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Dynamical properties of some classes of entire functions
Abstract
The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.
... Our main aim in this paper is to study the local connectivity of the Julia sets of some transcendental entire functions containing bounded type Siegel disks. Let B := {f is transcendental entire : S(f ) is bounded} be the Eremenko-Lyubich class (compare [EL92]). To obtain the local connectivity of the Julia set of f containing Siegel disks, we require that the dynamics of f on P(f ) is tame. ...
... By the finiteness of CV (f ) ∩ J(f ), it follows that f has only finitely many Siegel disks and they are all of bounded type. By Hence h has only finitely many singular values in C. According to [GK86] or [EL92], h has no wandering domains and hence f is also. □ ...
... It is well known that any limit function of {f •n | U } is a constant. Since f ∈ B, we cannot have f •n | U → ∞ [EL92]. By the finiteness of CV (f ) ∩ J(f ), there exists a subsequence {f •n k | U } whose limit is a finite point a, which must lie on the boundary of some bounded type Siegel disk ∆. ...
Based on the weak expansion property of a long iteration of a family of quasi-Blaschke products near the unit circle established recently, we prove that the Julia sets of a number of transcendental entire functions with bounded type Siegel disks are locally connected. In particular, if is of bounded type, then the Julia set of the sine function is locally connected. Moreover, we prove the existence of transcendental entire functions having Siegel disks and locally connected Julia sets with asymptotic values.
... First, in Sect. 1.1, we use Mitra's [44] machinery of describing holomorphic motions of a closed set to show that equivalence classes of a wide variety of entire maps are complex manifolds of possibly infinite dimension, generalising results of Eremenko and Lyubich [21]. Second, in Sect. ...
... [20,50,51]. In particular, it was shown by Eremenko and Lyubich [21,Section 2] (see also [28,Theorem 3.1]) that if f has finitely many singular values (critical values, asymptotic values, and accumulation points thereof) then M f is a complex manifold of dimension q + 2, where q is the number of singular values. In particular, M f itself can be considered a natural family. ...
... Let us assume now that S( f ) is a proper subset of C; the proof is essentially the same as that of [21,Lemma 2] with minor modifications. More specifically, let ...
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.
... Perhaps because wandering domains do not exist for rational maps [Sul85], nor for maps in the Speiser class S [EL92,GK86], these rare Fatou components have not been subject of attention until quite recently, when maps with infinitely many singular values (like Newton's method applied to entire functions) have started to emerge as interesting objects. Nevertheless, many recent breakthrough results about wandering domains have appeared in the last several years. ...
... Nevertheless, many recent breakthrough results about wandering domains have appeared in the last several years. After the classical result [EL92] which states that maps in class B cannot have wandering domains whose orbits converge to infinity uniformly (called escaping wandering domains), there was reasonable doubt of whether functions in class B could have wandering domains at all. This question was answered affirmatively by Bishop in [Bis15] who constructed a function in class B with an oscillating wandering domain, that is, a wandering domain whose orbits accumulate both at infinity and on a compact set (the collection of points that oscillate as such under f is termed the Bungee set BU (f ) -see [OS16] for details and properties of this set). ...
... We call this R-component V . Since R is in the escaping set of f (see [Laz17]), and f ∈ B, it follows that R ⊂ J (f ) (see [EL92]). Thus, f n (U ) ⊂ H := {z ∈ C : Im(z) > 0} for all n ∈ N. Denote by d H , d f n (U ) , d U the hyperbolic distance in H, f n (U ), U respectively. ...
We use the folding theorem of Bishop to construct an entire function f in class B and a wandering domain U of f such that f restricted to is univalent, for all . The components of the wandering orbit are bounded and surrounded by the postcritical set.
... The singular set S(f ) of an entire function f : C → C is the closure of the set of critical values and finite asymptotic values of f . Eremenko-Lyubich [15] introduced and studied class B consisting of all entire functions with bounded singular sets. It has as a subclass Speiser class S consisting of entire functions with finite singular sets. ...
