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On semiconjugation of entire functions
Abstract
Let f and h be transcendental entire functions and let
g be a continuous and open map of the complex plane into itself with
g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g. We show that if f satisfies
a certain condition, which holds, in particular, if f has no wandering domains, then
g−1(J(h))=J(f). Here
J(·) denotes the Julia set of a function. We conclude that if f
has no wandering domains, then h has no wandering domains. Further, we show that
for given transcendental entire functions f and h, there are only
countably many entire functions g such that
g[circ B: composite function (small circle)]f=h[circ B: composite function (small circle)]g.
... We make remarks on (1)(2)(3)(4)(5)(6)(7)(8). It is easy to see that (1-8) follows from ...
... In view of a result of Valiron (see [33,Theorem 2.7.5]), if f has no Borel directions of order λ − ω in Ω, then (1-9) holds for some a and b. Hence, (1)(2)(3)(4)(5)(6)(7)(8) can be replaced by the assumption that f has no Borel directions of order λ − ω in Ω. For completeness, let us give an example to show that the condition in (1-7) is necessary. ...
... THEOREM 1.8. For any given λ ∈ (1, ∞], there exists a meromorphic function of order and lower order equal to λ such that (1)(2)(3)(4)(5)(6)(7)(8) holds on the upper half plane and lower half plane, and its Julia set lies on the real axis and its Fatou set contains a sequence of annuli A(r n , dr n ) with r n → ∞ as n → ∞ for some d > 1. ...
This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.
... In the case of entire transcendental functions now we still do not know if any two commuting such maps have the same Julia set even though there has been a lot of research in this direction, see for example [10,18,97,137,138]. ...
... Finally, let us discuss the fast escaping set. The fast escaping set is an important subset of the escaping set and was rst dened by Bergweiler and Hinkkanen in [18]. ...
... Intuitively the fast escaping set is the set of points that escape to innity as fast as possible. In [18] it is also proved that ...
The work in this thesis revolves around the study of dynamical systems arising from iterating quasiregular maps. Quasiregular maps are a natural generalization of holomorphic maps in higher (real) dimensions and their dynamics have only recently started being systematically studied. We first study permutable quasiregular maps, i.e. maps that satisfy f ◦g = g ◦f, where we show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and g = φ ◦ f, where φ is a quasiconformal map, have the same Julia sets. Those results generalize well known theorems of Bergweiler, Hinkkanen and Baker on permutable entire functions. Next we study the dynamics of Zorich maps which are among the most important examples of quasiregular maps and can be thought of as analogues of the exponential map on the plane. For the exponential family Eκ : z 7→ κez, κ > 0, it has been shown that when κ > 1/e the Julia set of Eκ is the entire complex plane, essentially by Misiurewicz. Moreover, when n 0 < κ ≤ 1/e Devaney and Krych have shown that the Julia set of Eκ is an uncountable collection of disjoint curves. Bergweiler and Nicks have shown that a similar result is also true for Zorich maps. First we construct a certain "symmetric" family of Zorich maps, and we show that the Julia set of a Zorich map in this family is the whole of R3 when the value of the parameter is large enough, thus generalizing Misiurewicz's result. Moreover, we show that the periodic points of those maps are dense in R3 and that their escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential. On a similar note, we study the set of endpoints of the Julia sets of Zorich maps in the case that the Julia set is a collection of curves. We show that ∞ is an explosion point for the set of endpoints by introducing a topological model for the Julia sets of certain Zorich maps, similar to the so called straight brush of Aarts and Oversteegen. Moreover we introduce an object called a hairy surface which is a compactified version of the Julia set of Zorich maps and we show that those objects are not uniquely embedded in R3, unlike the corresponding two-dimensional objects which are all ambiently homeomorphic. Finally, we study the question of how a connected component of the inverse image of a domain under a quasiregular map covers the domain. We prove that the subset of the domain that is not covered can be at most of conformal capacity zero. This partially generalizes a result due to Heins. We also show that all points in this omitted set are asymptotic values.
... It is not even known if permutable transcendental entire functions have the same Julia set or not. However Bergweiler and Hinkkanen [9] in 1999, by introducing the so called fast escaping set A ( f ), managed to prove the following. ...
... We will give examples which show that this theorem cannot hold in the quasiregular case. To be more specific, by modifying an example given in [9], we are able to prove the following result. A natural thing to ask here is the following question: Is it possible for a polynomial to commute with an entire transcendental function? ...
