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Dynamics of transcendental functions
... In particular, if f (z) = z, then z is a fixed point of f . If |(f n ) ′ (z)| < 1, where (f n ) ′ represents complex differentiation of f n with respect to z, then periodic point z is called an attracting periodic point [2]. ...
... The limit function is a holomorphic function or the constant ∞. For z 0 ∈ D, if there exists a neighborhood N (z 0 ) ⊂ D of z 0 such that F is normal in N (z 0 ), then we say that F is normal at z 0 [2]. ...
... The Fatou and Julia set of f is defined by F (f ) = {z ∈ C ∞ : the sequence f n is well defined and normal at z} and J(f ) = C ∞ − F (f ), respectively [2]. Morosawa [3] gave some examples of Baker domains which lie in angular regions. ...
Here we discuss, for a given integer, the existence of transcendental entire function such that its number of periodic Fatou components lie in angular regions and their periodicity are related to the integer.
... Then h n (z) is called n th iteration of h with itself. 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]. ...
... 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]. ...
... 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]. 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. ...
... Then h n (z) is called n th iteration of h with itself. 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]. ...
... 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]. ...
... 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]. 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 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.
... In particular, if f (z) = z, then z is a fixed point of f . If |(f n ) ′ (z)| < 1, where (f n ) ′ represents complex differentiation of f n with respect to z, then periodic point z is called an attracting periodic point [2]. ...
... The limit function is a holomorphic function or the constant ∞. For z 0 ∈ D, if there exists a neighborhood N (z 0 ) ⊂ D of z 0 such that F is normal in N (z 0 ), then we say that F is normal at z 0 [2]. ...
... The Fatou and Julia set of f is defined by F (f ) = {z ∈ C ∞ : the sequence f n is well defined and normal at z} and J(f ) = C ∞ − F (f ), respectively [2]. Morosawa [3] gave some examples of Baker domains which lie in angular regions. ...
Here we discuss, for a given integer, the existence of transcendental entire function such that its number of periodic Fatou components lie in angular regions and their periodicity are related to the integer.
... In this regards, the chief aim of this paper is to prove the following result which we have considered a strongest result of transcendental semigroup dynamics. In classical holomorphic iteration theory, the stable basin is one of the above types but in transcendental iteration theory, the stable basin is not a Hermann because a transcendental entire function does not have Hermann ring [13,Proposition 4.2]. ...
... (2) supper attracting if it is a subdomain of supper attracting basin of each f ∈ S U (3) parabolic if it is a subdomain of parabolic basin of each f ∈ S U (4) Siegel if it is a subdomain of Siegel disk of each f ∈ S U (5) Baker if it is a subdomain of Baker domain of each f ∈ S U In classical transcendental iteration theory, the stable basin is one of the above types but not Hermann because a transcendental entire function does not have Hermann ring [13,Proposition 4.2]. ...
... Proof. By the theorem 1.1, F (S) = F (f ) for any f ∈ S. By the theorem 4.16 of [13], F (f ) does not contain any asymptotic value of f . Hence the result follows. ...
We mainly generalize the notion of abelian transcendental semigroup to nearly abelian transcendental semigroup. We prove that Fatou set, Julia set and escaping set of nearly abelian transcendental semigroup are completely invariant. We investigate no wandering domain theorem in such a transcendental semigroup. We also obtain results on a complete generalization of the classification of periodic Fatou components.
... First we recall the following so-called Classification Theorem. Theorem A ([11]) Let f be a transcendental entire function and D be a Fatou component of f . Then 1. D is a simply-connected wandering domain or 2. D is a multiply-connected wandering domain or 3. there exists a positive integer s such that f s (D) ⊂ U for some periodic Fatou component U , and U satisfies one and only one of the following possibilities: 1) U contains an attracting periodic point z o of period m. ...