... Let us consider here in this introduction the case where f −1 ({|z| > 1}) consists of only one tract Ω. Then, f |Ω has the particular form f = e τ , with τ a conformal map from Ω onto the half plane H = { z > 0} such that (1.1) ϕ := τ −1 : H → Ω extends continuously to infinity [15]. Although ∂Ω is an analytic curve, near infinity it often resembles more and more a fractal curve. ...
... This has a second important application: for entire functions of class S the Hölder tract property is in fact a property of an analytic family of functions and not only of a single function. More precisely, if g ∈ S then Eremenko-Lyubich [15] naturally associated to g an analytic family of entire functions M g ⊂ S. Proposition 10.1 states that every function of M g has Hölder tracts if a function, for example g, has. ...
We provide the full theory of thermodynamic formalism for a very general collection of entire functions in class . This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quasicircle, John or H\"older tracts. In this case we get a good control of the corresponding transfer operators, leading to full thermodynamic formalism along with its applications such as exponential decay of correlations, central limit theorem and a Bowen's formula for the Hausdorff dimension of radial Julia sets. Our approach applies in particular to every hyperbolic function from any Eremenko-Lyubich analytic family of Speiser class provided this family contains at least one function with H\"older tracts. The latter is, for example, the case if the family contains a Poincar\'e linearizer.
... In this section we present some known results about Speiser functions. In particular, the construction of the complex structure on the parameter space M f of f ∈ S is reviewed and restated briefly, following [EL92]. A characterization of the Misiurewicz condition is also presented in terms of non-recurrence of singular values. ...
... Then by definition P H ( f ) is compact, forward invariant, and contains neither critical points nor parabolic periodic points. Moreover, f ∈ S implies that it does not have wandering domains by [EL92] (see also [GK86] ...
... is the corresponding singular value of f a to s j (0). By Eremenko and Lyubich's construction of parameter spaces for Speiser functions [EL92], s j (a) is holomorphic in a for all j. We shall thus consider the evolution of the following sequence of maps ...
We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.
... In this section we present some known results about Speiser functions. In particular, the construction of the complex structure on the parameter space M f of f ∈ S is reviewed and restated briefly, following [EL92]. A characterization of the Misiurewicz condition is also presented in terms of non-recurrence of singular values. ...
... Then by definition P H (f ) is compact, forward invariant, and contains neither critical points nor parabolic periodic points. Moreover, f ∈ S implies that it does not have wandering domains by [EL92] (see also [GK86]). By [RvS11, Theorem 1.2] we can see that P H (f ) is a hyperbolic set. ...
... For an entire function f the escaping set I(f ) is the set of points tending to ∞ under iterates of f . The following result is due to Eremenko and Lyubich [EL92]. Theorem 6.1. ...
We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.
... In this note, we investigate the fundamental structure of the space of complex exponential maps z → E κ (z) = e z + κ with κ ∈ C. Translation by −κ conjugates such a map to e z+κ = λe z with λ = e κ . The space of complex exponential maps has been investigated since the mid-1980's by Baker and Rippon [BR], Eremenko and Lyubich [EL1,EL2,EL3], Devaney, Goldberg and Hubbard [DGH], and others. ...
... A hyperbolic component of period n is a maximal open set W ⊂ C such that for κ ∈ W , the map E κ has an attracting periodic orbit of period n; all other periodic orbits are then necessarily repelling. It is known from [EL3,BR,DGH] that every hyperbolic component is simply connected, and it comes with a holomorphic multiplier map µ: W → D * such that the attracting orbit of E κ has multiplier µ(κ). The map µ is a universal covering map. ...
We discuss the space of complex exponential maps \Ek\colon z\mapsto e^{z}+\kappa. We prove that every hyperbolic component W has connected boundary, and there is a conformal isomorphism \Phi_W\colon W\to\half^- which extends to a homeomorphism of pairs \Phi_W\colon(\ovl W,W)\to(\ovl\half^-,\half^-). This solves a conjecture of Baker and Rippon, and of Eremenko and Lyubich, in the affirmative. We also prove a second conjecture of Eremenko and Lyubich.