... Finally, let us discuss the fast escaping set. The fast escaping set, as we have already mentioned, was first defined by Bergweiler and Hinkkanen in [9]. Intuitively the fast escaping set is the set of points that escape to infinity as fast as possible. ...
Let f and g be two quasiregular maps in that are of transcendental type and also satisfy . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and , where is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.
... Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus in [12] and show that the multiply-connected wandering domain is a part of the fast escaping set under a growth condition of the maximal modulus. Finally we give examples of wandering domains escaping at arbitrary fast rate and slow rate. ...
... Next we investigate escaping speed of points in an escaping Fatou component under iterates. A fast escaping set of a transcendental entire function f , denoted by A(f ), was introduced by Bergweiler and Hinkkanen [12] and can be defined [29] by ...
... Other equivalent definitions of A(f ) (cf. [12] and [29]) can be defined with the Nevanlinna characteristic for an entire function f , but not for a meromorphic function with poles. In [41], we showed that (1) M (f ) is non-empty and completely invariant; Closure of a multiply connected Fatou component of an entire function must be in the fast escaping set A(f ), proved in [32]. ...
In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen when the maximal modulus of the meromorphic function is large enough. Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus in [12] and show that the multiply-connected wandering domain is a part of the fast escaping set under a growth condition of the maximal modulus. Finally we give examples of wandering domains escaping at arbitrary fast rate and slow rate.
... Although this fact was proved by both Fatou and Julia in 1923, the corresponding problem for transcendental functions -which could serve as a first step in classifying commuting analytic functions in a more general setting -is still open. After Baker proved this fact for commuting entire functions without escaping Fatou components in [1], much progress has been made for entire functions by considering the fast escaping set introduced by Bergweiler and Hinkkanen in [6]. This is the set A(f ) = z ∈ C : there exists l ∈ N such that |f n+l (z)| ≥ M (R, f n ) for all n ∈ N , ...
... where M (r, f ) denotes the maximum modulus M (r, f ) = max{|f (z)| : |z| = r} and R is chosen so that M (r, f ) > r for all r ≥ R. Bergweiler and Hinkkanen showed that, for commuting entire functions f and g, g −1 (A(f )) ⊂ A(f ) [6,Theorem 5] and used this to show that commuting implies J(f ) = J(g) whenever A(f ) ⊂ J(f ) and A(g) ⊂ J(g). More recently, Benini, Rippon and Stallard [4] showed that, for entire functions f and g, commuting implies having the same Julia set except possibly when f and g have simply connected fast escaping wandering domains. ...
... More recently, Benini, Rippon and Stallard [4] showed that, for entire functions f and g, commuting implies having the same Julia set except possibly when f and g have simply connected fast escaping wandering domains. The overall strategy used in each of the papers [1], [6] and [4] is to show that if f and g commute and the components of F (f ) and F (g) are only of certain types, then g (F (f )) ⊂ F (f ). It then follows that F (f ) ⊂ F (g), by Montel's theorem, and similarly F (g) ⊂ F (f ). ...
Let f and g be commuting meromorphic functions with finitely many poles. By studying the behaviour of Fatou components under this commuting relation, we prove that f and g have the same Julia set whenever f and g have no simply connected fast-escaping wandering domains. By combining this with a recent result of Tsantaris', we obtain the strongest statement (to date) regarding the Julia sets of commuting meromorphic functions. In order to highlight the difference to the entire case, we show that transcendental meromorphic functions with finitely many poles have orbits that alternate between approaching a pole and escaping to infinity at strikingly fast rates.
... or not. However Bergweiler and Hinkkanen [8] in 1999, by introducing the so called fast escaping set A (f ), managed to prove the following. Recently, Benini, Rippon and Stallard in [3] managed to improve the above theorem and include some cases where A(f ) ⊂ J (f ) and A(g) ⊂ J (g). ...
... We will give examples which show that this theorem cannot hold in the quasiregular case. To be more specific, by modifying an example given in [8], we are able to prove the following result. Theorem 1.5. ...
... Finally, let us discuss the fast escaping set. The fast escaping set, as we have already mentioned, was first defined by Bergweiler and Hinkkanen in [8]. For a transcendental entire function they defined it as ...
Let f and g be two quasiregular maps in that are of transcendental type and also satisfy . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and , where is a quasiconformal map, have the same Julia sets.