... If the set sing(f −1 ) is bounded, then we say f is of bounded type, in particular, if the set sing(f −1 ) is finite, then f is of finite type, and we denote this by f ∈ B and f ∈ S, respectively ([2]). We refer the reader to books [11] and [14], the survey article of Bergweiler [2], the papers of Fatou [8] and Julia [13] for more about the iteration theory of transcendental entire functions. Let f be a transcendental entire function and R > min z∈J(f ) |z| (Note: Since J(f ) = φ [11] for any transcendental entire function f , so R < +∞). ...
... We refer the reader to books [11] and [14], the survey article of Bergweiler [2], the papers of Fatou [8] and Julia [13] for more about the iteration theory of transcendental entire functions. Let f be a transcendental entire function and R > min z∈J(f ) |z| (Note: Since J(f ) = φ [11] for any transcendental entire function f , so R < +∞). Let us recall the following notations: ...
Let f and g be two distinct permutable transcendental entire functions. Suppose further that q(g)=aq(f)+b for some nonconstant polynomial q(z) and constants a(≠0), . In this article, we will investigate the dynamical properties of f and g and show that they have the same Fatou sets with the same components.
... The proof of the next result is motivated from the following lemma. Lemma 4.4 (Lemma 4.3 of [11]). Let f be a transcendental entire function and U be a component of F (f ). ...
... Lemma 4.2. Also from p. 64 of [11], U g is a Siegel disk for all g ? G U , and hence from the classification of periodic components of F(G) in [14], U ? F(G) is a Siegel disk. ...
We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.
... So, f k k ðzÞ ! a n as k ! 1 for all z 2 A n . We know the following from [ [16], Theorem 4.13]. ...
In this article, the dynamics of functions in the family has been studied. It is proved that the family has infinitely many singular values for every choice of non-zero . We discuss how the dynamics of a function belonging to the family changes as the parameter changes. Some specific values of the parameter are given for which the function has wandering domains. We show that when , the family has no invariant Baker domains. It is proved that for some complex parameter values, the Fatou set contains Siegel discs. Finally, we give a comparison of dynamics of the families , and through a table.
... To prove some important properties of the two sets mentioned above, it is necessary the concept of normal family and some results of Complex Analysis which drives us to the subject of holomorphic dynamics. We recommend the reader to revised [2] and [3] for starting the subject in holomorphic dynamics for rational and transcendental entire functions. ...
Due to the great growth of the world population and the increase in the standard of living in general,
the use of textile fibers has increased significantly in recent decades. In this context, this article allows
to show an analysis of the sustainability of the carpet production process and the perspectives that have
the process of recycling it in Mexico, focusing mainly on PVC recycling. This paper is one of the
results of one CONACYT-PEI Project, in which it objective is the design of a carpet recycling process
in Mexico. In order to show the potential and savings generated by the recycling of some compounds
such as PVC, Nylon, polyester and synthetic fibers, an analysis of the background on the process of
recycling of carpets and textiles worldwide and in the United States is performed. Likewise, an analysis
is carried out on the energy and water consumption of carpet production with virgin materials and the
recycling process. In order to present a context and show the viability of the process. This document is
focused particularly in the recycling process of the carpet backing. Finally, some criteria are taken to
obtain conclusions regarding the sustainability of the process.
... The main concern of transcendental iteration theory is to describe the nature of the components of Fatou sets and the structure and properties of the Julia sets, escaping sets, 2 Fast Escaping Set of Transcendental Semigroup and fast escaping sets. We refer to the monograph: Dynamics of Transcendental Entire Functions [7] and to the book: Holomorphic Dynamics [11] for basic facts concerning the Fatou set, Julia set and escaping set of a transcendental entire functions. We refer to [14,15,16] for facts and results concerning the fast escaping set of a transcendental entire functions. ...