... The set of finite singular values, denoted by S(f ), is the closure of the set of all critical and asymptotic values of f ; equivalently, it is the smallest closed set S with the property that f : C \ f −1 (S) → C \ S is a covering map. The well-studied Eremenko-Lyubich class B consists of those transcendental entire functions f such that S(f ) is bounded (see [7]). A transcendental entire function in the class B is hyperbolic if every singular value lies in the basin of an attracting periodic cycle; see [17] for this and other, equivalent, definitions. ...
... In this section we give a number of important definitions, together with several useful results on logarithmic transforms of functions of disjoint type. Many of the ideas in this section are well-known, and we refer to, for example, [7,15,22] and the survey [23] for more detail. ...
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.
... The next question one may ask is the uniqueness of the true form. This follows from the fact that two entire functions with only two singular values which are topologically equivalent in the sense of Eremenko and Lyubich [EL92] are in fact conformally equivalent [ERG15, Proposition 2.3]. Thus we see that the true form is unique up to affine maps in the plane. ...
... Remark 2.1. Transcendental entire functions in class S attract a lot of interest in transcendental dynamics since the work of Eremenko and Lyubich [EL92], mainly due to the fact that these functions have certain similarities to polynomials from a dynamical point of view. ...
Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function f with only two critical values and no asymptotic values such that is ambiently homeomorphic to the given tree. This can be viewed as a generalization of a result of Grothendieck to the case of infinite trees. Moreover, a similar idea leads to a new proof of a result of Nevanlinna and Elfving.
... In our context, we emphasize that a projectable function f (z ) has, in general, infinitely many poles and singular values accumulating at ∞, that is, f lies outside of the so-called Eremenko-Lyubich class B [28], as S(f ) ∩ C is not bounded. This is always the case if f is not 1-periodic (i.e. ...
... #S(g) < ∞), since g has no wandering components nor Baker domains (see e.g. [7,28]). The correspondence between critical and asymptotic values of f and g (remark 3.7 and proposition 3.8; see also proposition 3.6 and figure 2) leads to the following result. ...
We present a one-parameter family F λ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of F λ . Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f ( z ) in the complex plane that are semiconjugate, via the exponential, to some map g ( w ), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g , and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and . The family F λ is the simplest in this class, and its parameter space shows open sets of λ -values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.
... In our context, we emphasize that a projectable function f (z) has, in general, infinitely many poles and singular values accumulating at ∞, that is, f lies outside of the so-called Eremenko-Lyubich class B [28], as S(f ) ∩ C is not bounded. This is always the case if f is not 1-periodic, due to pseudoperiodicty: f (z + k) = f (z) + ℓk for all k ∈ Z. ...
... #S(g) < ∞), since g has no wandering components nor Baker domains (see e.g. [7,28]). The correspondence between critical and asymptotic values of f and g (Remark 3.9 and Proposition 3.10; see also Proposition 3.7 and Figure 2) leads to the following result. ...
We present a one-parameter family of transcendental entire functions with zeros, whose Newton's method yields wandering domains, coexisting with the basins of the roots of . Wandering domains for Newton maps of zero-free functions have been built before by, e.g., Buff and R\"uckert based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton's method projects to some map g which is defined at both 0 and . The family is the simplest in this class, and its parameter space shows open sets of -values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton's root-finding method fails.
... where E denotes the topological vector space of entire functions (armed with the compactopen topology) and QC(C, C) denotes the space of quasiconformal homeomorphisms of C. It is known that, if f has finitely many singular values (critical values, asymptotic values, and accumulation points thereof), then M f is a complex manifold of dimension q + 2, where q is the number of singular values (see, for instance, [14,Section 2] and [20, Theorem 3.1]). Our last theorem extends this result to functions with infinitely many singular values, as long as the latter remain discrete. ...
