, denoted U n . ...
The structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions, implying that functions with wandering domains - another hot topic of research in complex dynamics - have (for the most part) not been addressed in this context. Given an entire function f with a simply connected wandering domain U, we construct an object called a distortion sequence that, under some hypotheses, moves analytically as f moves within appropriate parameter families. In order to "ground" our discussion, we consider -- given an entire function f -- the set of entire functions quasiconformally equivalent to f. Generalising earlier results for the finite-type case, we show that admits the structure of a complex manifold (of possibly infinite dimension).
... The Mandelbrot set is generated by iterating a complex quadratic polynomial and has been studied for many years. [4][5][6] Iterations of quaternionic polynomials have also been used to generalize the Mandelbrot set in 3D. However, since quaternions have four dimensions, the visualization of the set is only possible through 3D slices. ...
In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, adds the angles and multiplies the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product.
The dynamic behavior of polynomials within the framework of the Fatou-Julia dichotomy is among the most fascinating topics of complex analysis. An introduction to the theory is the content of the last section of this chapter. The essential theoretical foundation is formed by the spherical normality theorem of Montel, also briefly referred to as Montel’s great theorem. As a further consequence of Montel’s great theorem, one obtains Picard’s great theorem, which in turn is undoubtedly one of the jewels of function theory.
Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich’s result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set of entire functions quasiconformally equivalent to f admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains—another hot topic of research in complex dynamics. Given an entire function f with a simply connected wandering domain U, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as f moves within appropriate parameter families.
In this chapter, we shall present various definitions and theorems useful for the understanding of the dynamics of the asymmetric interaction among objects (or nodes) which is observed in many fields of science as introduced in Chap. 1.
Fix an integer . The space of polynomial maps of degree d modulo conjugation by affine transformations is naturally an affine variety over of dimension . For each integer , the elementary symmetric functions of the multipliers at all the cycles with period induce a natural morphism defined on . In this article, we show that the morphism induced by the multipliers at the cycles with periods 1 and 2 is both finite and birational onto its image. In the case of polynomial maps, this strengthens results by McMullen and by Ji and Xie stating that is quasifinite and birational onto its image for all sufficiently large integers P. Our result arises as the combination of the following two statements: A sequence of polynomials over of degree d with bounded multipliers at its cycles with periods 1 and 2 is necessarily bounded in . A generic conjugacy class of polynomials over of degree d is uniquely determined by its multipliers at its cycles with periods 1 and 2.
Let f be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When f has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.
We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.
A linear differential operator with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, T has a unique minimal Hutchinson-invariant set in the complex plane. Using a geometric interpretation of its boundary in terms of envelops of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field ), and singular points of different types which we classify below. The latter decomposition of the boundary of is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field . We provide an upper bound for the number of local arcs in terms of degrees of P and Q. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.
We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the simpler case of iteration. A result of Fern\'andez, Meli\'an and Pestana gives a precise version of the classical Poincar\'e recurrence theorem for iterates of the boundary extension of an inner function that fixes~0. We generalise this to forward composition sequences where are inner functions that fix~0, giving conditions on the contraction of so that the radial boundary extension hits any shrinking target of arcs of a given size. Next, Aaronson, and also Doering and Ma\~n\'e, gave a remarkable dichotomy for iterates of any inner function, showing that the behaviour of the boundary extension is of two entirely different types, depending on the size of the sequence . In earlier work, we showed that one part of this dichotomy holds in the non-autonomous setting of forward compositions. It turns out that this dichotomy is closely related to the result of Fern\'andez, Meli\'an and Pestana, and here we show that a version of the second part of the dichotomy holds in the non-autonomous setting provided we impose a condition on the contraction of in relation to the size of the sequence . The techniques we use include a strong version of the second Borel--Cantelli lemma and strong mixing results of Pommerenke for contracting sequences of inner functions. We give examples to show that the contraction conditions that we need to impose in the non-autonomous setting are best possible.
We consider the geometric map , called Cayleyan, associating to a plane cubic E the adjoint of its dual curve. We show that and the classical Hessian map generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup generated by . We point out then how the dynamic behaviours of and differ drastically. Firstly, concerning the number of real periodic points: for these are infinitely many, for they are just 4. Secondly, the Julia set of is the whole projective line, unlike what happens for all elements of which are not iterates of .
In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T(z)=zn+mz+r, where n≥2 and m,r∈C, by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear.
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