... The main concern of transcendental iteration theory is to describe the nature of the components of Fatou sets and the structure and properties of the Julia sets, escaping sets, 2 Fast Escaping Set of Transcendental Semigroup and fast escaping sets. We refer to the monograph: Dynamics of Transcendental Entire Functions [7] and to the book: Holomorphic Dynamics [11] for basic facts concerning the Fatou set, Julia set and escaping set of a transcendental entire functions. We refer to [14,15,16] for facts and results concerning the fast escaping set of a transcendental entire functions. ...
... The main concern of such a transcendental iteration theory is to describe the nature of the components of Fatou set and the structure and properties of the Julia set, escaping set and fast escaping set. We use monograph: dynamics of transcendental entire functions [7] and book: holomorphic dynamics [10] for basic facts concerning the Fatou set, Julia set and escaping set of a transcendental entire function. We use [13,14,15] for facts and results concerning the fast escaping set of a transcendental entire function. ...
In this paper, we study fast escaping set of transcendental semigroup. We discuss some the structure and properties of fast escaping set of transcendental semigroup. We also see how far the classical theory of fast escaping set of transcendental entire function applies to general settings of transcendental semigroups and what new phenomena can occur.
... Next, we see a special subsemigroup of holomorphic semigroup that yields cofinite index. In classical holomorphic iteration theory, the stable basin is one of the above types but in transcendental iteration theory, the stable basin is not a Hermann because a transcendental entire function does not have Hermann ring [5,Proposition 4.2]. ...
We investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup S are non-empty.
... It is well-known from [1,2,4,8] that the classical escaping set of transcendental entire function f is completely invariant. ...
For a non-trivial transcendental semigroup, escaping set I(S) is in general S-forward invariant and it is S-completely invariant if semigroup S is abelian. In the contrary of this result, we investigate completely invariant escaping set K(S) in different way even if semigroup S is not abelian and we discuss some properties and structure of such type of escaping set. Also, we establish some relations between completely invariant escaping set K(S) and the general escaping set I(S).
... Let E(S) • = ∅, where E(S) • denotes the interior of E(S). Then there exists a disk D = {|z − z 0 | < r} ⊂ E(S) such that it intersects J(h) for some h ∈ S. Then by [2,Theorem 3.9], for each finite value a, there is sequence z k → z 0 ∈ J(h) and a sequence of positive integers n k → ∞ such that f n k (z k ) = a, (k = 1, 2, 3, . . .) except at most for a finite value. ...
In holomorphic semigroup dynamics, Julia set is in general backward invariant and so some fundamental results of classical complex dynamics can not be generalized to semigroup dynamics. In this paper, we define completely invariant Julia set of transcendental semigroup and we see how far the results of classical transcendental dynamics generalized to transcendental semigroup dynamics.
... The importance of singular values in the dynamics of transcendental functions can be seen in [1,4,7,17]. The dynamics of one-parameter family λ e z , that has only one singular value, is studied in detail, for instance see [2,3]. ...
The main goal of the present paper is to investigate the singular values of three-parameter families of transcendental (i) entire functions and ; , , (ii) meromorphic functions and , ; , . It is obtained that all the critical values of and lie in the right half plane for and in the left half plane for . It is also shown that all these critical values of and are interior and exterior of the open disk centered at origin and having radii and respectively. Further, it is described that the functions and both have infinitely many singular values.
... In that article, we conjectured that such a sequence existed for each choice of the pair (ψ, ρ), and we remarked that in view of (for example) [4], Theorem 2.6, this phenomenon would follow if each ψ is a repelling fixed point of zeta, hence an attracting fixed point of a branch of the inverse ζ (−1) . (Here branch has its usual meaning in complex analysis; it does not denote a branch of a backward orbit, as defined in [1], but it is true that φ ψ,ρ,ζ constitutes a branch of the backward orbit of zeta as defined there.) ...
We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically , we study the family of functions Vz : s → ζ(s) exp(zs). We observe convergence of Vz fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.