... Proof of Claim 5.1. We recall the linearinsing coordinates ψ : ∆ → D L of g (see page 14). From the construction, the dependence of the Beltrami coefficientsμ λ,n−1 = µ λ,n−1 + τ * µ λ,n−1 on λ n has two factors: one inside the unit disc given by µ λ,n−1 = ψ * ν n−1 , and another outside the unit disc given by τ * µ λ,n−1 = τ * ψ * ν n−1 . ...
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).
... We focus on a precise type of periodic Fatou components, Baker domains, in which iterates converge locally uniformly to infinity. Maps possessing Baker domains are not hyperbolic, nor bounded type (i.e. the set of singularities of the inverse branches of the function is unbounded [23]). In contrast with the other periodic Fatou components, in which the dynamics around the convergence point can be conjugate to some predetermined normal form, three different asymptotics are possible for Baker domains (see Theorem 2.4 and Remark 2.11). ...
... Therefore, the Fatou set F (h) is precisely A 0 and its preimages under h. Indeed, since h has only a finite number of singular values, it cannot have Baker nor wandering domains [23,Sect. 5], and the presence of any other invariant Fatou component (either a basin or a Siegel disk) would require an additional singular value (see e.g. [10,Thm. ...
We consider the transcendental entire function f ( z ) = z + e - z , which has a doubly parabolic Baker domain U of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infinity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in ∂ U whose orbit escapes to infinity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, and it is organized in curves encoded by some symbolic dynamics, whose closure is precisely ∂ U . We also prove that nevertheless, all escaping points in ∂ U are non-accessible from U , as opposed to points in ∂ U having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in ∂ U , answering a question posted in (Barański et al. in J Anal Math 137:679–706, 2019). None of these features are known to occur for a general doubly parabolic Baker domain.
... as J( f ) = ∂I( f ) and, in the cases we will consider, J( f ) = I( f ); [9]. ...
... Moreover, C \ K ∩ P( f ) = ∅ and each connected component of C \ K is simply-connected, since otherwise K would enclose a domain that escapes uniformly to infinity, contradicting that int(I( f )) = ∅ as f ∈ B, [9]. Thus, since f is an open map, all connected components of f −1 (C \ K) are simply-connected, which we label as U 0 , U 1 , . . ...
The set of points that escape to infinity under iteration of a cosine map, that is, of the form for , consists of a collection of injective curves, called dynamic rays . If a critical value of escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map . We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of , and conclude that no two of its dynamic rays land together.
... Moreover, for the exponential family from [42] it is true that I(f ) ⊂ J (f ) and ...
... The bungee set can be also dened for quasiregular maps (see [101]). So we ask It is also known that the escaping set of the exponential map E κ , for any value of the parameter κ ∈ C \ {0}, has zero Lebesgue measure (see [42,Theorem 7]). Question 4.12.5. ...
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.
... A holomorphic family of maps f λ : W λ → P 1 is natural [4,37] ...
We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another consequence, we also prove that there exist small perturbations of semi-parabolic Hénon maps for which the Hausdorff dimension of the forward Julia set is arbitrarily close to 4.
... The logarithmic coordinates are well suited for orbits staying "near ∞", in particular, for the escaping points. The most important feature of these coordinates is that for big enough R, F is expanding, and in a quite strong way [EL,Lemma 1]: ...