... For a transcendental entire function f , Eremenko [8] proved that the escaping set is always non-empty and in fact always meets the Julia set J(f ). The fast escaping set A(f ) was introduced by Bergweiler and Hinkkanen [5] and has been studied in detail by Rippon and Stallard [19,22,23,24]. This subset of I(f ) is often described as containing those points that escape to infinity 'as fast as possible' under iteration of the entire function f and it is again known that A(f ) ∩ J(f ) = ∅. ...
... This follows from results in [3]. In the entire case, Eremenko's original method for finding escaping points can be used to obtain such points; see the proof of [5,Lemma 2] and also [22,Theorem 2.5]. ...
This article concerns the iteration of quasiregular mappings on and entire functions on . It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of is asymptotic to the iterated maximum modulus .
... The strongest general results about Eremenko's conjecture have been obtained by using the subset of I(f ) called the fast escaping set A(f ), introduced in [6], which can be defined as follows; see [27]. First put (1.1) ...
... It was shown in [6] that Eremenko's construction in [9] of points in I(f ) actually gives points that are in A(f ). Points constructed in this way have particularly nice properties and as noted earlier we often refer to them as Eremenko points; see [25] and [5]. ...
Much recent work on the iterates of a transcendental entire function f has been motivated by Eremenko's conjecture that all the components of the escaping set I(f) are unbounded. Here we show that if I(f) is disconnected, then the set has uncountably many unbounded components for any open disc D that meets the Julia set of f. For the set , which is the `core' of the fast escaping set, we prove the much stronger result that for some either is connected and has the structure of an infinite spider's web or it has uncountably many components each of which is unbounded. There are analogous results for the intersections of these sets with the Julia set when no multiply connected wandering domains are present, but strikingly different results when they are present. In proving these, we obtain the unexpected result that multiply connected wandering domains can have complementary components with no interior, indeed uncountably many.
... The set A(f ) was introduced in [6] and consists of those points whose iterates under f eventually grow at least as fast as iterates of the maximum modulus, M (r) = max |z|=r |f (z)|. It can be defined as follows: ...
... and that I(f ) has no bounded components. (Recall that the fast escaping set A(f ) also has the properties listed in (4.4), and in addition A(f ) has no bounded components; see [6], [28] and [31].) For V (f ) and V + (f ), we have the following. ...
In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the {\it minimum} modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function.
... This set consists of the endpoints which do not belong to the fast escaping set A(f ). The set A(f ) is a subset of I(f ) which was introduced by Bergweiler and Hinkkanen in [6], and is defined as follows. First, for R > 0, we define the maximum modulus function by where R > 0 is such that M (r, f ) > r for r ≥ R. It is known [20,Theorem 2.2] that this definition is independent of the choice of R. Note that in this definition M n (R, f ) denotes n iterations of M (R, f ) with respect to the variable R. ...
... In particular, E(g) = π(E(f )). Note also that A(g) ⊂ π(A(f )) [6,Theorem 5]. It follows by Theorem A and Theorem C that ...
There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set of finite endpoints. It was recently shown that, for certain functions in the exponential family, there is a strong dichotomy between the topological properties of the set of endpoints which escape and those of the set of endpoints which do not escape. In this paper, we show that this result holds for large families of functions in the Eremenko-Lyubich class. We also show that this dichotomy holds for a family of functions, outside that class, which includes the much-studied Fatou function defined by Finally, we show how our results can be used to demonstrate that various sets are spiders' webs, generalising results such as those in a recent paper of the first author.
... It was shown in [12] that a Baker domain of a transcendental entire function cannot meet the fast escaping set A(f ); we defer a definition of this set to Section 2. Since, as mentioned earlier, a Baker domain of a transcendental entire function is full, one direction of the following result can be seen as a generalisation of this fact to quasiregular maps of transcendental type. The other direction, when combined with Lemma 5 below, shows that the hypothesis in the statement of Theorem 1 that U is full seems to be required. ...
... Suppose that x ∈ A(f ). Then, by (12), there exists ∈ N such that, for all sufficiently large k ∈ N, ...
We consider the iteration of quasiregular maps of transcendental type from to . We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from to with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from to which is equal to the identity map in a half-space.
... where R > min z∈J(f ) |z| and J(f ) is the Julia set, is called the fast escaping set. It was introduced in [9] and has also turned out to be very useful in transcendental dynamics. A thorough study of this set is given in [27] where it is also shown that ...
... The following lemma (see, e.g., [25, Lemma 2.2]) is a a consequence of Hadamard's three circles theorem; that is, the convexity of log M(r, f ) in log r. The next lemma can be found in [14] for the escaping set and in [9,27] for the fast escaping set. ...