... In that article, we conjectured that such a sequence existed for each choice of the pair (ψ, ρ), and we remarked that in view of (for example) [3], Theorem 2.6, this phenomenon would follow if each ψ is a repelling fixed point of zeta, hence an attracting fixed point of a branch of the inverse ζ (−1) . (Here branch has its usual meaning in complex analysis; it does not denote a branch of a backward orbit, as defined in [1], but it is true that φ ψ,ρ,ζ constitutes a branch of the backward orbit of zeta as defined there.) ...
We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically , we study the family of functions Vz : s → ζ(s) exp(zs). We observe convergence of Vz fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.
... Because λ ∈ Λ would be repelling, it would also be an attracting fixed point of a local branch of the functional inverse of ζ •L , and then the convergence would follow from standard results, for example, Theorem 2.6 of [5]. ...
We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero ρ (trivial or nontrivial) and each zeta fixed point ψ there is a nearly logarithmic spiral s ρ,ψ with center ψ containing ρ. (2) s ρ,ψ interpolates a subset B ρ,ψ of the backward zeta orbit of ρ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, ζ(B ρ,ψ) = B ρ,ψ ∪ {0}. (4) B ρ,ψ has nearly uniform angular distribution around the center of s ρ,ψ. We will make these statements precise.
... Because λ ∈ Λ would be repelling, it would also be an attracting fixed point of a local branch of the functional inverse of ζ •L , and then the convergence would follow from standard results, for example, Theorem 2.6 of [5]. ...
We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero ρ (trivial or nontrivial) and each zeta fixed point ψ there is a nearly logarithmic spiral s ρ,ψ with center ψ containing ρ. (2) s ρ,ψ interpolates a subset B ρ,ψ of the backward zeta orbit of ρ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, ζ(B ρ,ψ) = B ρ,ψ ∪ {0}. (4) B ρ,ψ has nearly uniform angular distribution around the center of s ρ,ψ. We will make these statements precise.
... Because λ ∈ Λ would be repelling, it would also be an attracting fixed point of a local branch of the functional inverse of ζ •L , and then the convergence would follow from standard results, for example, Theorem 2.6 of [5]. ...
We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero ρ (trivial or nontrivial) and each zeta fixed point ψ there is a nearly logarithmic spiral s ρ,ψ with center ψ containing ρ. (2) s ρ,ψ interpolates a subset B ρ,ψ of the backward zeta orbit of ρ comprising a set of zeros of all iterates of zeta. (4) If zeta is viewed as a function on sets, ζ(B ρ,ψ) = B ρ,ψ ∪ {0}. (5) B ρ,ψ has nearly uniform angular distribution around the center of s ρ,ψ. We will make these statements precise.
... Because λ ∈ Λ would be repelling, it would also be an attracting fixed point of a local branch of the functional inverse of ζ •L , and then the convergence would follow from standard results, for example, Theorem 2.6 of [5]. ...
We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero ρ (trivial or nontrivial) and each zeta fixed point ψ there is a nearly logarithmic spiral s ρ,ψ with center ψ containing ρ. (2) s ρ,ψ interpolates a subset B ρ,ψ of the backward zeta orbit of ρ comprising a set of zeros of all iterates of zeta. (4) If zeta is viewed as a function on sets, ζ(B ρ,ψ) = B ρ,ψ ∪ {0}. (5) B ρ,ψ has nearly uniform angular distribution around the center of s ρ,ψ. We will make these statements precise.
... The Julia sets are actively studied for many self-maps of C (polynomials, transcendental entire functions; see e.g. [1,4,7]), C 2 (generalized Hénon maps; see e.g. [8]) and C N (polynomial mappings with the Łojasiewicz exponents greater than 1; see [9]). ...
We gather known results on escape radii and add some facts and examples concerning the subject of complete invariance and compactness of filled Julia sets.