Given an entire function with finitely many singular values, one can construct a quasiregular function f by post-composing with a quasiconformal map equal to identity on some open set . It might happen that the f-orbits of all singular values of f are eventually contained in U. The goal of this article is to investigate properties of Thurston's pull-back map associated to such f, especially in the case when f is post-singularly infinite, that is, when acts on an infinite-dimensional Teichm\"uller space . The main result yields sufficient conditions for existence of a -invariant set such that its projection to the subspace of associated to marked points in is bounded in the Teichm\"uller metric, while the projection to the subspace associated to the marked points in U (generally there are infinitely many) is a small perturbation of identity. The notion of a ``fat spider'' is defined and used as a dynamically meaningful way define coordinates in the Teichm\"uller space. The notion of ``asymptotic area property'' for entire functions is introduced. Roughly, it requires that the complement of logarithmic tracts in U degenerates fast as U shrinks. A corollary of the main result is that for a finite order entire function, if the degeneration is fast enough and singular values of f escape fast, then f is Thurston equivalent to an entire function.
... As we have seen, f has two singularly values, 0 and 1, both of which are critical values. By [3] and [5], f has neither Baker domains nor wandering domains. Since entire functions have no Herman rings, every Fatou component of f must be eventually mapped into some periodic cycle of either Siegel disks, or parabolic components, or attracting components, or supper-attracting components. ...
We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-circles but the Julia set is non-locally connected.
... In [19] the class of functions f holomorphic and non-constant in a general hyperbolic domain, with the property that CV (f ) ∪ AV (f ) is bounded, was discussed and called the Eremenko-Lyubich class. Transcendental entire functions with this property have been very widely studied, particularly in complex dynamics; see, for example, [8,18,20,21]. Here we denote the Eremenko-Lyubich class in D by B D to avoid confusion with MacLane's class B, which we use later in the paper. ...
In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class .
... Note that the above same type of relation (Theorem 2.1) holds between F (S) and Proof. Eremenko and Lyubich's result [3] shows that I(f ) ⊂ J(f ) for each f ∈ S of bounded type. Poon's result shows [ ...
In this paper, we mainly study hyperbolic semigroups from which we get non-empty escaping set and Eremenko's conjecture remains valid. We prove that if each generator of bounded type transcendental semigroup S is hyperbolic, then the semigroup is itself hyperbolic and all components of I(S) are unbounded
... Now we introduce a key tool given by [EL92], the so-called logarithmic change of variable, to study functions in class B. We have the following result. Remark 2.2. ...
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.
... The proof will proceed by contradiction, so we suppose that such a measure exists and call it µ, while reserving λ for the Lebesgue measure of the plane. It follows from [4] that the set of points escaping to ∞ has zero Lebesgue's measure for every map in our family. It is not difficult to prove that for these functions the union P (f ) ∪ {∞} is not a metric attractor in sense of Milnor with respect to the measure λ on C. The results of [1] implies that f Λ is ergodic with respect to λ. ...
For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane.
... The escaping set I ( f ) = {z ∈ C : f n (z) → ∞} is one of the most studied objects in complex dynamics [6,7,12,14]. For any entire function f , it is easily seen to be an F σ δ -subset of the complex plane C. ...
We determine the exact Borel class of escaping sets in the exponential family . We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.
... (1) Proofs of nonexistence of wandering Fatou components for various classes of non-rational maps can be found in[19,20,22,17]. 4 e SÉRIE -TOME 56 -2023 -N o 6 ...
... The class of equivalence of a function f , denoted by [f ], is called the Hurwitz class of f or also it is called the full family of f . In [Eremenko & Lyubich, 1992], the authors show that the critical values and asymptotic values of f parameterize [f ] up to the action of Möbius transformations. Since ϕ and φ are in particular homeomorphisms, the number and the type of the critical values of f must be preserved. ...
For the class 𝒦 of meromorphic functions outside a compact countable set of essential singularities, we study the dynamics of some functions in 𝒦 for which the unit disc and its complement are invariant. These functions are products and compositions of the function E1 =exp(z−1 z+1 ) with Blaschke products, that include the family En =exp zn−1 zn+1, which are the main topic of the article. Slices of the space of parameters of En are given with a discussion of their main features. We construct a Poincaré extension of En to the hyperbolic three-dimensional space and to the 3-sphere. Also we study their dynamics.