Let f be an entire function and denote by be the spherical derivative of f and by the n-th iterate of f. For an open set U intersecting the Julia set J(f), we consider how fast and tend to . We also study the growth rate of the sequence for .
... Most of the works on I(f) are inspired by Eremenko's conjecture which states that every component of I(f) is unbounded [2]. Since then, an amount of work was done on Eremenko's conjecture by Bergweiler and Hinkkanen [3], Rippon and Stallard [4][5][6] and Evdoridou [7]. Recently, a counterexample to Eremenko's conjecture was provided in [8] proving the conjecture to be not true in general. ...
... This plays a major role in determining the dynamics of a function. Based on the rate of escape, Bergweiler and Hinkkanen [3], introduced the fast escaping set A(f). It is shown in [5] that I(f) always has at least one unbounded component. ...
In this paper, we investigate some properties of quite fast escaping set Q(f) and , a subset of Q(f), where f is a transcendental entire function. We first define level of , based on iterations of over r. Then, we show some relations between levels of and observe that every component of is unbounded. Finally, we give condition under which (0th level of ), , Q(f) and I(f) form a spider’s web.
... Of particular importance is the fast escaping set, A(f ), of points that escape to infinity 'as fast as possible' under iteration. This set, introduced by Bergweiler and Hinkkanen [BH99] in connection to permutable functions and absence of wandering domains, is defined in [RS12] as A(f ) . . = {z : there exists ∈ N such that |f n+ (z)| ≥ M n f (R), for n ∈ N}, ...
... = max |z|=r |f (z)|, for r > 0, is the maximum modulus function, M n f (r) denotes the n-th iterate of the function M f (r), and R > 0 can be taken to be any value such that M f (r) > r, for r ≥ R. It follows from [RS12, Theorem 1.2] that if U is a component of F (f ) such that U ∩ A(f ) = ∅, then U ⊂ A(f ), and we say that U is fast escaping. We note that all fast escaping components of F (f ) must be wandering domains; see [BH99,Lemma 4]. ...
We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide such functions with an orbit of unbounded fast escaping wandering domains, answering a question of Rippon and Stallard, that can be generated to have several different internal dynamics. In relation to a conjecture of Baker, we also construct functions of any order greater than 1/2 and smaller than 1 with an unbounded wandering domain, giving a partial answer to a question of Stallard.
... The conjecture has been proved by using the fast escaping set A ( f ), which consists of points whose iterates tend to infinity as fast as possible. This set was first introduced by Bergweiler and Hinkkanen (1999) and defined in the following form by Rippon and Stallard (2012). This paper is the most recent study in the field of transcendental dynamics and is primarily concerned with the partition of the fast escaping set A ( f ). ...
... For a TEF f, the following statements are hold. (2005) proved statement (1), (5) and (6); Bergweiler and Hinkkanen (1999) proved statements (2), (3), and (4); and Rippon and Stallard (2012) also proved statements (7), (8), (9), and (10) (7) is an important one and is considered the strongest result in the direction of Eremenko's conjecture. ...
For a transcendental entire function f, the set of form I ( f ) = {z ∈ : (z) → ∞ as n → ∞} is called an escaping set. The major open question in transcendental dynamics is the conjecture of Eremenko, which states that for any transcendental entire function f, the escaping set I ( f ) has no bounded component. This conjecture in a special case has been proved by defining the fast escaping set A ( f ), which consists of points that move to infinity as fast as possible. Very recent studies in the field of transcendental dynamics have concentrated on the partition of fast escaping sets into maximally and non-maximally fast escaping sets. It is well known that a fast escaping set has no bounded component, but in contrast, there are entire transcendental functions for which each maximally and non-maximally fast escaping set has uncountably many singleton components.
... This has prompted the introduction and study of numerous subsets of the escaping set. For example in [16], Bergweiler and Hinkkanen introduced the fast escaping set in the setting of entire functions. This set, informally, consists of points whose iterates escape to infinity 'as fast as possible' and has since been studied in detail in many different settings, for example see [56,77,91,97]. ...
... The above property has found numerous applications within covering results; for example see [16,89]. ...