... The compliment of Fatou set is called Julia set and this set is denoted by J ( f ). The basic properties and structure of these sets can be found in [1, 3, 4, 6, 7, 8]. This paper basically concerns on the structure of escaping set I( f ) and fast escaping set A( f ) introduced respectively by Eremenko [5], and Bergweiler and Hinkkanen [2]. ...
For a transcendental entire function f, the escaping set I (f) consists of points whose iterates tends to infinity under f and fast escaping set A(f) consists of points whose iterates tends to infinity as fast as possible. In this article, we examine the conditions that both sets have the structure of infinite spider's web.
... In particular, U is called a Baker domain if f nk (z) → ∞ as n → ∞ for z ∈ U and, moreover, U is simply connected [1, Theorem 3.1]. For more details on the subject we refer the reader to [4,5,15]. ...
Let g and h be transcendental entire functions and let f be a
continuous map of the complex plane into itself with Then
g and h are said to be semiconjugated by f and f is called a
semiconjugacy. We consider the dynamical properties of semiconjugated
transcendental entire functions g and h and provide several conditions
under which the semiconjugacy f carries Fatou set of one entire function into
the Fatou set of other entire function appearing in the semiconjugation. We
have also shown that under certain condition on the growth of entire functions
appearing in the semiconjugation, the number of asymptotic values of the
derivative of composition of the entire functions is bounded.
... When there is no confusion, we briefly write F and J instead of F(f) and J(f). Clearly F(f) is open and it is well-known that J(f) is a nonempty perfect set which either coincides with C or is nowhere dense in C For the basic results in the dynamical system theory of transcendental functions, we refer the reader to the books [9] and [13]. OPEN QUESTION 1 (Baker [1]). ...
Let f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.
... In a more general case, let f(z) be a transcendental meromorphic function in C with finitely many poles. If J(f) has only bounded components, then d(J(f), P\J(f)) = 0, since J(f) is not uniformly perfect by Theorem 1 in [19] PROOF: If / is entire in C or is a holomorphic self map of the punctured plane C\{0}, then the conclusion is true, see [7, 10]. [4] Now, we consider / € PM but not being included in the above two ...
Pommerenke and Beardon [5, 12, 13] introduced and studied the uniformly perfect sets. Following Pommerenke, many authors have researched the topic, and many papers on the uniformly perfect sets have appeared in the literature. It was proved that the Julia sets of rational functions are uniformly perfect, see [8, 9, 11], and also [13]. However, unlike the case of rational functions, the Julia sets of a transcendental meromorphic function may not be uniformly perfect, see the example in Section 1. But, it is interesting to discuss when the Julia sets of transcendental meromorphic functions will be uniformly perfect. Zheng [17, 18, 19] studied uniformly perfect boundaries of stable domains in the iteration of meromorphic functions. In this paper, we shall study the Julia sets of functions from the class PM, which a is more general class of functions than meromorphic functions. We shall also prove that the Julia sets of the skew product associated with a finitely generated semigroup of rational functions, each generator having degree not less than 2, are uniformly perfect.
... Background on iteration theory of rational maps can be found for example in [7], [18] or [22]. For transcendental maps, the reader can check the survey in [8] or the book [16]. ...
We prove some results concerning the possible configurations of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period and give a sufficient condition for a configuration to be realizable.
... A well known property of the Julia set of an entire or rational function f is that J(f )=J(f n ). Other basic knowledge of iterations of rational or transcendental functions can be found in [5, 6, 13]. In 1922–23, Julia [16] and Fatou [12] proved that for any 2 rational functions f and g of degree at least 2 such that f and g are permutable, i.e. f @BULLET g = g @BULLET f , then their Julia sets will be the same. ...
In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of
degree at least 2 such that f(g(z)) = g(f(z)),
have the same Julia set. Baker then
asked whether the result remains true for nonlinear entire functions. In this paper,
we shall show that the answer to Baker's question is true for almost all nonlinear
entire functions. The method we use is useful for solving functional equations. It
actually allows us to find out all the entire functions g which permute with a given
f which belongs to a very large class of entire functions.
