Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.
In this paper, we have investigated the Bungee set of composition of two transcendental entire functions. We have provided a class of permutable entire functions for which their Bungee sets are equal. Moreover, we have obtained a result on permutable entire functions for which the escaping set of the composite entire function equals the union of the escaping sets of the two functions. In addition, we establish important relations between the Bungee set, Escaping set and Filled Julia set of composite entire functions with that of its individual functions.
We give criteria for the escaping set and the Julia set of an entire function to have positive measure. The results are applied to Poincaré functions of semihyperbolic polynomials and to the Weierstraß $σ$-function.
We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest. It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire functi…
We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show that the result holds under suitable growth conditions involving the minimum and maximum modulus.
Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ ϕ= ψ\circ f$, where $ϕ,ψ:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.
Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a question of Eremenko.
We show that the fast escaping set $A(f)$ of a transcendental entire function $f$ has a structure known as a spider's web whenever the maximum modulus of $f$ grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko.
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.
A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its…
We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders' webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders' webs to give new results concerning functions with no unbounded Fatou components.
Let $f$ be a transcendental entire function. The fast escaping set, $A(f)$, plays a key role in transcendental dynamics. The quite fast escaping set, $Q(f)$, defined by an apparently weaker condition is equal to $A(f)$ under certain conditions. Here we introduce $Q_2(f)$ defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy $Q_2(f)=A(f)$. We also show that the finite composition of such functions satisfies $Q_2(f)=A(f)$. Finally, we construct a function for which $Q_2(f) \neq Q(f)= A(f)$.
Let $f$ be a transcendental entire function. The quite fast escaping set, $Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast escaping set, $A(f),$ under certain conditions. In this paper we generalise these sets by introducing a family of sets $Q_m(f)$, $m \in \mathbb{N}.$ We also give one regularity and one growth condition which imply that $Q_m(f)$ is equal to $A(f)$ and we show that all functions of finite order and positive lower order satisfy $Q_m(f)=A(f)$ for any $m$. Finally, we relate the new regularity condition to a sufficient condition for $Q_2(f)=A(f)$ …
Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make progress on fundamental questions concerning the iteration of $f$. In this paper we establish new relationships between these three concepts. We prove that a certain weak regularity condition is necessary and sufficient for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\neq A(f)$. We also apply a result of Beurling that relates the size of the minim…
Beginning with Devaney, several authors have studied transcendental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such curves are often called Devaney hairs. We show that, in many cases, every point in such a curve, apart from possibly a finite endpoint of the curve, belongs to the fast escaping set. We also give an example of a Devaney hair which lies in a logarithmic tract of a transcendental entire function and contains no fast escaping points.
We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is es…
For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU(f), then most boundary points of U (in the sense of harmonic measure) lie in BU(f). We prove this using a new result concerning the set of limit points of the iterates of f on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.
Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite …
We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set.
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 examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a long-standing question of Rippon and Stallard. Moreover, these examples cover all possible types of simply connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering d…
The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.
Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.
The fast escaping set, A(f), of a transcendental entire function f has begun to play a key role in transcendental dynamics. In many cases A(f) has the structure of a spider's web, which contains a sequence of fundamental loops. We investigate the structure of these fundamental loops for functions with a multiply connected Fatou component, and show that there exist transcendental entire functions for which some fundamental loops are analytic curves and approximately circles, while others are geometrically highly distorted. We do this by introducing a real-valued function which measures the rate…
We construct a transcendental entire $f:\mathbb{C}\rightarrow\mathbb{C}$ such that (1) $f$ has bounded singular set, (2) $f$ has a wandering domain, and (3) each singular value of $f$ escapes to infinity under iteration by $f$.
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.
We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two.
We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders' webs of positive area. This result is unusual in that most of these functions lie outside the Eremenko-Lyubich class B. This is also the first result on the area of a spider's web.
We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.
It is known that, for many transcendental entire functions in the Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight brush.
We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity `slowly', and which have Hausdorff dimension equal to 1. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.
We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ 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 $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a suffic…
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…
This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. 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 $f:\mathbb{R}^d\to\mathbb{R}^d$ 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 $f^n$ …
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…
We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.
Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $σ$-compact, resolving a question of Rippon from 2009.
We modify a construction of Kisaka and Shishikura to show that there exists an entire function which has both a simply connected and a multiply connected wandering domain. Moreover, these domains are contained in the set of fast escaping points.
Let $f$ be Fatou's function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a `spider's web' and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with infinity form a totally disconnected set. We also give a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider's web and we point out that the same property holds for families of functions.
For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new connected structure known as an infinite spider's web. We investigate a connection between these two topological structures for a certain class of sums of exponentials.
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 $I(f)\setminus D$ has uncountably many unbounded components for any open disc $D$ that meets the Julia set of $f$. For the set $A_R(f)$, which is the `core' of the fast escaping set, we prove the much stronger result that for some $R>0$ either $A_R(f)$ is connected and has the structure of an infinite spider's web or it has uncountably many components each …
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, w…
Let $f$ and $g$ be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of $f$ and $g$ are equal; in particular, we show that $J(f)=J(g)$ provided that neither $f$ nor $g$ has a simply connected wandering domain in the fast escaping set.
We construct an entire function in the Eremenko-Lyubich class $\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\B$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.
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.
The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite critical and asymptotic values of f is bounded. It is shown that if f is in B and of finite order of growth, if infinity is not an asymptotic value of f and if the multiplicities of the poles of f are uniformly bounded, then the Hausdorff dimension of I(f) is strictly smaller than 2. In fact, we give a sharp bound for the Hausdorff dimension of I(f) in terms o…
Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Mathematics and Computer Science, University of Barcelona（巴塞罗那大学数学与计算机科学学院）；Institute for Mathematics, Physics and Mechanics（数学、物理与力学研究所）
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-…
We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such domain can be realized as an escaping or an oscillating wandering domain. As a consequence we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function.
In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes …
In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C- action defined on standard probability space. In the same paper he asked about the minimal possible growth of measurably entire functions. In this work we show that for every arbitrary free C- action defined on a standard probability space there exists a measurably entire function whose growth does not exceed exp (exp[log^p |z|]) for any p > 3. This complements a recent result by Buhovski, Glücksam, Logunov, and Sodin (arXiv:1703.08101) who showed that such functions cann…
Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpe…
We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metr…
Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which …
We prove a form of the $\cos πρ$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but…