This thesis is concerned with the iterative behaviour of quasimeromorphic mappings of transcendental type, which form higher-dimensional analogues of transcendental meromorphic functions on the complex plane. We extend classical Julia theory and results on escaping points from complex dynamics to the new setting. This complements recent dynamical advancements for quasiregular mappings, which are higher-dimensional analogues of holomorphic functions on the complex plane. First, we define the Julia set for quasimeromorphic mappings of transcendental type and investigate its properties through two cases based on the cardinality of the backward orbit of infinity. To this end, we construct an example of a quasiregular mapping in dimension 3 with exactly one zero, subsequently showing that both cases arise. We then generalise an important growth result by Bergweiler to quasiregular mappings defined near an essential singularity. From this we show that many classical properties of the Julia set hold in our case; this includes proving a cardinality conjecture that remains open for general quasiregular mappings. Next, we study the existence of escaping and non-escaping points in the new Julia set. In particular, following work by Nicks, we show that there exist points that escape arbitrarily slowly to infinity under iteration. Moreover we prove some basic relationships between the Julia set, the escaping set, the set of points whose orbit is bounded, and the set of points whose orbit is neither bounded nor tends to infinity. Finally, motivated by the work of Bolsch, we consider a class of mappings that is closed under composition and contains all quasimeromorphic mappings. Adapting previous methods, we show that the above results for quasimeromorphic mappings of transcendental type continue to hold for their iterates in a natural way. We also define a generalised escaping set, consisting of points whose orbits accumulate to some essential singularities or their pullbacks, and prove some existence results regarding points with specified accumulation sets.
... The rate at which the points of I(f ) escape to infinity also plays an important role. Bergweiler and Hinkkanen in [3] defined the fast escaping set, denoted by A(f ) as, ...
... Clearly A(f ) ⊂ I(f ). It was shown in [3] that A(f ) = ∅ and also that, ...
Let f be a transcendental entire function and let I(f) be the points which escape to infinity under iteration. Bergweiler and Hinkkanen introduced the fast escaping sets A(f) and subsequently, Rippon and Stallard introduced 'Levels' of fast escaping sets A L R (f). These sets under some restriction have the properties of "infinite spider's web" structure. Here we give some topological properties of the infinite spider's web and show some of the transcendental entire functions whose levels of the fast escaping sets have infinite spider's web structure.
... There are also points in I(f ) which move towards ∞ at a fast rate. In this regard Bergweiler and Hinkkanen [3] defined A(f ) which was also used in [11] as: ...
... Clearly A(f ) ⊂ I(f ), and is called fast escaping set. It was shown in [3] that for a transcendental entire function f, A(f ) = ∅ and ∂A(f ) = J(f ). In fact discovery of the set A(f ), gave new directions to transcendental dynamics and lead to solve various problems, like moving a step towards Eremenko's conjecture [4] (all the components of I(f ) are unbounded) by proving that I(f ) has at least one unbounded component namely A(f ). ...
Levels of fast escaping sets were discussed by Rippon and Stallard. Here we have defined a set B R ( f ) analogous to 0 th level of fast escaping set by using maximum term and formation of spider’s web structure has been discussed for this set.
... This conjecture 1.1 has been proved for the fast escaping set A(f ), which consists of points that tend to infinity as fast as possible under iteration. This set is a subset of I(S), and it was introduced first time by Bergweiler and Hinkkanen [3] and defined in the following form by Rippon and Stallard [15]. where M (r) = max |z|=r |f (z)|, r > 0 and M n (r) denotes nth iteration of M (r) with respect to r. R > 0 can be taken to be any number such that M (r) > r for r ≥ R. are respectively known as Speiser class and Eremenko-Lyubich class. ...
... The principal aim of this paper is to show how far classical transcendental dynamics applies to transcendental semigroup dynamics and what new phenomena appear in transcendental semigroup settings. Note that the fast escaping set A(f ) now plays a key role in classical transcendental dynamics which was introduced first time by Bergweiler and Hinkkanen [3] and studied in more depth by Rippon and Stallard [15]. The set A(f ) has some properties exactly similar to those of Theorem 2]) and ([15, Theorem 1.2]). ...
... This conjecture 1.1 has been proved for the fast escaping set A(f ), which consists of points that tend to infinity as fast as possible under iteration. This set is a subset of I(S), and it was introduced first time by Bergweiler and Hinkkanen [3] and defined in the following form by Rippon and Stallard [15]. where M (r) = max |z|=r |f (z)|, r > 0 and M n (r) denotes nth iteration of M (r) with respect to r. R > 0 can be taken to be any number such that M (r) > r for r ≥ R. are respectively known as Speiser class and Eremenko-Lyubich class. ...