... In the following, let λ and c be such that λ = e −c . Proof of Theorem 1: In [8] it has been proved that if λ 1 and λ 2 are in the unit circle, then g λ 1 and g λ 2 are quasiconformally conjugate. Hence, it suffices to show the claim for a real λ. ...
We consider the family of transcendental entire functions given by
. If Re c > 0, then fc features a Baker domain as the only component of the Fatou set, while the functions fc show a different dynamical behaviour if
. We describe the dynamical planes of these functions and show that the Julia sets converge in the limit process
with respect to the Hausdorff metric, where
and
. We use this to show that Baker domains of any type (concerning a classification of König) are not necessarily stable under perturbation.
... Moreover, it is well known that J(f ) is the smallest closed completely invariant set which contains at least three points. For the details of the general theory of iterations of meromorphic functions, we refer the reader to a series of papers by Baker, Kotus and L ¨ u78910, the survey article [13], as well as the book [26]. For the iteration theory of rational functions, we refer to the books [12], [27] or [37]. ...
In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function f belongs to the class S and its Julia set is locally connected, then the residual Julia set of f is empty if and only if its Fatou set F (f) has a completely invariant component or consists of only two components. We also show that if f is a meromorphic function which is not of the form α + (z − α) −k e g(z) , where k is a natural number, α is a complex number and g is an entire function, then f has buried components provided that f has no completely invariant components and its Julia set J(f) is disconnected. Moreover, if F (f) has an infinitely connected component, then the singleton buried components are dense in J(f). This generalizes a result of Baker and Domínguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.
Wiman-Valiron theory and results of Macintyre about "flat regions" describe the asymptotic behavior of entire functions in certain disks around points of maximum modulus. We estimate the size of these disks for Macintyre's theory from above and below.
For a transcendental entire function with sufficiently small growth, Baker raised the question whether it has no unbounded Fatou components. We have shown that if the function is of order strictly less than half, minimal type, then it has no unbounded Fatou components. This, in particular gives a partial answer to Baker's question. In addition, we have addressed Wang's question on Fejér gaps. Certain results about functions with Fabry gaps and of infinite order have also been generalized.
We study some relation between escaping sets of two permutable entire functions. In addition, we investigate the dynamical properties of the map $f(z)=z+1+e^{-z}.
We discuss the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory and provide some condition for the complete
invariance of escaping set and Julia set of transcendental semigroups. Some
results on limit functions and postsingular set have been discussed. A class of
hyperbolic transcendental semigroups and semigroups having no wandering domains
have also been provided.
Let f be a nonconstant meromorphic function. The sequence of the iterates of f is denoted by {f^0}=id,{f^1}=f,\cdots,{f^{n + 1}} = {f^n}(f), \cdots
The aim of this paper is to study the singular values and fixed points of the one parameter family of generating functions h λ (z)=λze z e z -1, λ∈ℝ∖{0}, which arises from the generalized Bernoulli generating function, the Apostol-Bernoulli generating function or the Stirling generating function. It is found that the function h λ (z) has infinitely many singular values. Further, it is shown that all critical values of h λ (z) are lying outside the open disk centered at the origin an having radius λ. Moreover, for λ<0, the real fixed points of h λ (z) and their nature are investigated. Finally, the results found here are compared with the dynamical properties of the functions λtanz, E λ (z)=λe z -1 z and f λ (z)=λz z+4e z for λ>0.
This paper describes a driver HomMap to the standard local bifurcation software AUTO for numerical analysis of homoclinic and heteroclinic bifurcations in maps and periodically forced systems. The driver can detect loci of homoclinic points if the unstable or stable manifolds are of dimension one, and treat problems of two or more dimensions. The algorithms used and their implementations in AUTO with HomMap are explained. Three examples are given, namely the Hénon map and the single and coupled Duffing oscillators, in order to demonstrate the usefulness and capabilities of the driver.
