Baker Domains Research Articles

Lower order and limiting directions of Julia sets of solutions to second order differential equations

In this paper, we consider the properties of entire solutions to second order differential equation(⁎)f″+Af′+Bf=0, where A(z) and B(z)≢0 are entire functions. Under certain assumptions on A(z) and B(z), we prove that every non-trivial solution f of equation (*) is of infinite lower order, and then obtain the measure estimation of the limiting directions of Julia sets for those infinite lower order entire solutions. The existence of Baker domain for f(n) is also discussed.

A model for boundary dynamics of Baker domains

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 partial 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 partial U . We also prove that nevertheless, all escaping points in partial U are non-accessible from U, as opposed to points in partial U having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in partial 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.

Baker domains and non-convergent deformations

For an entire transcendental function f with a non-completely invariant Baker domain U , we study the pinching process of paths in U with certain restrictions, that we call Baker laminations. We show that if some curve in the Baker lamination of f joins a point in the boundary of U with infinity, then the deformation does not converge. Thus, in this particular case, the boundary of the space of deformations of f is incomplete.

On Quasi-nested wandering domains

In this paper, the nature of the singularity of a meromorphic functions of the form f (z) = 1 h(z) + a for a ? C and h is an entire function having a Baker wandering domain, lying over the Baker omitted value is discussed. Various dynamical issues relating to the singular values of f have been studied. Also following are shown in this paper. If a be the Baker omitted value of f then f has a Quasi-nested wandering domain U if and only if there exists {nk}k>0 such that each Unk surrounds a and Unk ? a as k ? ?. If f is a function having Quasi-nested wandering domain then all the Fatou components of f are bounded. In particular, f has no Baker domain. Also existence of Quasi-nested wandering domain ensures that the Julia component containing ? i.e., J? is a singleton buried component. At the end of the paper a result about the non existence of Quasi-nested wandering domain is given.

On Fatou Sets Containing Baker Omitted Value

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

Carathèodory convergence for Leau and Baker domains

ABSTRACT Consider a meromorphic function f with a countable compact set of essential singularities and whose Fatou set consists only of finitely many cycles of Leau domains and finitely many families of invariant Baker domains and their preimages. We provide sufficient conditions on a sequence of meromorphic functions with only Leau domains and uniformly convergent on compact sets to f, so that the Julia sets of converge in the Hausdorff metric to the Julia set of f. In particular, the Leau domains of approximate in the sense of Carathèodory, either the Leau or the Baker domains of f.

Singular orbits and Baker domains

We show that there is a transcendental meromorphic function with an invariant Baker domain U such that every singular value of f is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that U can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljević and the author from 2013, and complements recent work of Barański et al concerning this question.

On the connectivity of the escaping set in the punctured plane

We consider the dynamics of transcendental self-maps of the punctured plane, mathbb {C}^*=mathbb {C}{setminus } {0}. We prove that the escaping set I(f) is either connected, or has infinitely many components. We also show that I(f)cup {0,infty } is either connected, or has exactly two components, one containing 0 and the other infty . This gives a trichotomy regarding the connectivity of the sets I(f) and I(f)cup {0,infty }, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in mathbb {C}^* by constructing the first such example. We also prove that if f has a doubly connected Baker domain, then its closure contains both 0 and infty , and hence I(f)cup {0,infty } is connected.

Escaping Fatou components of transcendental self-maps of the punctured plane

AbstractWe study the iteration of transcendental self-maps of $\mathcal{C}^*:\=\mathcal{C}\{0}$, that is, holomorphic functions $\fnof:\mathcal{C}^*:\rarr\mathcal{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0},\infin$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathcal{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

The iterated minimum modulus and conjectures of Baker and Eremenko

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 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.

Escaping points in the boundaries of Baker domains

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ has harmonic measure zero. We also present some extensions of the results to the case when $f$ has infinite degree on $U$, including classical Fatou example.

Fatou components and singularities of meromorphic functions

AbstractWe prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

Simply connected Baker domains of some meromorphic functions

The first example of a meromorphic function with non-invariant Baker domains was published by Baker et al. (1991) [2]. In this article we answer some questions concerning connectivity of Baker domains of that meromorphic function and location of the iterates of critical values. We analyze dynamics on the Fatou set and connectivity of the Julia set. We extend some of the achieved results to the family fλ,μ(z)=λexp(z)+μz of transcendental meromorphic functions, where λ,μ∈C∖{0}. We also prove that some ‘non-parabolic’ bifurcations appear many times in this family.

Boundaries of univalent Baker domains

Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture.

Dynamics of transcendental H\xe9non maps

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.

Families of Baker domains for meromorphic functions with countable many essential singularities

Given a transcendental meromorphic function that satisfies an asymptotic behaviour near infinity, Rippon and Stallard showed that such function has a family of invariant Baker domains associated to its essential singularity at infinity. In this work, we extend this result to meromorphic functions with a countable number of essential singularities. Furthermore, for any we provide a closed form description of a large class of functions with countable many essential singularities that exhibit p families of invariant Baker domains. This closed form allow us to provide explicit examples of the results.

Periodic domains of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. 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 the 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 $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ 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 bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$.

The dynamics of semigroups of transcendental entire functions I

We consider the dynamics associated with an arbitrary semigroup of transcendental entire functions. Fatou-Julia theory is used to investigate the dynamics of these semigroups. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroup case. We also investigate the dynamics of conjugate semigroups, abelian transcendental semigroups and wandering and Baker domains of transcendental semigroups.

Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

A transcendental entire function fa(z) = z +e z +a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials Pa,d(z) = (1+a/d)z +(1+z/d) d+1 +a, which converges uniformly on compact sets to fa as d → ∞. We show its dynamical convergence under a certain assumption, even though fa has a Baker domain or a wandering domain. We also investigate the parameter spaces of fa and Pa,d.