... The principal aim of this paper is to show how far classical transcendental dynamics applies to transcendental semigroup dynamics and what new phenomena appear in transcendental semigroup settings. Note that the fast escaping set A(f ) now plays a key role in classical transcendental dynamics which was introduced first time by Bergweiler and Hinkkanen [3] and studied in more depth by Rippon and Stallard [15]. The set A(f ) has some properties exactly similar to those of Theorem 2]) and ([15, Theorem 1.2]). ...
... It has been shown in [10,11] that I(f ) = ∅ and the escaping set is strongly related to the Julia set, via J(f ) ∩ I(f ) = ∅ and J(f ) = ∂I(f ). Since then, properties of the escaping set have been extensively studied; see for example [13,29,30,32,33] The fast escaping set A(f ) ⊂ I(f ) was introduced by Bergweiler and Hinkkanen [6] for transcendental entire functions. Subsequently, it was asked whether all escaping points could be fast escaping. ...
... 6 Proof of Theorem 1.4 and counterexamples 6.1 Sufficient conditions for Theorem 1.4(i) Let f be a K-quasimeromorphic mapping of transcendental type with at least one pole. To prove Theorem 1.4(i), we shall provide sufficient conditions for the existence of infinitely many points in BO(f )∩J(f ) and BU(f )∩J(f ). ...
We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.
... The Eremenko's conjecture 1.1 in general case has been proved by using the fast escaping set A(f ), which consists of points whose iterates tends to infinity as fast as possible. This set is a subset of escaping set and it was introduced first time by Bergweiler and Hinkkanen [3] and defined in the following form by Rippon and Stallard [14]. For a transcendental entire function f , the fast escaping set is a set of the form: ...
... In this paper, we introduce fast escaping set in transcendental semigroup settings which is a main concern of our study. Note that the fast escaping set A(f ) in classical transcendental dynamics introduced first time by Bergweiler and Hinkkanen [3] and studied in more depth by Rippon and Stallard [14]. Indeed, it is a set consisting of points whose iterates tends to infinity as fast as possible and now plays a key role in classical transcendental dynamics. ...
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.
... A subset A(f ) of I(f ) called the fast escaping set was introduced in [8]. It also plays an important role in transcendental dynamics; see, e.g., [30,31]. ...
We give criteria for the escaping set and the Julia set of an entire function to have positive measure. The results are applied to Poincar\'e functions of semihyperbolic polynomials and to the Weierstra{\ss} -function.
... A key tool in the proof of the above results is the fast escaping set. This was first defined, for a transcendental entire function, in [11], and a detailed study of this set was given in [34]. See also [8,10], which studied the fast escaping set of a quasiregular map of R d of transcendental type. ...
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.
... Our final result is the following, which gives a simple topological characterisation of an I(f ) spider's web, and also an A(f ) spider's web, for a transcendental entire function f . Here A(f ) is the so-called fast escaping set, which was introduced in [BH99], and can be defined, see [RS12], by; ...
Suppose that f is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider's web. We use our results to give a large class of functions in the Eremenko-Lyubich class for which the escaping set is not a spider's web. Finally we give a novel topological criterion for certain sets to be a spider's web.
... A priori, the set J s (F ) may be empty for some external addresses in s ∈ Σ. Rippon and Stallard showed that, for a general transcendental entire function f , the components of the fast escaping set A(f ) ⊆ I(f ), which was previously introduced by Bergweiler and Hinkkanen [BH99], are all unbounded [RS05b]. Using similar ideas, Rempe showed that if f ∈ B (and the same argument follows for class B log ), then every tract T contains an unbounded closed connected set A consisting of points that escape within T [Rem08, Theorem 2.4]. ...
We study the escaping set of functions in the class , that is, holomorphic functions for which both zero and infinity are essential singularities, and the set of singular values of f is contained in a compact annulus of . For functions in the class , escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of (and hence, in the class ), then we show that every escaping point of f can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary , we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to according to e under iteration by f.
... In fact, we prove the following more general result. We prove Corollary 4.5 by using various properties of the subset of I(f ) known as the fast escaping set A(f ), which was introduced in [6]. In particular, we use the facts that J(f ) = ∂A(f ), that all components of A(f ) are unbounded, and that all components of A(f ) ∩ J(f ) are unbounded whenever f has no multiply connected Fatou components. ...
We investigate the connectedness properties of the set of points where the iterates of an entire function f are unbounded. In particular, we show that is connected whenever iterates of the minimum modulus of f tend to infinity. For a general transcendental entire function f , we show that is always connected and that, if is disconnected, then it has uncountably many components, infinitely many of which are unbounded.
