Prime Ends and Local Connectivity

Let U be a simply connected domain on the Riemann sphere whose complement K contains more than one point. We establish a characterization of local connectivity of K at a point in terms of the prime ends whose impressions contain this point. Invoking a result of Ursell and Young, we obtain an alternative proof of a theorem of Torhorst, which states that the impression of a prime end of $U$ contains at most two points at which $K$ is locally connected.Comment: 13 pages, 2 figures. V6: Added additional remarks on the mathematical work of Marie Torhorst. Summary of previous versions: V5 - Updated the article by adding a historical note. V4 - Final preprint prior to publication. V3 - Proof of the Ursell-Young theorem and illustrations added. V2 - The original short note was expanded to a full article. V1 - original short not

The exponential map is chaotic: An invitation to transcendental dynamics

We present an elementary and conceptual proof that the complex exponential map is "chaotic" when considered as a dynamical system on the complex plane. (This result was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.Comment: 22 pages, 4 figures. (Provisionally) accepted for publication the American Mathematical Monthly. V2: Final pre-publication version. The article has been revised, corrected and shortened by 14 pages; see Version 1 for a more detailed discussion of further properties of the exponential map and wider transcendental dynamic

Escaping endpoints explode

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 to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem.Comment: 35 pages. To appear in Comput. Methods Funct. Theory. V2: Authors' final accepted manuscript. Revisions and clarifications have been made throughout from V1. This includes improvements in the proof of Proposition 6.11 and Theorem 8.1, as well as corrections in Remarks 7.1 and 7.3 concerning differing definitions of escaping endpoints in greater generalit

Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions

We show that there exists a hyperbolic entire function of finite order of growth such that the hyperbolic dimension---that is, the Hausdorff dimension of the set of points in the Julia set of whose orbit is bounded---is equal to two. This is in contrast to the rational case, where the Julia set of a hyperbolic map must have Hausdorff dimension less than two, and to the case of all known explicit hyperbolic entire functions. In order to obtain this example, we prove a general result on constructing entire functions in the Eremenko-Lyubich class with prescribed behavior near infinity, using Cauchy integrals. This result significantly increases the class of functions that were previously known to be approximable in this manner. Furthermore, we show that the approximating functions are quasiconformally conjugate to their original models, which simplifies the construction of dynamical counterexamples. We also give some further applications of our results to transcendental dynamics.Comment: 36 pages, 1 figure. V3: Minor revisions from V2. To appear in Proceedings of the London Mathematical Societ

Absence of wandering domains for some real entire functions with bounded singular sets

Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real constant and {\lambda} is positive. We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f). As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane. Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.Comment: 25 pages, 1 figure. To appear in Mathematische Annalen. (V2: Final preprint version. Figure added; some general revision and corrections throughout.

Density of hyperbolicity for classes of real transcendental entire functions and circle maps

We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salom\~ao and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.Comment: 46 pages, 3 figures. V5: Final peer-reviewed accepted manuscript, to appear in Duke Mathematical Journal. Only minor changes from the previous (significantly revised) version V

Eventual hyperbolic dimension of entire functions and Poincaré functions of polynomials

Let $P \colon \mathbb{C} \to \mathbb{C}$ be an entire function. A Poincar\'e function $L \colon \mathbb{C} \to \mathbb{C}$ of $P$ is the entire extension of a linearising coordinate near a repelling fixed point of $P$. We propose such Poincar\'e functions as a rich and natural class of dynamical systems from the point of view of measurable dynamics, showing that the measurable dynamics of $P$ influences that of $L$. More precisely, the hyperbolic dimension of $P$ is a lower bound for the hyperbolic dimension of $L$. Our results allow us to describe a large collection of hyperbolic entire functions having full hyperbolic dimension, and hence no natural invariant measures. (The existence of such examples was only recently established, using very different and much less direct methods.) We also give a negative answer to a natural question concerning the behaviour of eventual dimensions under quasiconformal equivalence

A landing theorem for entire functions with bounded post-singular sets

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called "periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called "dreadlocks". We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock

On Connected Preimages of Simply-Connected Domains Under Entire Functions

Let f be a transcendental entire function, and let U,V⊂C be disjoint simply-connected domains. Must one of f−1(U) and f−1(V) be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains. (A domain U⊂C is completely invariant under f if f−1(U)=U.) It was recently observed by Julien Duval that Baker's argument, which has also been used in later generalisations and extensions of Baker's result, contains a flaw. We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function f(z)=ez+z, there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property. Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f) is bounded; in other words, f belongs to the Eremenko–Lyubich class. On the other hand, if S(f) is finite (or if certain additional hypotheses are imposed), many of the original results do hold. For the convenience of the research community, we also include a description of the error in Baker's proof, and a summary of other papers that are affected

Hyperbolic entire functions with bounded Fatou components

We show that an invariant Fatou component of a hyperbolic transcenden- tal entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our re- sults are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values