... That the space of rational maps is finite dimensional is crucial: shortly after the preprint version of [106] appeared, Baker constructed entire functions f : C → C which do have wandering domains. Sullivan's no wandering theorem has also been extended to the setting of entire maps (and similar spaces) with a finite number of singular values, see for example [37] and [43]. A very elegant proof of the no wandering domains theorem due to McMullen -which circumvents the use of the MRMT and uses an infinitesimal deformations argument more in line with Ahlfors' original proof of his finiteness theorem -can be found in [78, p. 90]. ...
In this expository paper, we provide the readers with an overview of Dennis Sullivan's major contributions to the area of Dynamical Systems.
... If g P B, then there exists r ą 0 such that Spgq Ă D r . Then g´1pDr q consists of mutually disjoint unbounded Jordan domains Ω r with real analytic boundaries such that g : Ω Ñ Dr is a covering map (see [8]). Thus, an entire function g in class B has only logarithmic singularities over infinity. ...
Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions f having Hausdorff dimension of the Julia set \HD (\J _f)<2 and hyperbolic dimension \HypDim(f)<\HD(\J_f).
... The escaping set I(f ) = {z ∈ C : f n (z) → ∞} is one of the most studied objects in complex dynamics [6,7,14,12]. For any entire function f , it is easily seen to be an F σδ -subset of the complex plane C. ...
We determine the exact Borel class of the points whose iterates under tend to infinity. We also show that the sets of non-escaping Julia points for many of these functions are topologically equivalent.
... A commonly used tool for studying functions in class B is the logarithmic change of coordinates. This technique was firstly used in this context in [EL92,§2]; for more details, see [Six18,§5], [RRRS11,§2] or [Rem16,§3]. ...
A hyperbolic transcendental entire function with connected Fatou set is said to be of disjoint type. The Julia set of a disjoint-type function of finite order forms a Cantor bouquet; in particular, it is a collection of arcs, each connecting a finite endpoint to infinity. It is a natural question whether the latter property already implies that the Julia set is a Cantor bouquet. We give a negative answer to this question. We also provide a new characterisation of Cantor bouquet Julia sets in terms of the certain absorbing sets for the set of escaping points, and use this to give a new intrinsic description of a class of entire functions previously introduced by the first author. Finally, the main known sufficient condition for Cantor bouquet Julia sets is the so-called head-start condition of Rottenfusser et al. Under a mild geometric condition, we prove that this condition is also necessary.
... Error functions belong to the larger Speiser class-the family of entire functions with finitely many critical and asymptotic values. This family is studied in [GK86], [EL92] and several others. The simplest of the Speiser class is the family of exponential functions. ...
In this article, for degree , we construct an embedding of the connectedness locus of the polynomials into the connectedness locus of degree 2d+1 bicritical odd polynomials.
In the theory of dynamical systems, attractors represent states toward which a system tends to evolve. The basins of attraction are the regions of initial conditions leading to a particular attractor. In this paper, a single parameter family of even meromorphic functions involving cosine function for , is considered. The dynamics of functions is investigated. Also, it is shown that there exists such that the Fatou set (or stable set) of is the basin of attraction of an attracting fixed point and the Julia set (or the set of chaotic points) of is disconnected for .
Quasiconformal maps provide a powerful tool for complex analysis and complicated dynamics studies. In this article, the iterations of quasiconformal maps are investigated, showing that the iterative dynamics can be very complicated. One kind of quasiconformal map in the sense of Beltrami has a global attractor and a growing horseshoe, where the latter means that the number of folds of the horseshoe is increasing as a parameter is varied. Further, the generalized Hénon maps are represented as generalized quasiconformal maps in certain parameter regions. It is shown that this class of generalized quasiconformal maps is different from the maps in the sense of Beltrami or of Beltrami-David. Moreover, a natural class of invariant measures are constructed for the generalized Hénon maps. Based on the characteristics of the quasiconformal maps, some new properties of the generalized Hénon maps are revealed and analyzed, which are related to the existence of the wandering domains.