... At the same time, Bergweiler and Hinkkanen [16] introduced the so-called fast escaping set ...
... The proofs in [8,16] actually show that not only the escaping set I(f ) is nonempty, but that this in fact holds for a subset A(f ) of I(f ) called the fast escaping set. This set was first considered in [10]. ...
It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of can take any value in the interval [1,3]. The Hausdorff dimension of the Julia set of such a map is estimated under some growth condition.
... The fast escaping set A( f ) was introduced by Bergweiler and Hinkkanen [4], and was studied further by Rippon and Stallard [12]. It is defined as follows. ...
We prove that the meandering set for fa(z)=ez+a is homeomorphic to the space of irrational numbers whenever a belongs to the Fatou set of fa. This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of fa has topological dimension zero for all attracting and parabolic parameters, including all a∈(-∞,-1]. Similar results are obtained for Fatou’s function f(z)=z+1+e-z.
... Since then several other examples of wandering domain were given by many authors, for instance see, [2], [8] and it's references. If f and g are two transcendental entire functions then their composition is also a transcendental entire function, also dynamics of composition of these function is related to each other, [3], [4], [7]. Infact dynamics of fog and gof are very much similar, but this can not be expected for individual functions. ...
Dynamics of composition of entire functions is well related to it's factors, as it is known that for entire functions f and g , fog has wandering domain if and only if gof has wandering domain. However the Fatou components may have different structures and properties. In this paper we have shown the existence of domains with all possibilities of wandering and periodic in given angular region θ .
... is the fast escaping set of f , introduced by Bergweiler and Hinkkanen in [BH99] and then studied by Rippon and Stallard in [RS12]. Here R > 0 is a large fixed number, M f (r) = max |z|=r |f (z)| for r > 0 and M n f denotes the n-th iterate of M f (·). ...
We determine the Hausdorff and packing dimension of sets of points which escape to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.
... Definitions of A(f ) and J m (f ). The fast escaping set A(f ) was introduced by Bergweiler and Hinkkanen [4], and was studied further by Rippon and Stallard [12]. It is defined as follows. ...
For each let be the complex exponential mapping . We show the radial Julia set has topological dimension zero for all attracting and parabolic parameters, including all . Moreover, we prove that the meandering Julia set is homeomorphic to the irrationals, extending recent results by Vasiliki Evdoridou and Lasse Rempe-Gillen. This has several consequences for the topologies of the escaping and fast escaping sets and their endpoints.
... The fast escaping set of a transcendental entire function is introduced in [5]. It is natural to ask whether there exists a fast escaping point whose orbit go along singular directions. ...
This paper consists of tow parts. One is to study the existence of a point a in the intersection of Julia set and escaping set such that is a singular direction if is a limit point of under some growth condition of a meromorphic function. The other is to study the connection between the Fatou set and singular direction. We prove that the absent of singular direction deduces the non-existence of annuli in the Fatou set.
... In transcendental dynamics, the rates of escape for the escaping set is very useful. The fast escaping set [5] is given by ...
A generalized family of non-polynomial entire functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to 1, there exist multiply connected wandering domains, and the dynamics can be completed described. For any , there is a function taken from this family with the order of growth s. This is the first example, where the Julia set has packing dimension 1, and arbitrarily positive or even infinite order of growth. This gives an answer to an open problem in [Problem 4,J.Bishop, A transcendental Julia set of dimension 1, Invent. Math., 212 (2018) 407--460].
... The fast escaping set A(f ) is a subset of the escaping set. It was introduced by Bergweiler and Hinkkanen [5], and is defined by A(f ) := {z : there exists l ∈ N such that |f n (z)| ≥ M n−l (R, f ) for n > l}, where R is chosen such that M (r, f ) > r for r ≥ R. ...
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f have finite Lebesgue measure. Essentially, these conditions are designed such that for some and all z outside a set of finite Lebesgue measure.
... If f and g are two transcendental entire functions, then their composition is also a transcendental entire function and this composition contains infinitely many fixed points [8]. Also the dynamics of composition of these functions is related to each other [1,9,10]. In fact, dynamics of f g and g f are very much similar, but this cannot be expected for individual functions. ...
In this paper, we have shown the existence of wandering domains as well as periodic domains with all possible combination under composition of transcendental entire functions in angular region using approximation theory.
... Many of the major results on the structure of I(f ) have been obtained by studying the fast escaping set A(f ), introduced in [3], which can be defined as follows; see [20]. First put (1.1) ...
