9 research outputs found
Dimensions of slowly escaping sets and annular itineraries for exponential functions
We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity âslowlyâ, 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
Julia and escaping set spiders' webs of positive area
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 Î. This is also the first result on the area of a spider's web
Geometrically finite transcendental entire functions
For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle -tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c-Brandt for more restrictive classes of entire functions
On Connected Preimages of Simply-Connected Domains Under Entire Functions
Let f be a transcendental entire function, and let U,VâC be disjoint simply-connected domains. Must one of fâ1(U) and fâ1(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 UâC is completely invariant under f if fâ1(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 EremenkoâLyubich 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 and the EremenkoâLyubich class: Class B or not class B?
Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the EremenkoâLyubich classB of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class B, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain
Geometrically finite transcendental entire functions
For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle -tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c for more restrictive classes of entire functions
On fundamental loops and the fast escaping set
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 spiderâs 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 of escape of points in A(f), and show that this function has a number of interesting properties
Geometrically finite transcendental entire functions
For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle -tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevi\'c for more restrictive classes of entire functions
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Dynamics of generalised exponential maps
Since 1984, many authors have studied the dynamics of maps of the form Δa(z) = ez - a, with a > 1 . It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions.
It is rather surprising that many of the interesting dynamical properties of the maps Δa actually arise from their elementary function theoretic structure, rather than as a result of analyticity. We show this by studying a large class of continuous â2 maps, which, in general, are not even quasiregular, but are somehow analogous to Δa . We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of Δa , including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set