We study the dynamics of an arbitrary semigroup of transcendental entire
functions using Fatou-Julia theory. Several results of the dynamics associated
with iteration of a transcendental entire function have been extended to
transcendental semigroups. We provide some conditions for connectivity of the
Julia set of the transcendental semigroups. We also study finitely generated
transcendental semigroups, abelian transcendental semigroups and limit
functions of transcendental semigroups on its invariant Fatou components.
We consider the dynamical properties of transcendental entire functions and
their compositions. We give several conditions under which Fatou set of a
transcendental entire function f coincide with that of where g
is another transcendental entire function. We also prove some result giving
relationship between singular values of transcendental entire functions and
their compositions.
We consider entire transcendental functions f with an invariant (or periodic) Baker domain U. First, we classify these domains into three types (hyperbolic, simply parabolic and doubly parabolic) according to the surface they induce when we quotient by the dynamics. Second, we study the space of quasiconformal deformations of an entire map with such a Baker domain by studying its Teichmuller space. More precisely, we show that the dimension of this set is infinite if the Baker domain is hyperbolic or simply parabolic, and from this we deduce that the quasiconformal deformation space of f is infinite dimensional. Finally, we prove that the function f(z) = z +e z, which possesses infinitely many invariant Baker domains, is rigid, i.e., any quasiconformal deformation of f is affinely conjugate tof.
Let f,h and g be transcendental entire functions satisfying h(f) = g(h). Suppose that fB hS. In this paper we shall prove that h −1(J(g))=J(f).
Wiman–Valiron theory and results of Macintyre about ‘flat regions’ describe the asymptotic behaviour of entire functions in certain discs around points of maximum modulus. We estimate the size of these discs for Macintyre's theory from above and below.
It is known that the dynamics of f and g vary to a large extent from that of
its composite entire functions. We have shown using Approximation theory of
entire functions, the existence of entire functions f and g having infinite
number of domains satisfying various properties and relating it to their
composition. We have explored and enlarged all the maximum possible ways of the
solution in comparison to the past result worked out.
Boundedness of components of the Fatou sets of iteration of tran-scendental entire or meromorphic functions are investigated in this paper.
We consider the Teichmüller space of a general entire transcendental function f : C → C regardless of the nature of the set of singular values of f (critical values and asymptotic values). We prove that, as in the known case of periodic points and criti-cal values, asymptotic values are also fixed points of any quasiconformal automorphism that commutes with f and which is homotopic to the identity, rel. the ideal bound-ary of the domain. As a consequence, the general framework of McMullen and Sullivan [McMullen & Sullivan 1998] for rational functions applies also to entire functions and we can apply it to study the Teichmüller space of f , analyzing each type of Fatou component separately. Baker domains were already considered in citefh, but wandering domains are new. We provide different examples of wandering domains, each of them adding a differ-ent quantity to the dimension of the Teichmüller space. In particular we give examples of rigid wandering domains.
Let g and h be two transcendental entire functions. Suppose that the Fatou set F(g ∘ h) contains multiply connected components. In this article, we will consider the growth of the functions g and h.
Let f be an entire function. A point zo is called a critical point of f if f′(zo)=0, and f(zo) is called a critical value (or an algebraic singularity) of f. Next aε
is said to be an asymptotic value (or a transcendental singularity) of f if there exists a curve λ:[0,1]→ℂ such that, relations between critical points of f, g and f o g and also in the case when the two functions f and g are semi-conjugated with another entire function.
In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying
and , for some transcendental entire function h , rational function and a function , which is analytic in the range of h , then . Then as an application of this result, we show that if , where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or , where p, q are nonconstant polynomials, such that f(g)=g(f) for a nonconstant entire function g , then J(f)=J(g) .
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