We show that for many families of transcendental entire functions f the property that as , for some , where , implies that the escaping set I(f) of f has the structure of a spider's web. In particular, in this situation I(f) is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.
... This was shown by considering the fast escaping set A(f ). This set was first introduced by Bergweiler and Hinkkanen in [4] and can be defined by A(f ) = {z : there exists L ∈ N such that |f n+L (z)| ≥ M n (r) for n ∈ N} as in [16], where M (r) = max |z|=r |f (z)| for r > 0 ...
Let f be a transcendental entire function. By a result of Rippon and Stallard, there exist points whose orbit escapes arbitrarily slowly. By using a range of techniques to prove new covering results, we extend their theorem to prove the existence of points which escape arbitrarily slowly within logarithmic tracts and tracts with certain boundary properties. We then give examples to illustrate our results in a variety of tracts.
... To prove (4), suppose that f ′ a (z) = 0, f ′′ a (z) = 0, then ...
Let f be a transcendental entire function with at least one critical point and let be an entire function such that (i) as (ii) has at least one zero, (iii) has no common zeros with In this paper we show that if g is any transcendental entire function permutable with then g and have identical Julia sets as long as a is outside some denumerable subset of ; this answers Baker's question for almost all nonlinear entire functions.
It is shown that the Hausdorff dimension of the fast escaping set of a quasiregular self-map of ℝ3 can take any value in the interval [1, 3]. The Hausdorff dimension of the Julia set of such a map is estimated under some growth condition.
We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that for some and all z outside a set of finite Lebesgue measure.
We consider the class of real transcendental entire functions f of finite order with only real zeros and show that if the iterated minimum modulus tends to , then the escaping set of f has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function f.
Let f f be a transcendental entire function and let I ( f ) I(f) be the set of points whose iterates under f f tend to infinity. We show that I ( f ) I(f) has at least one unbounded component. In the case that f f has a Baker wandering domain, we show that I ( f ) I(f) is a connected unbounded set.
Let f be a transcendental entire function. For let denote the iterate of f. Let as and Then the set denoted by BU(f) is called Bungee set of f. In this paper we give an alternate definition for BU(f) which is very easy to work with, and we illustrate it by proving some properties of Bungee sets of composite transcendental entire functions and also of Bungee sets of permutable transcendental entire functions.
We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.
The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.
Applications de type polynomial. Familles analytiques de telles applications. Resultats negatifs. Familles a un parametre d'applications de degre 2. Petites copies de M dans M. Carrottes pour le dessert
Fonctions entieres a domaines multiplement connexes de normalite. Ensembles de Fatou-Julia pour des fonctions qui commuttent. Exemples de type d'Herman. Classes de fonctions entieres sans domaines d'errance
Let f be a nonconstant and nonlinear entire function, g an analytic self-map of ℂ∖{0}, and suppose that exp∘f=g∘exp. It is shown that z is in the Julia set of f if and only if e z is in the Julia set of g.
Suppose that f and g are transcendental entire
functions of finite order, without
wandering domains, and suppose that f[circ B: composite function (small circle)]g=g[circ B: composite function (small circle)]f.
Then the Julia sets of f and g coincide.
If f is a transcendental entire function and D is a non-wandering component of the set of normality of the iterates of f such that fn → ∞ in D then log |fn(z)| = O(n) as n → ∞ for z in D. For a wandering component the convergence of fn to ∞ in D may be arbitrarily fast.
In the theory of the iteration of a rational function or transcendental entire function R ( z ) of the complex variable z we study the sequence of natural iterates, { R n ( z ): n = 0, 1,…}, of R , where
The domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F -set, F ( R ), of R , R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.
Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pnP(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.
Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains.
This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains). Comment: 38 pages. Abstract added in migration. See http://analysis.math.uni-kiel.de/bergweiler/ for recent comments and corrections
- L Carleson
- T W Gamelin
L. Carleson and T. W. Gamelin. Complex dynamics (Springer, 1993).
Dynamics in one complex variable: introductory lectures Stony Brook Institute for Mathematical Sciences On an integral function of an integral function
- J Milnor
J. Milnor. Dynamics in one complex variable: introductory lectures. Stony Brook Institute for
Mathematical Sciences, Preprint 1990/5.
[22] G. P ´
olya. On an integral function of an integral function. J. London Math. Soc. 1 (1926),
12–15.
Rational iteration (de Gruyter
- N Steinmetz
N. Steinmetz. Rational iteration (de Gruyter, 1993).