We give an elementary proof of the fact that the Julia set of e z − 1 is an uncountable family of mutually separated curves (a Cantor bouquet). Our proof reveals many interesting aspects of the function's dynamics, such as the existence of cyclic orbits of all periods.
Let be a transcendental entire function of finite order which has an attracting periodic point of period at least 2. Suppose that the set of singularities of the inverse of is finite and contained in the component of the Fatou set that contains . Under an additional hypothesis, we show that the intersection of with the escaping set of has Hausdorff dimension 1. The additional hypothesis is satisfied for example if has the form with polynomials and and a constant . This generalizes a result of Barański, Karpińska, and Zdunik dealing with the case .
We give the first example of a non-invariant cycle of Baker domains of infinite connectivity for non-entire meromorphic functions. We also prove the necessary and sufficient condition for a cycle of Baker domains to be infinitely connected in terms of critical points for the family , where and are defined in the paper.
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.
Iteration of the function is investigated in this article for . It is proved that for every , the Fatou set of has a completely invariant Baker domain ; we call it the primary Fatou component. The rest of the article deals with when it is topologically hyperbolic. For all real or such that for some integer k and , the Fatou set of is the union of and another completely invariant Baker domain. It is proved that if , then the Fatou set is the union of and infinitely many invariant attracting domains (along with their pre-images). Each such attracting domain U has exactly one invariant access to infinity and is unbounded in a special way; is unbounded whereas for every , is bounded. If then it is found that the primary Fatou component is the only Fatou component and the Julia set is disconnected. For every natural number k, there exists a complex number namely, such that the Fatou set of has k many wandering domains with distinct grand orbits. These wandering domains are found to be escaping. The Fatou set of is the union of and the grand orbits of these k wandering domains.
In this article, for degree d ≥ 1 d\geq 1 , we construct an embedding Φ d \Phi _d of the connectedness locus M d + 1 \mathcal {M}_{d+1} of the polynomials z d + 1 + c z^{d+1}+c into the connectedness locus of degree 2 d + 1 2d+1 bicritical odd polynomials.
In this paper, we obtained a four terms asymptotic formula of the Hausdorff dimension of the radial Julia set Jr(fλ) for the exponential function fλ(z) = λez as λ → 0. Moreover, we also discussed the asymptotic behavior of the Hausdorff dimension of the radial Julia set of cosine functions fa,b(z) = aez + be−z as a, b → 0 in various ways.
We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.
We study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.
Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions f f having Hausdorff dimension of the Julia set HD ( J f ) > 2 \operatorname {HD} (\mathcal {J}_f)>2 and hyperbolic dimension H y p D i m ( f ) > H D ( J f ) \mathrm {HypDim}(f)>\mathrm {HD}(\mathcal {J}_f) .
Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko–Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.
We explore geometric properties of the Mandelbrot set , and the corresponding Julia sets , near the main cardioid. Namely, we establish that: (a) is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; (b) The Julia sets are also locally connected and have positive area; (c) is self-similar near Siegel parameters of periodic type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed by the authors jointly with N. Selinger in [DLS] as a global transcendental family. It is the first occasion when external rays and puzzles of limiting transcendental maps are applied to study the Polynomial dynamics.
The Speiser class S S is the set of all entire functions with finitely many singular values. Let S q ⊂ S S_q\subset S be the set of all transcendental entire functions with exactly q q distinct singular values. The Fatou-Shishikura inequality for f ∈ S q f\in S_q gives an upper bound q q of the sum of the numbers of its Cremer cycles and its cycles of immediate attractive basins, parabolic basins, and Siegel disks. In this paper, we show that the inequality for f ∈ S q f\in S_q is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some T ∈ S q T\in S_q realizes it. In our construction, T T is a structurally finite transcendental entire function.
In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if f is of finite order, has bounded criticality on its Julia set J(f), and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then J(f) is a collection of dynamic rays or hairs, which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of f on its Julia set.
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